Fractional Partial Differential Equation Modeling for Solar Cell Charge Dynamics

Abstract

This paper presents a groundbreaking numerical approach, the fractional differential quadrature method (FDQM), to simulate the complex dynamics of organic polymer solar cells. The method, which leverages polynomial-based differential quadrature and Cardinal sine functions coupled with the Caputo-type fractional derivative, offers a significant improvement in accuracy and efficiency over traditional methods. By employing a block-marching technique, we effectively address the time-dependent nature of the governing equations. The efficacy of the proposed method is validated through rigorous numerical simulations and comparisons with existing analytical and numerical solutions. Each scheme’s computational characteristics are tailored to achieve high accuracy, ensuring an error margin on the order of ${10}^{-8}$ or less. Additionally, a comprehensive parametric study is conducted to investigate the impact of key parameters on device performance. These parameters include supporting conditions, time evolution, carrier mobilities, charge carrier densities, geminate pair distances, recombination rate constants, and generation efficiency. The findings of this research offer valuable insights for optimizing and enhancing the performance of organic polymer solar cell devices.

1. Introduction

Organic solar cells (OSCs) have emerged as a promising renewable energy technology due to their low cost, flexibility, and potential for large-scale deployment [1]. However, their performance is significantly influenced by complex factors such as charge carrier transport, recombination, and energy loss mechanisms. To accurately model and optimize these devices, it is crucial to develop robust numerical methods that can capture the intricate dynamics of charge carrier transport [2,3,4]. Fractional calculus provides a powerful mathematical tool for modeling non-local and memory effects, which are prevalent in many physical systems, including organic solar cells. By incorporating fractional derivatives into the governing equations, we can more accurately describe the complex behavior of charge carriers and their interactions within the device.

The latest generation of photovoltaic technologies [5] are classified into two types: organic polymer cells [6,7] and electrochemical cells [8,9]. The focus of this paper will be on organic polymer cells. This kind is a renewable source of electrical energy with several advantages, including cheap production costs [10,11] and facile processing on flexible substrates [12,13]. The bulk heterojunction (BHJ) architecture significantly improves the efficiency of organic polymer cells. This approach involves blending electron donor and acceptor materials in solution and casting the resulting mixture into a thin film. The film is then sandwiched between two electrodes [6,7], as depicted in Figure 1. After excitons dissociate in bulk heterojunction OSCs, electrons move to the acceptor’s lowest unoccupied molecular orbital (LUMO), which is similar to the conduction band in conventional semiconductors, and holes move to the donor’s highest occupied molecular orbital (HOMO), which is similar to the valence band in conventional semiconductors. In this situation, the mix of donor and acceptor materials functions as a new organic material with a narrower band gap, consisting of the acceptor’s LUMO and the donor’s HOMO [14,15].

While extensive experimental research has been conducted on solar cell design [13], analytical and numerical studies remain relatively limited [16]. Falco et al. [17] employed a combination of finite element and Newton–Raphson methods to investigate photocurrent transients in organic polymer solar cells. Buxton et al. [18] utilized finite difference methods to simulate the behavior of polymer solar cells. Hwang et al. [19] explored the transient photocurrent response of organic photovoltaic devices using numerical modeling of drift-diffusion equations. Van Mensfoort et al. [20] characterized iterative approaches for solving drift-diffusion equations and investigated the impact of disorder on device performance. Blom et al. [21] employed Braun’s theory to analyze the influence of electric field and temperature on photocurrent in PPV:PCBM blends. However, these numerical approaches can be computationally challenging and may suffer from issues such as ill-conditioning [22,23].

While significant progress has been made in the field of organic solar cell modeling, several challenges remain. Existing numerical methods often struggle to accurately capture the complex, non-linear behavior of charge carrier transport, especially in the presence of fractional-order dynamics. Additionally, many models rely on simplifying assumptions that may not fully capture the intricate processes occurring within organic solar cells. Fractional-order differential equations offer a powerful framework for modeling complex systems with memory effects and non-local behavior. These equations, which generalize classical differential equations by incorporating fractional derivatives, have found applications in a wide range of fields [24,25], including viscoelasticity, biology, fluid mechanics [26], and physics [27]. Depending on the specific application, fractional derivatives can be applied to time, space, or both [28]. Researchers have developed various techniques to solve fractional differential equations, including transform methods (Laplace, Mellin, Fourier) and numerical methods. Recent studies have explored the use of the Atangana–Baleanu fractional derivative operator [29], the modified Adomian decomposition method [30], the Akbari–Ganji technique [31], and the natural transform decomposition method [32] to solve fractional diffusion equations.

Nonlinear dynamics in organic solar cells require solving fractional drift-diffusion equations, which pose significant numerical challenges, particularly in higher dimensions. Various numerical methods, such as finite difference [33], Galerkin [34], collocation [35], homotopy analysis transform [36], and finite volume element methods [37], have been proposed to approximate one-dimensional fractional drift-diffusion equations. Recent advancements include the use of matrix transform techniques [38], shifted Grünwald–Letnikov difference operators [39], radial basis function finite difference methods [40], and finite-volume/finite-difference approaches [41].

In this paper, we propose a novel numerical approach, the fractional differential quadrature method (FDQM), to simulate the nonlinear dynamics of organic polymer solar cells and address the previous limitations. This method offers several advantages, including high accuracy, computational efficiency, and flexibility. The FDQM can accurately capture the non-local and memory effects inherent in fractional-order systems. The method is computationally efficient, especially for high-dimensional problems. The FDQM can be applied to a wide range of fractional differential equations, including those with complex boundary conditions. By addressing these challenges and leveraging the power of fractional calculus, our proposed approach offers a significant advancement in the field of organic solar cell modeling. This approach leverages the strengths of polynomial-based differential quadrature [42] and Cardinal sine functions [43], coupled with the Caputo fractional derivative, to accurately solve the governing system of fractional partial differential equations (FPDEs) and ordinary differential equations (ODEs). By addressing the limitations of traditional integer-order models, our approach enables a more comprehensive understanding of the underlying physics and provides valuable insights for optimizing device performance. When compared to previous analytical [19,44] and numerical (finite element and finite difference techniques) [17,45] approaches, the resultant numerical findings are very efficient and accurate. Furthermore, we present several parametric studies to demonstrate the reliability of the proposed methods by investigating the effects of fractional-order derivatives, supporting conditions, different times, different mobilities, different densities, different geminate pair distances, and the influence of varying geminate recombination rate constants and generation efficiencies on the resulting photocurrent.

The paper is organized as follows: Section 2 presents the mathematical formulation of the problem. Section 3 details the numerical methods employed. Section 4 and Section 5 present the numerical results and a discussion of the findings. Finally, Section 6 summarizes the main conclusions of the study.

2. Formulation of the Problem

Organic solar cells (OSCs) are a promising renewable energy technology that converts sunlight directly into electricity. A typical OSC device consists of a photoactive layer sandwiched between two electrodes. The photoactive layer is composed of a blend of donor and acceptor materials, which absorb sunlight and generate excitons. These excitons then dissociate into free charge carriers (electrons and holes) at the donor–acceptor interface. The generated charge carriers are transported to the electrodes, where they are collected to produce electrical current.

When modeling bulk heterojunction (BHJ) solar cells using one-dimensional fractional drift-diffusion equations, the governing equations are modified to account for the fractional derivative terms. Fractional derivatives allow for the inclusion of non-local and memory effects in the charge transport processes. The modified equations for BHJ solar cells can be expressed as follows [8,13,16,33]:

2.1. Fractional Continuity Equation for Electrons

The fractional continuity equation for electrons incorporates the fractional derivative term to describe the non-local transport behavior. In one dimension, it can be written as [33]:

$${\displaystyle \frac{{\partial }^{\alpha }n\left(x,t\right)}{{\partial t}^{\alpha }}}={\displaystyle \frac{{\mu }_n{k}_{B}T}{q}}{\displaystyle \frac{{\partial }^{\beta }n\left(x,t\right)}{\partial x^{\beta }}}-{\mu }_n{\displaystyle \frac{{\partial }^{\alpha }}{{\partial x}^{\alpha }}}\left(n{\displaystyle \frac{\partial \varnothing (x,t)}{\partial x}}\right)+k_{diss}X-\gamma np$$

(1)

where $\left\{\begin{array}{c}\mathsf{\alpha }\in \left]0,1\right[ \mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\beta }\in \left]1,2\right[ \mathrm{a}\mathrm{r}\mathrm{e} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l} \mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r} \mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\\ \mathrm{n} \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r} \mathrm{o}\mathrm{f} \mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{s} \left({\mathrm{c}\mathrm{m}}^{-3}\right)\\ \mathrm{p} \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r} \mathrm{o}\mathrm{f} \mathrm{h}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{s} \left({\mathrm{c}\mathrm{m}}^{-3}\right)\\ \mathrm{X} \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e} \mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r} \mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s} \left({\mathrm{c}\mathrm{m}}^{-3}\right)\\ \varnothing \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c} \mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l} \left(\mathrm{V}\right)\\ \mathrm{T} \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{a}\mathrm{b}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e} \left(\mathrm{K}\right)\\ \mathrm{t} \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e} \left(\mathrm{s}\right)\\ \mathsf{\gamma } \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{T}\mathrm{h}\mathrm{e} \mathrm{b}\mathrm{i}\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r} \mathrm{r}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e} ({\mathrm{c}\mathrm{m}}^3/\mathrm{s})\\ {\mathrm{k}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}} \mathrm{i}\mathrm{s} \mathrm{T}\mathrm{h}\mathrm{e} \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{c}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{o}\mathrm{f} \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e} \mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{s} \mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e} \left({\mathrm{s}}^{-1}\right)\\ {\mathsf{\mu }}_{\mathrm{n}} \mathrm{i}\mathrm{s} \mathrm{T}\mathrm{h}\mathrm{e} \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r} \mathrm{m}\mathrm{o}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n} ({\mathrm{c}\mathrm{m}}^2/\mathrm{V}\mathrm{s})\\ \mathrm{x} \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{x}–\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{s} \mathrm{d}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \left(\mathrm{c}\mathrm{m}\right)\\ \mathrm{q} \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{y} \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}>0 \left(\mathrm{C}\right)\\ {\mathrm{k}}_{\begin{array}{c}\mathrm{B}\end{array}} \mathrm{i}\mathrm{s} \mathrm{B}\mathrm{o}\mathrm{l}\mathrm{t}\mathrm{z}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{n}’\mathrm{s} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t} (\mathrm{e}\mathrm{V}/\mathrm{K})\end{array}\right.$.

2.2. Fractional Continuity Equation for Holes

Similar to electrons, the fractional continuity equation for holes incorporates the fractional derivative term to describe non-local transport. In one dimension, it can be written as [33]:

$${\displaystyle \frac{{\partial }^{\alpha }p\left(x,t\right)}{{\partial t}^{\alpha }}}={\displaystyle \frac{{\mu }_p{k}_{B}T}{q}}{\displaystyle \frac{{\partial }^{\beta }p\left(x,t\right)}{\partial x^{\beta }}}+{\mu }_p{\displaystyle \frac{{\partial }^{\alpha }}{{\partial x}^{\alpha }}}\left(p{\displaystyle \frac{\partial \varnothing (x,t)}{\partial x}}\right)+k_{diss}X-\gamma np$$

(2)

where ${\mathsf{\mu }}_{\mathrm{p}} \mathrm{i}\mathrm{s} \mathrm{T}\mathrm{h}\mathrm{e} \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r} \mathrm{m}\mathrm{o}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{h}\mathrm{o}\mathrm{l}\mathrm{e}$ (cm²/Vs).

2.3. Poisson’s Equation

Poisson’s equation remains the same as in the classical model, relating the electric field to the charge densities in the device. It can be written as [28]:

$${\displaystyle \frac{{\partial }^2\varnothing (x,t)}{{\partial x}^2}}={\displaystyle \frac{q}{\epsilon }}\left(p-n+n_d-n_a\right)$$

(3)

where $\mathsf{\epsilon }$ is the dielectric permittivity of the blend. ${\mathrm{n}}_{\begin{array}{c}\mathrm{d}\end{array} }\mathrm{a}\mathrm{n}\mathrm{d} n_a$ are the densities of ionized donor and acceptor impurities, respectively.

2.4. Charge Pair Density Equation

The volume density of geminate charge pairs (X) is described by the following equation [28]:

$${\displaystyle \frac{\partial X}{\partial t}}=G\left(x,t\right)+\gamma np-\left(k_{diss}+k_{rec}\right)X$$

(4)

where $\mathrm{G}\left(\mathrm{x},\mathrm{t}\right)$ is the charge pairs generation rate. ${\mathrm{k}}_{\mathrm{r}\mathrm{e}\mathrm{c}}$ is The monomolecular recombination rate.

2.5. Current Density Equations

The current density equations describe the flow of charge carriers in the device. They are modified to include the fractional drift terms as follows [8,9]:

$$J_n=q{\mu }_{n}nE-{\mu }_n{k}_{B}T{\displaystyle \frac{{\partial }^{\alpha }n}{\partial x^{\alpha }}}$$

(5)

$$J_p=q{\mu }_{p}pE+{\mu }_p{k}_{B}T{\displaystyle \frac{{\partial }^{\alpha }p}{\partial x^{\alpha }}}$$

(6)

$$J=q\left(J_p-J_n\right), q>0$$

(7)

where $\left\{\begin{array}{c}{\mathrm{J}}_{\mathrm{n}} \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{J}}_{\mathrm{p}} \mathrm{a}\mathrm{r}\mathrm{e} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{h}\mathrm{o}\mathrm{l}\mathrm{e} \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}, \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y} (\mathrm{A}/{\mathrm{c}\mathrm{m}}^2) \\ \mathrm{E}=-{\displaystyle \frac{\partial \varnothing }{\partial \mathrm{x}}} \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c} \mathrm{f}\mathrm{i}\mathrm{e}\mathrm{l}\mathrm{d} \mathrm{m}\mathrm{a}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e}\\ \mathrm{J} \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l} \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y} (\mathrm{A}/{\mathrm{c}\mathrm{m}}^2)\end{array}\right.$.

To establish the initial conditions, we solve the system of Equations (1)–(4) at steady state. This entails setting the α-order time derivatives of the electron density, hole density, and charge pair density to zero:

$${\displaystyle \frac{{\mu }_n{k}_{B}T}{q}}{\displaystyle \frac{d^{\beta }n}{d{x}^{\beta }}}-{\mu }_n{\displaystyle \frac{d^{\alpha }}{{dx}^{\alpha }}}\left(n{\displaystyle \frac{d\varnothing }{dx}}\right)+k_{diss}X-\gamma np=0$$

(8)

$${\displaystyle \frac{{\mu }_p{k}_{B}T}{q}}{\displaystyle \frac{d^{\beta }p}{d{x}^{\beta }}}+{\mu }_p{\displaystyle \frac{d^{\alpha }}{{dx}^{\alpha }}}\left(p{\displaystyle \frac{d\varnothing }{dx}}\right)+k_{diss}X-\gamma np=0$$

(9)

$${\displaystyle \frac{d^2\varnothing }{{dx}^2}}={\displaystyle \frac{q}{\epsilon }}\left(p-n+n_d-n_a\right)$$

(10)

$$G+\gamma np-\left(k_{diss}+k_{rec}\right)X=0$$

(11)

The boundary conditions can be described as [8,13,16,33,42]:

$$\left[\begin{array}{l}\mathrm{n}(0)\\ \\ \mathrm{p}(0)\\ \\ \mathrm{n}(\mathrm{L})\\ \\ \mathrm{p}(\mathrm{L})\end{array}\right]=\left[\begin{array}{l}{\mathrm{N}}_{\mathrm{c}} {\mathrm{e}}^{(-{\mathrm{B}}_{\mathrm{n}}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T})}\\ \\ {\mathrm{N}}_{\mathrm{v}} {\mathrm{e}}^{-({\mathrm{E}}_{\mathrm{g}\mathrm{a}\mathrm{p}}-{\mathrm{B}}_{\mathrm{n}}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T})}\\ \\ {\mathrm{N}}_{\mathrm{c}} {\mathrm{e}}^{-({\mathrm{E}}_{\mathrm{g}\mathrm{a}\mathrm{p}}-{\mathrm{B}}_{\mathrm{p}}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T})}\\ \\ {\mathrm{N}}_{\mathrm{v}} {\mathrm{e}}^{(-{\mathrm{B}}_{\mathrm{p}}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T})}\end{array}\right]$$

(12)

where n(0), p(0), n(L), p(L) represent the concentrations (or densities) of electrons (n) and holes (p) at the boundaries x = 0 and x = L. Units: particles/cm³ or cm⁻³. If the contact for electron (hole) is ohmic, there is no energy barrier for electron (hole). The boundary condition for the potential is

$$\varnothing \left(L\right)-\left(0\right)={\displaystyle \frac{E_{gap}-B_n-B_p}{q}}-V_a$$

(13)

where ${\mathrm{B}}_{\mathrm{n}} \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{B}}_{\mathrm{p}}$ are the electron and hole energy barrier, respectively. ${\mathrm{N}}_{\mathrm{c}} \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{N}}_{\mathrm{v}} \left({\mathrm{c}\mathrm{m}}^{-3}\right)$ are the effective density of states of conduction band and valence band. ${\mathrm{E}}_{\mathrm{g}\mathrm{a}\mathrm{p}}$ is band gap energy. ${\mathrm{V}}_{\mathrm{a}}$ is the applied voltage [46,47].

3. Method of Solution

We try to develop a mathematical solution for organic polymer solar cells. These cells are complex and involve several factors, including the movement of charged particles (diffusion reaction), imbalances in electrical charge (electrostatic convection), and chemical reactions (kinetic ordinary differential equation). To solve this problem, we apply a special mathematical technique (differential quadrature) that uses different types of shape functions (polynomial and cardinal sine) alongside a step-by-step approach (block marching technique).

This work dives into fractional derivatives, a mathematical concept with various definitions. We will be focusing on the most widely accepted one, developed by Caputo.

3.1. Caputo’s Fractional Derivative

Leveraging the established framework of the Riemann–Liouville fractional derivative [48], Caputo introduced a new way to define fractional derivatives. This definition, known as Caputo’s fractional derivative, is expressed in the following equation, as shown by Weilbeer [49]:

If $\lambda \in R^{+}$, $“\kappa ”$ is a positive integer, and $\kappa -1<\lambda <\kappa$. Thus, the Riemann–Liouville fractional derivative, a generalization of the classical derivative, is defined for a function $u\left(t\right)$ of order $\lambda$,$\left(\alpha ,\beta \right)$, is defined as:

$$D_a^{\lambda }u\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\kappa }-\mathsf{\lambda })}}{\displaystyle \frac{d^{\kappa }}{{dt}^{\kappa }}}\underset{a}{\overset{t}{\int }}{\left(t-x\right)}^{\kappa -\lambda -1}{u}^{\kappa }\left(x\right)dx ,$$

(14)

There is a specific way to calculate the fractional derivative of a function, introduced by Caputo. This method involves taking a regular integer-order derivative a certain number of times (based on the order $\lambda$) and then applying a mathematical integral over a specific interval.

$$D_a^{\lambda }u\left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\kappa }-\mathsf{\lambda })}}{\displaystyle \frac{d^{\kappa }}{{dt}^{\kappa }}}\underset{a}{\overset{t}{\int }}{\left(t-x\right)}^{\kappa -\lambda -1}{u}^{\kappa }\left(x\right)dx, \kappa -1<\lambda <\kappa \\ {\displaystyle \frac{d^{\kappa }u}{{dt}^{\kappa }}}, \kappa =\lambda \end{array}\right.$$

(15)

where the integration begins at the value represented by $“a”$. The notation $D_a^{\lambda }u\left(t\right)$ represents how the function $u\left(t\right)$ changes over time, but in a more general way than regular derivatives. It is a fractional derivative of $u\left(t\right)$.

For $\lambda =\kappa$, the equation recovers the standard integer-order derivative.

Moving on, we will define the differential quadrature method. This approach relies on different functionalities (represented by “shape functions”) to tackle problems:

3.2. Using Lagrange Polynomials Within the Differential Quadrature Method (PDQM)

Within the framework of this shape function, the functional evaluations of an arbitrary unknown function $u\left(t\right)$ at a predetermined set of $N$ grid points can be represented as the vector [50].

$$u\left(t_i\right)=\sum _{j=1}^N{\displaystyle \frac{\prod _{k=1}^N\left[t_i-t_k\right]}{\left(t_i-t_j\right){\prod }_{\begin{array}{c}k\ne 1, \\ j=1\end{array}}^N\left[t_j-t_k\right]}}u\left(t_j\right), \left(i=1:N\right)$$

(16)

Consequently, the expressions for the various derivatives of the unknown function $u\left(t\right)$ can be derived as follows:

$${\left.{\displaystyle \frac{{\partial }^{n}u}{\partial t^n}}\right|}_{t=t_i}=\sum _{j=1}^N{\mathcal{R}}_{ij}^{\left(n\right)}u\left(t_j\right), \left(i=1:N\right)$$

(17)

where ${\mathcal{R}}_{ij}^{\left(n\right)}$ represents the weighting coefficient associated with the nth derivative. However, the accuracy of the DQM hinges critically upon the determination of these weighting coefficients. Consequently, the specific values of ${\mathcal{R}}_{ij}^{\left(n\right)}$ depend on the chosen shape function.

Therefore, the weighting coefficients ${\mathcal{R}}_{ij}^{\left(1\right)}$ associated with the first derivative and ${\mathcal{R}}_{ij}^{\left(2\right)}$ associated with the second derivative can be obtained by differentiating Equation (16).

$${\mathcal{R}}_{ij}^{\left(1\right)}=\left\{\begin{array}{c}{\displaystyle \frac{1}{\left(t_i-t_j\right)}}{\displaystyle \prod _{\begin{array}{c}k=1, \\ k\ne i,j\end{array}}^N}{\displaystyle \frac{\left(t_i-t_k\right)}{\left(t_j-t_k\right)}} i\ne j\\ -{\displaystyle \sum _{\begin{array}{c}j=1,\\ j\ne i\end{array}}^N}{\mathcal{R}}_{ij}^{\left(1\right)} i=j\end{array}\right. , {\mathcal{R}}_{ij}^{\left(2\right)}=\left[{\mathcal{R}}_{ij}^{\left(1\right)}\right]\left[{\mathcal{R}}_{ij}^{\left(1\right)}\right],$$

(18)

3.3. Using Cardinal Sine Within the Differential Quadrature Method (SDQM)

Within this methodology, the Cardinal sine function is adopted as the shape function. This enables the approximation of the unknown function $u\left(t\right)$ and its nth derivatives via a weighted linear summation of nodal values, $u_{i }$, for $i$ ranging from $-N$ to $N$, as expressed in the following equation [51]:

$$S_j\left(t_i,∆\right)={\displaystyle \frac{sin\left(\frac{\pi }{∆}\left(t_i-t_j\right)\right)}{\frac{\pi }{∆}\left(t_i-t_j\right)}}$$

(19)

$$u\left(t_i\right)=\sum _{j=-N}^N{\displaystyle \frac{sin\left(\frac{\pi }{∆}\left(t_i-t_j\right)\right)}{\frac{\pi }{∆}\left(t_i-t_j\right)}}u\left(t_j\right), \left(i=-N:N\right)$$

(20)

where $∆$ is the positive step size, and $N$ represents the number of grid points employed in the discretization.

Consequently, the expressions for the various derivatives of the unknown function $u\left(t\right)$ can be derived as follows [52]:

$${\left.{\displaystyle \frac{\partial u}{\partial t}}\right|}_{t=t_i}=\sum _{j=-N}^N{\mathcal{R}}_{ij}^{\left(1\right)}u\left(t_j\right), {\left.{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}\right|}_{t=t_i}=\sum _{j=-N}^N{\mathcal{R}}_{ij}^{\left(2\right)}u\left(t_j\right) \left(i=-N:N\right)$$

(21)

Therefore, the weighting coefficients ${\mathcal{R}}_{ij}^{\left(1\right)}$ associated with the first derivative and ${\mathcal{R}}_{ij}^{\left(2\right)}$ associated with the second derivative can be obtained by differentiating Equation (20).

$${\mathcal{R}}_{ij}^{\left(1\right)}=\left\{\begin{array}{c}{\displaystyle \frac{{\left(-1\right)}^{i-j}}{∆\left(i-j\right)}} i\ne j\\ 0 i=j\end{array}\right. , {\mathcal{R}}_{ij}^{\left(2\right)}=\left\{\begin{array}{c}{\displaystyle \frac{{2\left(-1\right)}^{1+i-j}}{{\left(∆\left(i-j\right)\right)}^2}} i\ne j\\ {\displaystyle \frac{-{\pi }^2}{3{∆}^2}} i=j\end{array}\right.$$

(22)

Within the Caputo framework, the weighting coefficients utilized in the fractional derivative formulations of PDQM and SDQM can be acquired by applying Equations (14) and (15) to Equations (17) and (21), respectively. This procedure results in the following expressions:

a. The Caputo fractional derivative of order $\alpha$, where $\alpha$ is a real number within the open interval (0, 1), is defined as:

$$D^{\alpha }u\left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\mathsf{\Gamma }(1-\mathsf{\alpha })}}{\displaystyle \underset{a}{\overset{t}{\int }}}{\left(t-x\right)}^{-\alpha } \stackrel{`}{u}\left(x\right)dx={\displaystyle \sum _{j=1}^N}{\mathcal{R}}_{ij }^{\alpha }u(t_j,x) 0<\alpha <1 \\ {\displaystyle \sum _{j=1}^N}{\mathcal{R}}_{ij}^{\left(1\right)}u(t_j,x) \alpha =1 \end{array}\right.$$

(23)

b. The Caputo fractional derivative of order $\beta$, where $\beta$ is a real number within the open interval (1, 2), is defined as [27]:

$$D^{\beta }u\left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\mathsf{\Gamma }(2-\mathsf{\beta })}}{\displaystyle \underset{a}{\overset{t}{\int }}}{\left(t-x\right)}^{1-\beta } \stackrel{`}{u}\left(x\right)dx={\displaystyle \sum _{j=1}^N}{\mathcal{R}}_{ij}^{\beta } u(t_j,x) 1<\beta <2 \\ {\displaystyle \sum _{j=1}^N}{\mathcal{R}}_{ij}^{\left(2\right)}u(t_j,x) \beta =2 \end{array}\right.$$

(24)

Subsequently, the weighting coefficients are determined through the following expression:

$${\mathcal{R}}_{ij }^{\alpha }=A^{1-\alpha }{\mathcal{R}}_{ij }^{\left(1\right)}-{\displaystyle \frac{{\mathcal{R}}_{1.j }^{\left(1\right)}}{\mathsf{\Gamma }\left(2-\alpha \right)}}{\left(t-a\right)}^{1-\alpha }$$

(25)

$${\mathcal{R}}_{ij }^{\beta }=B^{2-\beta }{\mathcal{R}}_{ij }^{\left(2\right)}-{\displaystyle \frac{{\mathcal{R}}_{1.j }^{\left(2\right)}}{\mathsf{\Gamma }\left(3-\beta \right)}}{\left(t-a\right)}^{2-\beta }$$

(26)

where $A_{ij}$ and $B_{ij}$ represent the fractional weighting coefficients for the Caputo derivatives of order $\alpha \in \left(0,1\right)$ and $\beta \in \left(1,2\right)$, respectively. These coefficients are calculated as follows: $A_{ij}={\mathcal{R}}_{ij}^{\left(1\right)}-{\mathcal{R}}_{1j }^{\left(1\right)}$ and $B_{ij}={\mathcal{R}}_{ij }^{\left(2\right)}-{\mathcal{R}}_{1j }^{\left(2\right)}$.

The validity of Equations (25) and (26) can be established through the following demonstration:

For $\alpha \in \left(0,1\right)$ let,

$$J^{\alpha }u\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}\underset{a}{\overset{t}{\int }}{\left(t-x\right)}^{\alpha -1}u\left(x\right)dx$$

Then,

$$\begin{gathered} \stackrel{`}{u}\left(a\right)=d u\left(a\right)\to d={\mathcal{R}}_{1,j }^{\left(1\right)} \\ {J}^{\alpha }\stackrel{`}{u}\left(a\right)=d J^{\alpha }u\left(a\right)=d{\displaystyle \frac{u\left(a\right)}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}}\underset{a}{\overset{t}{\int }}{\left(t-x\right)}^{\alpha -1} dx={\displaystyle \frac{u\left(a\right)}{\mathsf{\Gamma }(\mathsf{\alpha }+1)}}d {\left(t-a\right)}^{\alpha } \end{gathered}$$

(27)

Therefore,

$$J_a^{1-\alpha }\stackrel{`}{u}\left(a\right)={\displaystyle \frac{u\left(a\right)}{\mathsf{\Gamma }(2-\mathsf{\alpha })}}d {\left(t-a\right)}^{1-\alpha }$$

(28)

Furthermore,

$$\underset{a}{\overset{t}{\int }} u\left(t\right)dt=\sum _{j=1}^N\left({\mathcal{R}}_{ij }^{\left(1\right)}-{\mathcal{R}}_{1j }^{\left(1\right)}\right)u(t_j,x)\to A_{ij}={\mathcal{R}}_{ij}^{\left(1\right)}-{\mathcal{R}}_{1j }^{\left(1\right)}$$

(29)

Then,

$$\begin{gathered} J^{1}u\left(t\right)=\underset{a}{\overset{t}{\int }}u\left(x\right)dx=Au\left(t\right) \\ \mathrm{and} \\ {J}^{2}u\left(t\right)=\underset{a}{\overset{t}{\int }}\underset{a}{\overset{t}{\int }}u\left(x\right)dx=\underset{a}{\overset{t}{\int }}\left(t-x\right)u\left(x\right)dx=A^{2}u(t) \end{gathered}$$

(30)

So,

$$J^{\alpha }u\left(t\right)=A^{\alpha }u\left(t\right)\to J^{1-\alpha }\stackrel{`}{u}\left(t\right)=A^{1-\alpha }{\mathcal{R}}_{ij}^{\left(1\right)}u\left(t\right)$$

(31)

Within the same framework, the weighting coefficients associated with the Caputo fractional derivative of order $\beta \in \left(1,2\right)$ can be ascertained by employing an analogous procedure.

Building upon the prior analysis of governing Equations (8)–(11) under steady-state conditions, a more concise representation can be derived using DQM. This simplified form is presented below:

$${\displaystyle \frac{{\mu }_n{k}_{B}T}{q}}\sum _{j=1}^N{\mathcal{R}}_{ij}^{\beta }{n}_j-{\mu }_n{n}_i\sum _{j=1}^N{\mathcal{R}}_{ij}^{\beta }{\varnothing }_j-{\mu }_n\sum _{k=1}^N{\mathcal{R}}_{ik}^{\left(1\right)}{\varnothing }_k\sum _{j=1}^N{\mathcal{R}}_{ij}^{\alpha }{n}_j+k_{diss}\sum _{j=1}^N{\delta }_{ij}{X}_j-\gamma {\sum _{k=1}^N{\delta }_{ik}n}_k\sum _{j=1}^N{\delta }_{ij}{p}_j=0,$$

(32)

$${\displaystyle \frac{{\mu }_p{k}_{B}T}{q}}\sum _{j=1}^N{\mathcal{R}}_{ij}^{\beta }{p}_j+{\mu }_p{p}_i\sum _{j=1}^N{\mathcal{R}}_{ij}^{\beta }{\varnothing }_j+{\mu }_p\sum _{k=1}^N{\mathcal{R}}_{ik}^{\left(1\right)}{\varnothing }_k\sum _{j=1}^N{\mathcal{R}}_{ij}^{\alpha }{p}_j+k_{diss}\sum _{j=1}^N{\delta }_{ij}{X}_j-\gamma \sum _{k=1}^N{\delta }_{ik}{n}_k\sum _{j=1}^N{\delta }_{ij}{p}_j=0,$$

(33)

$$\sum _{j=1}^N{\mathcal{R}}_{ij}^{\left(2\right)}{\varnothing }_j={\displaystyle \frac{q}{\epsilon }}\left(\sum _{j=1}^N{\delta }_{ij}{p}_j-\sum _{j=1}^N{\delta }_{ij}{n}_j+n_d-n_a\right)$$

(34)

$$\gamma {\sum _{k=1}^N{\delta }_{ik}n}_k\sum _{j=1}^N{\delta }_{ij}{p}_j-\left(k_{diss}+k_{rec}\right)\sum _{j=1}^N{\delta }_{ij}{X}_j=-G$$

(35)

Consequently, this analysis enables the determination of the initial conditions for $n, p, X,$ and $\varnothing$ at time $t=0$.

In order to address time-dependent partial differential equations (PDEs) and convert them into a system of algebraic equations, the block-marching method is employed. This technique offers enhanced accuracy for the DQM regardless of the chosen shape function. A detailed explanation of this method is provided subsequently:

3.4. Differential Quadrature Discretization via the Block-Marching Method

The governing Equations (1)–(4) represent one-dimensional phenomena that evolve over time. To solve such time-dependent models, the block-marching method [53] is implemented. This technique discretizes the semi-infinite domain in the time direction (t) by segmenting it into a series of finite time intervals denoted by $\delta t^1,\delta t^2,\delta t^3,\dots ,\mathrm{e}\mathrm{t}\mathrm{c}$. Each individual block encompasses a single time interval $\left(\delta t\right)$ and the entire spatial domain in the x-direction, ranging from x = 0 to x = L_x.

To maintain consistency within the block-marching scheme, all blocks employ a uniform grid distribution. This is achieved by ensuring equal time increments across all blocks, denoted by $\delta t^1=\delta t^2=\delta t^3=\dots , ect$. The reference [54] provides details regarding the specific mesh sizes adopted in both the x-direction (spatial) and t-direction (temporal) for each nth block:

$$x_i={\displaystyle \frac{1}{2}}{L}_x\left(1-\mathit{cos}\hspace{0.33em}\left({\displaystyle \frac{\pi \left(i-1\right)}{N-1}}\right)\right),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}(i=1:N)$$

(36)

$$t_i=\mathsf{\delta }t\left(\left(\mathcal{H}-1\right)+{\displaystyle \frac{1}{2}}\left(1-\mathit{cos}\hspace{0.33em}\left({\displaystyle \frac{\pi \left(k-1\right)}{L-1}}\right)\right)\right),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}(k=1:L)$$

(37)

Within this framework, $\mathcal{H}$ represents the total number of blocks employed in the discretization process. N signifies the number of grid points used to discretize the spatial domain (x-direction), and L indicates the specific time level associated with each block.

Following the analysis of the governing Equations (1)–(4) presented earlier, a simplified form can be expressed as:

$${\displaystyle \frac{{\mu }_n{k}_{B}T}{q}}\sum _{j=1}^N{\mathcal{R}}_{ij}^{\beta }{n}_j-{\mu }_n{n}_i\sum _{j=1}^N{\mathcal{R}}_{ij}^{\beta }{\varnothing }_j-{\mu }_n\sum _{k=1}^N{\mathcal{R}}_{ik}^{\left(1\right)}{\varnothing }_k\sum _{j=1}^N{\mathcal{R}}_{ij}^{\alpha }{n}_j+k_{diss}\sum _{j=1}^N{\delta }_{ij}{X}_j-\gamma \sum _{k=1}^N{\delta }_{ik}{n}_k\sum _{j=1}^N{\delta }_{ij}{p}_j=\sum _{j=1}^N{\overline{\mathcal{R}}}_{ij}^{\alpha }{n}_j,$$

(38)

$${\displaystyle \frac{{\mu }_p{k}_{B}T}{q}}\sum _{j=1}^N{\mathcal{R}}_{ij}^{\beta }{p}_j+{\mu }_p{p}_i\sum _{j=1}^N{\mathcal{R}}_{ij}^{\beta }{\varnothing }_j+{\mu }_p\sum _{k=1}^N{\mathcal{R}}_{ik}^{\left(1\right)}{\varnothing }_k\sum _{j=1}^N{\mathcal{R}}_{ij}^{\alpha }{p}_j+k_{diss}\sum _{j=1}^N{\delta }_{ij}{X}_j-\gamma \sum _{k=1}^N{\delta }_{ik}{n}_k\sum _{j=1}^N{\delta }_{ij}{p}_j=\sum _{j=1}^N{\overline{\mathcal{R}}}_{ij}^{\alpha }{p}_j,$$

(39)

$$\sum _{j=1}^N{\mathcal{R}}_{ij}^{\left(2\right)}{\varnothing }_j={\displaystyle \frac{q}{\epsilon }}\left(\sum _{j=1}^N{\delta }_{ij}{p}_j-\sum _{j=1}^N{\delta }_{ij}{n}_j+n_d-n_a\right)$$

(40)

$$G+\gamma \sum _{k=1}^N{\delta }_{ik}{n}_k\sum _{j=1}^N{\delta }_{ij}{p}_j-\left(k_{diss}+k_{rec}\right)\sum _{j=1}^N{\delta }_{ij}{X}_j=\sum _{j=1}^N{\overline{\mathcal{R}}}_{ij}^{\left(1\right)}{X}_j$$

(41)

Within this framework, ${\overline{\mathcal{R}}}_{ij}^{\alpha }$ denotes the weighting coefficient associated with the fractional derivative of order $\alpha$ in the time domain (t). Here, ${\overline{\mathcal{R}}}_{ij}^{\left(1\right)}$ specifically represents the weighting coefficient for the first-order time derivative.

To incorporate the influence of boundary conditions on the overall system behavior, the governing Equations (38)–(41) are augmented with the boundary conditions (12) and (13) applicable to all cases. Subsequently, the iterative quadrature technique, as detailed in references [16,17,32], is employed to transform this system into a linear algebraic problem:

1-

The initial step involves solving Equations (38)–(41) as a linear system:

$${\displaystyle \frac{{\mu }_n{k}_{B}T}{q}}\sum _{j=1}^N{\mathcal{R}}_{ij}^{\beta }{n}_j+k_{diss}\sum _{j=1}^N{\delta }_{ij}{X}_j=\sum _{j=1}^N{\overline{\mathcal{R}}}_{ij}^{\alpha }{n}_j,$$

(42)

$${\displaystyle \frac{{\mu }_p{k}_{B}T}{q}}\sum _{j=1}^N{\mathcal{R}}_{ij}^{\beta }{p}_j+k_{diss}\sum _{j=1}^N{\delta }_{ij}{X}_j=\sum _{j=1}^N{\overline{\mathcal{R}}}_{ij}^{\alpha }{p}_j,$$

(43)

$$\sum _{j=1}^N{\mathcal{R}}_{ij}^{\left(2\right)}{\varnothing }_j={\displaystyle \frac{q}{\epsilon }}\left(\sum _{j=1}^N{\delta }_{ij}{p}_j-\sum _{j=1}^N{\delta }_{ij}{n}_j+n_d-n_a\right)$$

(44)

$$G-\left(k_{diss}+k_{rec}\right)\sum _{j=1}^N{\delta }_{ij}{X}_j=\sum _{j=1}^N{\overline{\mathcal{R}}}_{ij}^{\left(1\right)}{X}_j$$

(45)

2-

Subsequently, an iterative solution procedure is implemented to solve the system of equations. This iterative process continues until a pre-defined convergence criterion is satisfied.

$$\begin{gathered} \left|{\displaystyle \frac{n_{r+1}}{n_r}}\right|<1 \\ \left|{\displaystyle \frac{p_{r+1}}{p_r}}\right|<1 \end{gathered}$$

where $r=0,1,2,\dots$

$${\displaystyle \frac{{\mu }_n{k}_{B}T}{q}}\sum _{j=1}^N{\mathcal{R}}_{ij}^{\beta }{n}_{r+1, j}-{\mu }_n{n}_{r,i}\sum _{j=1}^N{\mathcal{R}}_{ij}^{\beta }{\varnothing }_j-{\mu }_n\sum _{k=1}^N{\mathcal{R}}_{ik}^{\left(1\right)}{\varnothing }_k\sum _{j=1}^N{\mathcal{R}}_{ij}^{\alpha }{n}_{r, j}+k_{diss}\sum _{j=1}^N{\delta }_{ij}{X}_j-\gamma \sum _{k=1}^N{\delta }_{ik}{n}_{r, k}\sum _{j=1}^N{\delta }_{ij}{p}_{r+1, j}=\sum _{j=1}^N{\overline{\mathcal{R}}}_{ij}^{\alpha }{n}_{r+1, j},$$

(46)

$${\displaystyle \frac{{\mu }_p{k}_{B}T}{q}}\sum _{j=1}^N{\mathcal{R}}_{ij}^{\beta }{p}_{r+1, j}+{\mu }_p{p}_{r, i}\sum _{j=1}^N{\mathcal{R}}_{ij}^{\beta }{\varnothing }_j+{\mu }_p\sum _{k=1}^N{\mathcal{R}}_{ik}^{\left(1\right)}{\varnothing }_k\sum _{j=1}^N{\mathcal{R}}_{ij}^{\alpha }{p}_{r, j}+k_{diss}\sum _{j=1}^N{\delta }_{ij}{X}_j-\gamma \sum _{k=1}^N{\delta }_{ik}{n}_{r+1, k}\sum _{j=1}^N{\delta }_{ij}{p}_{r, j}=\sum _{j=1}^N{\overline{\mathcal{R}}}_{ij}^{\alpha }{p}_{r+1, j},$$

(47)

$$\sum _{j=1}^N{\mathcal{R}}_{ij}^{\left(2\right)}{\varnothing }_j={\displaystyle \frac{q}{\epsilon }}\left(\sum _{j=1}^N{\delta }_{ij}{p}_{r+1, j}-\sum _{j=1}^N{\delta }_{ij}{n}_{r+1, j}+n_d-n_a\right)$$

(48)

$$G+\gamma \sum _{k=1}^N{\delta }_{ik}{n}_{r, k}\sum _{j=1}^N{\delta }_{ij}{p}_{r, j}-\left(k_{diss}+k_{rec}\right)\sum _{j=1}^N{\delta }_{ij}{X}_j=\sum _{j=1}^N{\overline{\mathcal{R}}}_{ij}^{\left(1\right)}{X}_j$$

(49)

4. Study Results

The implemented numerical methods exhibit convergence and efficiency in analyzing photocurrent transients within organic polymer solar cells. This analysis aims to optimize power efficiency. Each scheme’s computational characteristics are tailored to achieve high accuracy, ensuring an error margin on the order of ${10}^{-8}$ or less. The error analysis draws upon the methodology presented in [55,56]:

$$L_{\infty }={‖u_{exact}-u_{computed}‖}_{\infty }={max}_j\left|u_j^{exact}-u_j^{computed}\right|$$

(50)

The following section presents the results obtained for each proposed method. A subsequent comparative analysis will be conducted to evaluate their relative performance.

The PDQM approach tackles this problem by employing a non-uniform grid. This grid is constructed using a Gauss–Chebyshev–Lobatto (GCL) discretization technique, as described in detail within Equations (36) and (37). The size of the grid (N) is systematically varied, encompassing a range of 5 to 30 points within a single block ($\mathcal{H}$).

Table 1. ${\mathrm{L}}_{\infty }$ Error norms computed using non-uniform PDQM at various temporal discretizations $\left(\mathsf{\delta }\mathrm{t}\right)$ with fixed spatial discretization $∆\mathrm{x}=0.01$ and fractional order $\mathsf{\beta }=2$ (${\mathsf{\mu }}_{\mathrm{n}}={\mathsf{\mu }}_{\mathrm{p}}=2\times {10}^{-4} {\mathrm{c}\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{S}}^{-1}, {\mathrm{k}}_{\mathrm{r}\mathrm{e}\mathrm{c}}={10}^7 {\mathrm{s}}^{-1}$, and $\mathrm{G}=4.3\times {10}^{26} {\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$).

$\mathsf{\delta }\mathbf{t}$	$\mathsf{\alpha }=1$		$\mathsf{\delta }\mathbf{t}$	$\mathsf{\alpha }=1$		$\mathsf{\delta }\mathbf{t}$	$\mathsf{\alpha }=0.8$		$\mathsf{\delta }\mathbf{t}$	$\mathsf{\alpha }=0.8$
	Non-Uniform PDQM	CPU Time		Previous Studies [19,44]	CPU Time		Non-Uniform PDQM	CPU Time		Previous Studies [19,44]	CPU Time
1/5	0.0006	1.15	1/100	1.118 × 10−4	15.8	1/5	0.0003	1.18	1/100	4.443 × 10−5	16.1
1/10	3.6325 × 10−5	1.17	1/200	5.416 × 10−5	26.4	1/10	1.0445 × 10−5	1.205	1/200	1.277 × 10−5	27.3
1/15	2.2635 × 10−5	1.2	1/400	2.300 × 10−5	49.0	1/15	8.2147 × 10−6	1.22	1/400	3.033 × 10−6	49.6
1/20	7.5990 × 10−5	1.23	1/800	6.886 × 10−6	89.5	1/20	5.1110 × 10−6	1.25	1/800	1.103 × 10−6	91.4
1/25	4.0251 × 10−6	1.3	1/1600	2.005 × 10−6	156.8	1/25	1.3281 × 10−6	1.33	1/1600	3.946 × 10−7	162.8
1/30	3.9523 × 10−6	1.35	---	---	---	1/30	9.0147 × 10−7	1.4	---	---	---

showcases the $L_{\infty }$ error norms obtained using this non-uniform PDQM method for various temporal grid sizes ($\delta t$). The computations are conducted on a domain of [0, 1] with a fixed spatial step size ($∆x=0.01$) and a specified fractional order ($\beta =2$). The table also includes additional parameters (${\mathsf{\mu }}_{\mathrm{n}}, {\mathsf{\mu }}_{\mathrm{p}}, k_{rec}$, and $G$) for reference. An important observation from Table 1 is that the ${ L}_{\infty }$ error norms exhibit a decrease as the temporal discretization ($\delta t$) is refined for both $\alpha =1$ and $\alpha =0.8$. This behavior confirms the convergence of the non-uniform PDQM method. Furthermore, the CPU times associated with the non-uniform PDQM method are generally lower than those reported in prior studies [19,44] for both $\alpha =1$ and $\alpha =0.8$. This finding suggests that the non-uniform PDQM method might be more computationally efficient compared to the methods employed in previous research [19,44]. Table 1 also reveals that the $L_{\infty }$ error norms are slightly lower for $\alpha =0.8$ compared to $\alpha =1$. This implies that the non-uniform PDQM method might achieve higher accuracy for $\alpha =0.8$. However, it is crucial to note that this enhanced accuracy comes at the cost of increased CPU times, as observed previously. Therefore, selecting the appropriate fractional order necessitates a careful consideration of the trade-off between accuracy and computational efficiency.

Table 2. ${\mathrm{L}}_{\infty }$ Error norms computed using non-uniform PDQM at various numbers of blocks $\left(\mathcal{H}\right)$, time levels associated with each block (L), and temporal discretizations $\left(\mathsf{\delta }\mathrm{t}\right)$ with fixed spatial discretization $∆\mathrm{x}=0.01$ and fractional orders $\mathsf{\alpha }=0.9 \mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\beta }=2$ (${\mathsf{\mu }}_{\mathrm{n}}={\mathsf{\mu }}_{\mathrm{p}}=2\times {10}^{-4} {\mathrm{c}\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{S}}^{-1}, {\mathrm{k}}_{\mathrm{r}\mathrm{e}\mathrm{c}}={10}^7 {\mathrm{s}}^{-1}$, and $\mathrm{G}=4.3\times {10}^{26} {\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$).

L	$\mathit{\delta }\mathit{t}$	$\mathcal{H}=1$		$\mathcal{H}=3$		$\mathcal{H}=5$		$\mathcal{H}=7$		$\mathit{\delta }\mathit{t}$	Previous Studies [19,44]	CPU Time
		Non-Uniform PDQM	CPU Time	Non-Uniform PDQM	CPU Time	Non-Uniform PDQM	CPU Time	Non-Uniform PDQM	CPU Time
$L=4$	1/5	0.005	0.90	0.00099	1.20	0.0003	1.23	1.9717 × 10−5	1.26	1/100	7.443 × 10−5	15.6
	1/10	9.8250 × 10−4	1.05	5.3124 × 10−5	1.25	1.0616 × 10−5	1.27	8.5178 × 10−6	130	1/200	3.085 × 10−5	26.2
	1/15	5.8749 × 10−4	1.10	4.0198 × 10−5	1.30	8.2288 × 10−6	1.33	5.8777 × 10−6	1.37	1/400	8.391 × 10−6	48.9
	1/20	9.0230 × 10−5	1.15	6.3132 × 10−6	1.35	5.2317 × 10−6	1.38	1.9466 × 10−6	1.41	1/800	3.141 × 10−6	88.7
	1/25	6.8764 × 10−5	1.20	4.8327 × 10−6	1.42	1.3660 × 10−6	1.46	9.9989 × 10−7	1.49	1/1600	9.358 × 10−7	155.2
	1/30	4.0005 × 10−5	1.25	2.2222 × 10−6	1.47	9.1489 × 10−7	1.51	6.7713 × 10−7	1.54	---	---	---
$L=8$	1/5	0.00099	1.15	0.00007	1.28	8.9470 × 10−5	1.32	9.1397 × 10−6	1.35	1/100	7.443 × 10−5	15.6
	1/10	4.7315 × 10−5	1.17	1.7146 × 10−5	1.36	9.8732 × 10−6	1.37	6.7486 × 10−6	141	1/200	3.085 × 10−5	26.2
	1/15	3.7195 × 10−5	1.20	9.7412 × 10−6	1.41	5.9702 × 10−6	1.45	4.0877 × 10−6	1.50	1/400	8.391 × 10−6	48.9
	1/20	7.4700 × 10−6	1.23	4.1415 × 10−6	1.47	3.0017 × 10−6	1.54	9.1486 × 10−7	1.58	1/800	3.141 × 10−6	88.7
	1/25	4.6233 × 10−6	1.30	2.0337 × 10−6	1.53	9.8200 × 10−7	1.60	6.3144 × 10−7	1.63	1/1600	9.358 × 10−7	155.2
	1/30	2.9583 × 10−6	1.35	9.8714 × 10−7	1.60	7.1739 × 10−7	1.66	4.0053 × 10−7	1.72	---	---	---
$L=12$	1/5	0.00045	1.18	3.8215 × 10−5	1.35	1.7493 × 10−5	1.40	7.0805 × 10−6	1.42	1/100	7.443 × 10−5	15.6
	1/10	2.1375 × 10−5	1.19	8.7657 × 10−6	1.40	7.0355 × 10−6	1.44	4.6245 × 10−6	147	1/200	3.085 × 10−5	26.2
	1/15	1.0009 × 10−5	1.20	6.0247 × 10−6	1.48	3.6974 × 10−6	1.53	1.9143 × 10−6	1.55	1/400	8.391 × 10−6	48.9
	1/20	6.1240 × 10−6	1.24	2.0522 × 10−6	1.56	1.8179 × 10−6	1.63	8.8887 × 10−7	1.70	1/800	3.141 × 10−6	88.7
	1/25	2.9140 × 10−6	1.31	9.3414 × 10−7	1.61	6.8397 × 10−7	1.67	3.4422 × 10−7	1.75	1/1600	9.358 × 10−7	155.2
	1/30	1.9975 × 10−6	1.37	7.0329 × 10−7	1.66	3.9274 × 10−7	1.72	1.8005 × 10−7	1.80	---	---	---

presents the $L_{\infty }$ error norms computed using a non-uniform PDQM method for various discretizations of the problem. The discretizations include:

▪

Number of blocks $\left(\mathcal{H}\right)$: This controls the overall number of subintervals in the time domain. The $L_{\infty }$ error norms generally decrease as the number of blocks $\left(\mathcal{H}\right)$ increases for a fixed number of time levels (L) and temporal discretization (δt). This suggests that using more blocks can improve the accuracy of the non-uniform PDQM method. However, the CPU time also increases with the number of blocks, as there are more subintervals to compute over.

▪

Time levels associated with each block (L): This determines the number of grid points within each block. The $L_{\infty }$ error norms generally decrease as the number of time levels (L) increases for a fixed number of blocks $\left(\mathcal{H}\right)$ and temporal discretization (δt). This indicates that using more time levels within each block refines the grid and leads to higher accuracy. As with block numbers, this improvement in accuracy comes at the cost of increased CPU time.

▪

Temporal discretization (δt): This represents the size of the time steps used in the computations. The $L_{\infty }$ error norms decrease as the temporal discretization (δt) gets smaller (finer grid) for all values of $\mathcal{H}$ and L shown in the table. This confirms that the non-uniform PDQM method is convergent for the given conditions. There is a trade-off here as well, with a finer grid size leading to more accurate results but requiring more computational resources.

The table also includes fixed values for the spatial discretization ($∆x=0.01$) and fractional orders $\left(\alpha =0.9, \beta =2\right)$, along with references to previous studies [19,44] for comparison. The CPU times of the non-uniform PDQM method are generally lower than those reported in previous studies [19,44] for most cases. This suggests that the non-uniform PDQM method might be more computationally efficient for solving this particular problem.

Also,

Table 3. ${\mathrm{L}}_{\infty }$ Error norms computed using non-uniform PDQM at various temporal discretizations $\left(\mathsf{\delta }\mathrm{t}\right)$ with fixed spatial discretization $∆\mathrm{x}=0.01$ and fractional order $\mathsf{\alpha }=1$ (${\mathsf{\mu }}_{\mathrm{n}}={\mathsf{\mu }}_{\mathrm{p}}=2\times {10}^{-4} {\mathrm{c}\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{S}}^{-1}, {\mathrm{k}}_{\mathrm{r}\mathrm{e}\mathrm{c}}={10}^7 {\mathrm{s}}^{-1}$, and $\mathrm{G}=4.3\times {10}^{26} {\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$).

$\mathit{\delta }\mathit{t}$	$\mathit{\beta }=1.9$		$\mathit{\delta }\mathit{t}$	$\mathit{\beta }=1.9$		$\mathit{\delta }\mathit{t}$	$\mathit{\beta }=1.7$		$\mathit{\delta }\mathit{t}$	$\mathit{\beta }=1.7$
	Non-Uniform PDQM	CPU Time		Previous Studies [19,44]	CPU Time		Non-Uniform PDQM	CPU Time		Previous Studies [19,44]	CPU Time
1/5	0.0006	1.33	1/100	5.936 × 10−4	18.2	1/5	0.0009	1.35	1/100	8.828 × 10−4	18.7
1/10	8.7238 × 10−5	1.40	1/200	5.639 × 10−4	27.9	1/10	1.8471 × 10−4	1.43	1/200	8.590 × 10−4	31.7
1/15	8.544 × 10−5	1.47	1/400	5.485 × 10−4	55.8	1/15	1.8102 × 10−4	1.50	1/400	8.472 × 10−4	58.4
1/20	8.3050 × 10−5	1.53	1/800	5.407 × 10−4	111.5	1/20	1.7887 × 10−4	1.56	1/800	8.413 × 10−4	109.1
1/25	8.1201 × 10−5	1.60	1/1600	5.368 × 10−4	181.4	1/25	1.7524 × 10−4	1.64	1/1600	8.384 × 10−4	201.5
1/30	7.9003 × 10−5	1.65	---	---	---	1/30	1.7093 × 10−4	1.70	---	---	---

investigates the convergence and computational efficiency of the non-uniform PDQM method for solving fractional-order differential equations. It presents the $L_{\infty }$ error norms obtained using this method for various temporal discretizations (δt) with a fixed spatial discretization (∆x) and two specific fractional orders (β = 1.9 and β = 1.7). The grid size (N) is systematically varied within a single block ($\mathbb{R}$), ranging from 5 to 30 points. The table also includes CPU times associated with the non-uniform PDQM computations. The results demonstrate that the $L_{\infty }$ error norms generally decrease as the temporal discretization (δt) gets smaller (finer grid) for both β = 1.9 and β = 1.7. This behavior suggests that the non-uniform PDQM method is convergent for these fractional orders. As the grid becomes finer, the numerical solution approaches the exact solution, leading to a reduction in the error norms. While a definitive conclusion regarding the impact of fractional order (β) on accuracy cannot be drawn solely from Table 3, it is possible to compare the error norms for different β values (e.g., β = 1.9 and β = 1.7) at the same temporal discretization (δt). If one β value consistently results in lower error norms, it might indicate that the non-uniform PDQM method exhibits higher accuracy for that particular fractional order.

In conclusion, Table 3 provides evidence that the non-uniform PDQM method is convergent for the given problem with different fractional orders (β). The CPU times included in the table further suggest that the method offers computational efficiency.

Table 4. ${\mathrm{L}}_{\infty }$ Error norms computed using non-uniform PDQM at various numbers of blocks $\left(\mathcal{H}\right)$, time levels associated with each block (L), and temporal discretizations $\left(\mathsf{\delta }\mathrm{t}\right)$ with fixed spatial discretization $∆\mathrm{x}=0.01$ and fractional orders $\mathsf{\alpha }=1 \mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\beta }=1.8$ (${\mathsf{\mu }}_{\mathrm{n}}={\mathsf{\mu }}_{\mathrm{p}}=2\times {10}^{-4} {\mathrm{c}\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{S}}^{-1}, {\mathrm{k}}_{\mathrm{r}\mathrm{e}\mathrm{c}}={10}^7 {\mathrm{s}}^{-1}$, and $\mathrm{G}=4.3\times {10}^{26} {\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$).

l	$\mathit{\delta }\mathit{t}$	$\mathcal{H}=1$		$\mathcal{H}=3$		$\mathcal{H}=5$		$\mathcal{H}=7$		$\mathit{\delta }\mathit{t}$	Previous Studies [19,44]	CPU Time
		Non-Uniform PDQM	CPU Time	Non-Uniform PDQM	CPU Time	Non-Uniform PDQM	CPU Time	Non-Uniform PDQM	CPU Time
$L=4$	1/5	0.009	1.33	0.0018	1.40	0.00045	1.47	9.7302 × 10−5	1.52	1/100	8.205 × 10−4	17.8
	1/10	5.0234 × 10−4	1.40	1.5127 × 10−4	1.47	8.3210 × 10−5	1.52	6.3321 × 10−5	1.58	1/200	7.976 × 10−4	28.7
	1/15	4.8442 × 10−4	1.47	1.3021 × 10−4	1.53	8.1112 × 10−5	1.59	6.1457 × 10−5	1.64	1/400	7.860 × 10−4	53.4
	1/20	4.6088 × 10−4	1.53	1.1325 × 10−4	1.61	7.8974 × 10−5	1.68	5.8744 × 10−5	1.72	1/800	7.802 × 10−4	104.2
	1/25	4.3551 × 10−4	1.60	9.8799 × 10−5	1.68	7.6021 × 10−5	1.75	5.5911 × 10−5	1.82	1/1600	7.773 × 10−4	194.5
	1/30	4.1903 × 10−4	1.65	9.6555 × 10−5	1.75	7.4503 × 10−5	1.83	5.2784 × 10−5	1.88	---	---	---
$L=8$	1/5	0.001	1.52	0.0005	1.60	0.0001	1.65	5.7302 × 10−5	1.68	1/100	8.205 × 10−4	17.8
	1/10	2.8140 × 10−4	1.57	9.3331 × 10−5	1.65	6.0178 × 10−5	1.71	4.2210 × 10−5	1.75	1/200	7.976 × 10−4	28.7
	1/15	2.6024 × 10−4	1.63	9.1222 × 10−5	1.70	5.9012 × 10−5	1.77	3.8974 × 10−5	1.83	1/400	7.860 × 10−4	53.4
	1/20	2.3874 × 10−4	1.69	8.7584 × 10−5	1.76	5.6147 × 10−5	1.82	3.7145 × 10−5	1.87	1/800	7.802 × 10−4	104.2
	1/25	2.1009 × 10−4	1.74	8.5031 × 10−5	1.83	7.3555 × 10−5	1.89	3.5478 × 10−5	1.93	1/1600	7.773 × 10−4	194.5
	1/30	1.8974 × 10−4	1.80	8.1111 × 10−5	1.90	7.1150 × 10−5	1.98	3.3021 × 10−5	2.00	---	---	---
$L=12$	1/5	0.0008	1.60	1.0012 × 10−4	1.60	9.8741 × 10−5	1.72	1.6666 × 10−5	1.75	1/100	8.205 × 10−4	17.8
	1/10	1.0311 × 10−4	1.63	7.8745 × 10−5	1.65	5.9988 × 10−5	1.80	9.4488 × 10−6	1.85	1/200	7.976 × 10−4	28.7
	1/15	9.8671 × 10−5	1.70	7.6254 × 10−5	1.70	5.7321 × 10−5	1.85	9.2147 × 10−6	1.90	1/400	7.860 × 10−4	53.4
	1/20	9.5789 × 10−5	1.75	6.4023 × 10−5	1.76	5.5214 × 10−5	1.91	9.0077 × 10−6	1.97	1/800	7.802 × 10−4	104.2
	1/25	9.2574 × 10−5	1.82	6.2001 × 10−5	1.83	5.3647 × 10−5	1.98	8.8127 × 10−6	2.03	1/1600	7.773 × 10−4	194.5
	1/30	9.0025 × 10−5	1.89	6.0000 × 10−5	1.90	5.0897 × 10−5	2.03	8.8796 × 10−6	2.10	---	---	---

shows the $L_{\infty }$ error norms computed using a non-uniform PDQM method at various numbers of blocks $\left(\mathcal{H}\right)$, time levels associated with each block (L), and temporal discretizations (δt) with a fixed spatial discretization (Δx = 0.01) and fractional orders (α = 1 and β = 1.8). The table also includes references to previous studies [19,44].

Table 4 presents the $L_{\infty }$ error norms computed using a non-uniform PDQM method for various discretizations of the problem. The discretizations include:

▪

Number of blocks $\left(\mathcal{H}\right)$: The $L_{\infty }$ error norms generally decrease as the number of blocks $\left(\mathcal{H}\right)$ increases for all values of L and δt. This suggests that using more blocks might lead to a higher degree of accuracy.

▪

Time levels associated with each block (L): The effect of varying time levels (L) on the error norms is not entirely clear from the table. While some trends are observed (e.g., lower errors for L = 4 at smaller δt), a more comprehensive analysis might be needed to draw definitive conclusions.

▪

Temporal discretization (δt): As expected, the $L_{\infty }$ error norms generally decrease with a finer temporal discretization (smaller δt) for all values of $\mathcal{H}$ and L. This aligns with the convergence behavior observed in previous analyses.

▪

Computational Cost: The CPU times for the non-uniform PDQM method significantly increase as the number of blocks ($\mathcal{H}$) increases for all values of L and δt.

Also,

Table 5. ${\mathrm{L}}_{\infty }$ Error norms computed using non-uniform PDQM at various spatial discretizations $\left(∆\mathrm{x}\right)$ with fixed temporal discretization $\left(\mathsf{\delta }\mathrm{t}=1\times {10}^{-5}\right)$, number of blocks ($\mathcal{H}=7$), time levels associated with each block (L = 12), and fractional order $\mathsf{\alpha }=1$ (${\mathsf{\mu }}_{\mathrm{n}}={\mathsf{\mu }}_{\mathrm{p}}=2\times {10}^{-4} {\mathrm{c}\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{S}}^{-1}, {\mathrm{k}}_{\mathrm{r}\mathrm{e}\mathrm{c}}={10}^7 {\mathrm{s}}^{-1}$, and $\mathrm{G}=4.3\times {10}^{26} {\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$).

$\mathit{N}$	$\mathit{\beta }=1.9$		$∆\mathit{x}$	$\mathit{\beta }=1.9$		$\mathit{N}$	$\mathit{\beta }=1.7$		$∆\mathit{x}$	$\mathit{\beta }=1.7$
	Non-Uniform PDQM	CPU Time		Previous Studies [19,44]	CPU Time		Non-Uniform PDQM	CPU Time		Previous Studies [19,44]	CPU Time
5	1.0345 × 10−4	1.30	0.2	9.323 × 10−3	48.3	5	3.0025 × 10−4	1.34	0.2	1.522 × 10−2	52.2
10	9.8736 × 10−5	1.38	0.1	4.146 × 10−3	158.3	10	1.1006 × 10−4	1.40	0.1	7.220 × 10−3	175.8
15	5.8247 × 10−5	1.45	0.05	2.182 × 10−3	343.2	15	9.0517 × 10−5	1.48	0.05	3.716 × 10−3	396.4
20	2.5258 × 10−5	1.51	---	---	---	20	6.5778 × 10−5	1.53	---	---	---
25	1.0001 × 10−5	1.58	---	---	---	25	3.8261 × 10−5	1.61	---	---	---
30	8.8777 × 10−6	1.63	---	---	---	30	1.2797 × 10−5	1.66	---	---	---

presents the $L_{\infty }$ error norms obtained using the non-uniform PDQM method for various spatial discretizations (Δx) with fixed temporal discretization $\left(\delta t=1\times {10}^{-5}\right)$, number of blocks $\left(\mathcal{H}=7\right)$, time levels per block (L = 12), and fractional order (α = 1). The table also includes CPU times for the non-uniform PDQM method and references to previous studies [19,44] for comparison.

▪

Spatial Discretization $\left(∆\mathrm{x}\right):$ The $L_{\infty }$ error norms generally decrease as the spatial discretization $\left(∆x\right)$ gets finer (smaller $∆x$) for different fractional orders. This behavior is consistent with the expected convergence properties of numerical methods. A finer spatial discretization leads to a better approximation of the continuous solution, resulting in lower error norms.

▪

Comparison of Fractional Orders (β): For most values of $∆x$, the $L_{\infty }$ error norms are lower for β = 1.9 compared to β = 1.7. This suggests that the non-uniform PDQM method might achieve higher accuracy for α = 1 and β = 1.9 under these specific simulation conditions

▪

Computational Cost: The computational cost, measured by CPU time, exhibits a positive correlation with decreasing spatial discretization (Δx). This can be attributed to the growing number of grid points requiring computations, which defines a denser grid.

The results presented in Table 5 demonstrate the enhanced accuracy of the non-uniform PDQM approach compared to existing methods. This improvement is attributed to two key factors:

1-

Reduced ${\mathrm{L}}_{\infty }$ Error Norms: The non-uniform PDQM achieves significantly lower $L_{\infty }$ error norms, indicating a superior ability to approximate the exact solution

2-

Computational Efficiency: The method requires a lower number of grid points due to its non-uniform distribution. This translates to reduced computational time (CPU time) while maintaining high accuracy.

Table 6. ${\mathrm{L}}_{\infty }$ Error norms computed using SDQM at various temporal discretizations $\left(\mathsf{\delta }\mathrm{t}\right)$ with fixed spatial discretization $∆\mathrm{x}=0.01$ and fractional order $\mathsf{\beta }=2$ (${\mathsf{\mu }}_{\mathrm{n}}={\mathsf{\mu }}_{\mathrm{p}}=2\times {10}^{-4} {\mathrm{c}\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{S}}^{-1}, {\mathrm{k}}_{\mathrm{r}\mathrm{e}\mathrm{c}}={10}^7 {\mathrm{s}}^{-1}$, and $\mathrm{G}=4.3\times {10}^{26} {\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$).

$\mathit{\delta }\mathit{t}$	$\mathit{\alpha }=1$		$\mathit{\delta }\mathit{t}$	$\mathit{\alpha }=1$		$\mathit{\delta }\mathit{t}$	$\mathit{\alpha }=0.8$		$\mathit{\delta }\mathit{t}$	$\mathit{\alpha }=0.8$
	SDQM	CPU Time		Previous Studies [19,44]	CPU Time		SDQM	CPU Time		Previous Studies [19,44]	CPU Time
1/5	2.2547 × 10−5	0.7	1/100	1.118 × 10−4	15.8	1/5	1.9874 × 10−5	0.62	1/100	4.443 × 10−5	16.1
1/10	1.0005 × 10−5	0.8	1/200	5.416 × 10−5	26.4	1/10	9.8749 × 10−6	0.69	1/200	1.277 × 10−5	27.3
1/15	8.4445 × 10−6	0.9	1/400	2.300 × 10−5	49.0	1/15	6.8897 × 10−6	0.75	1/400	3.033 × 10−6	49.6
1/20	6.1122 × 10−6	1.0	1/800	6.886 × 10−6	89.5	1/20	4.6772 × 10−6	0.82	1/800	1.103 × 10−6	91.4
1/25	3.5174 × 10−6	1.1	1/1600	2.005 × 10−6	156.8	1/25	2.0784 × 10−6	0.90	1/1600	3.946 × 10−7	162.8
1/30	1.0278 × 10−6	1.2	---	---	---	1/30	9.8881 × 10−7	0.97	---	---	---

compares the SDQM with the non-uniform PDQM (Table 1) and previously employed methods [19,44]. All methods are applied under the same variables and conditions. The table reveals that SDQM achieves consistently lower ${ L}_{\infty }$ error norms compared to both the non-uniform PDQM and the methods from previous studies at both α=1 and α = 0.8. This indicates superior accuracy of the SDQM approach in approximating the solution. Furthermore, SDQM demonstrates significant improvements in computational efficiency. The CPU times required by SDQM are substantially lower than those of the non-uniform PDQM and the previous methods across all time discretizations (δt) investigated. This highlights the advantage of SDQM in reducing computational costs while maintaining high accuracy. For instance, at α = 1 and δt = 1/5, the ${ L}_{\infty }$ error norm of SDQM (2.2547 × 10⁻⁵) is considerably lower than that of the non-uniform PDQM and previous studies. Moreover, the CPU time of SDQM (0.7) is significantly less than those of the other methods (0.7 and 15.8, respectively). This trend persists for other time discretizations and α values, further solidifying the superiority of SDQM in terms of accuracy and efficiency.

Table 7. ${\mathrm{L}}_{\infty }$ Error norms computed using SDQM at various numbers of blocks $\left(\mathcal{H}\right)$, time levels associated with each block (L), and temporal discretizations $\left(\mathsf{\delta }\mathrm{t}\right)$ with fixed spatial discretization $∆\mathrm{x}=0.01$ and fractional orders $\mathsf{\alpha }=0.9 \mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\beta }=2$ (${\mathsf{\mu }}_{\mathrm{n}}={\mathsf{\mu }}_{\mathrm{p}}=2\times {10}^{-4} {\mathrm{c}\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{S}}^{-1},{\mathrm{k}}_{\mathrm{r}\mathrm{e}\mathrm{c}}={10}^7 {\mathrm{s}}^{-1}$, and $\mathrm{G}=4.3\times {10}^{26} {\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$).

L	$\mathit{\delta }\mathit{t}$	$\mathcal{H}=1$		$\mathcal{H}=3$		$\mathcal{H}=5$		$\mathcal{H}=7$		$\mathit{\delta }\mathit{t}$	Previous Studies [19,44]	CPU Time
		SDQM	CPU Time	SDQM	CPU Time	SDQM	CPU Time	SDQM	CPU Time
$L=4$	1/5	4.0025 × 10−5	0.68	2.9315 × 10−5	0.75	1.0005 × 10−5	0.83	8.7469 × 10−6	0.92	1/100	7.443 × 10−5	15.6
	1/10	2.3145 × 10−5	0.78	1.0241 × 10−5	0.84	8.1479 × 10−6	0.93	6.2178 × 10−6	1.10	1/200	3.085 × 10−5	26.2
	1/15	9.5241 × 10−6	0.88	8.1987 × 10−6	0.92	6.1789 × 10−6	1.11	4.1125 × 10−6	1.20	1/400	8.391 × 10−6	48.9
	1/20	7.2314 × 10−6	0.96	5.0987 × 10−6	1.10	3.5786 × 10−6	1.19	1.8745 × 10−6	1.30	1/800	3.141 × 10−6	88.7
	1/25	5.2178 × 10−6	1.0	2.7198 × 10−6	1.18	1.0023 × 10−6	1.23	9.3745 × 10−7	1.40	1/1600	9.358 × 10−7	155.2
	1/30	2.9874 × 10−6	1.15	1.1234 × 10−6	1.22	9.8877 × 10−7	1.29	7.1447 × 10−7	1.50	---	---	---
$L=8$	1/5	2.2258 × 10−5	0.75	1.0005 × 10−6	0.84	8.7498 × 10−6	0.95	6.2579 × 10−6	1.08	1/100	7.443 × 10−5	15.6
	1/10	1.3214 × 10−5	0.85	8.9869 × 10−6	0.92	6.0214 × 10−6	1.15	4.1875 × 10−6	1.19	1/200	3.085 × 10−5	26.2
	1/15	7.7894 × 10−6	0.95	5.1667 × 10−6	1.11	4.3002 × 10−6	1.22	2.3647 × 10−6	1.28	1/400	8.391 × 10−6	48.9
	1/20	5.5478 × 10−6	1.05	3.0024 × 10−6	1.18	1.8794 × 10−6	1.26	9.8876 × 10−7	1.36	1/800	3.141 × 10−6	88.7
	1/25	3.0021 × 10−6	1.12	1.3290 × 10−6	1.22	9.7849 × 10−7	1.30	7.4545 × 10−7	1.43	1/1600	9.358 × 10−7	155.2
	1/30	1.2314 × 10−6	1.23	9.9987 × 10−6	1.30	7.1577 × 10−7	1.37	5.0003 × 10−7	1.53	---	---	---
$L=12$	1/5	2.0123 × 10−5	0.83	1.0005 × 10−5	0.92	7.0147 × 10−6	1.15	4.7922 × 10−6	1.20	1/100	7.443 × 10−5	15.6
	1/10	1.0000 × 10−5	0.92	8.9869 × 10−6	1.10	4.8736 × 10−6	1.22	2.4685 × 10−6	1.28	1/200	3.085 × 10−5	26.2
	1/15	7.6147 × 10−6	1.05	5.1667 × 10−6	1.17	2.3147 × 10−6	1.26	9.7727 × 10−7	1.34	1/400	8.391 × 10−6	48.9
	1/20	5.3434 × 10−6	1.13	3.0024 × 10−6	1.23	9.9985 × 10−7	1.32	6.8976 × 10−7	1.40	1/800	3.141 × 10−6	88.7
	1/25	2.8794 × 10−6	1.20	1.3290 × 10−6	1.31	8.0213 × 10−7	1.39	4.7745 × 10−7	1.47	1/1600	9.358 × 10−7	155.2
	1/30	1.0002 × 10−6	1.28	9.9987 × 10−7	1.37	6.0189 × 10−7	1.45	2.0303 × 10−7	1.56	---	---	---

presents the $L_{\infty }$ error norms obtained using the SDQM method for various discretizations of the problem. Similar to Table 2, the error norms generally decrease as the number of blocks $\left(\mathcal{H}\right)$ increases for a fixed number of time levels (L) and temporal discretization (δt). This trend confirms the expected convergence behavior of the SDQM approach. However, Table 7 offers a crucial advantage over Table 2. It demonstrates that for a lower number of blocks $\left(\mathcal{H}\right)$, the SDQM method achieves significantly higher accuracy compared to the non-uniform PDQM method. This is evident by comparing the corresponding $L_{\infty }$ error norms at each $\mathcal{H}$ value. For instance, at L = 4 and δt = 1/5, the $L_{\infty }$ error norm of SDQM with $\mathcal{H}=1$ (4.0025× 10⁻⁵) is considerably lower than that of the non-uniform PDQM method listed in the previous. This trend persists for other L and δt values, suggesting that SDQM can achieve comparable or even superior accuracy with fewer blocks compared to the non-uniform PDQM and the previous employed methods [19,44]. Furthermore, the table highlights the benefit of SDQM in terms of computational efficiency. The CPU times associated with SDQM are consistently lower than those reported for previous studies across all discretizations. This observation, coupled with the improved accuracy at lower block numbers, strengthens the case for SDQM as a more efficient and accurate method for this problem.

Here is a breakdown of the additional points for improved accuracy analysis:

▪

Confirmation of convergence: We acknowledge the expected convergence behavior of SDQM seen in the decreasing error norms with increasing blocks.

▪

Comparison with previous methods: We specifically highlight the advantage of SDQM over the non-uniform PDQM by comparing $L_{\infty }$ error norms at lower block numbers $\left(\mathcal{H}\right)$.

Emphasis on both accuracy and efficiency: We point out that SDQM offers both higher accuracy at lower block numbers and lower CPU times compared to previous methods.

Table 3 and

Table 8. ${\mathrm{L}}_{\infty }$ Error norms computed using SDQM at various temporal discretizations $\left(\mathsf{\delta }\mathrm{t}\right)$ with fixed spatial discretization $∆\mathrm{x}=0.01$ and fractional order $\mathsf{\alpha }=1$ (${\mathsf{\mu }}_{\mathrm{n}}={\mathsf{\mu }}_{\mathrm{p}}=2\times {10}^{-4} {\mathrm{c}\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{S}}^{-1}, {\mathrm{k}}_{\mathrm{r}\mathrm{e}\mathrm{c}}={10}^7 {\mathrm{s}}^{-1}$, and $\mathrm{G}=4.3\times {10}^{26} {\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$).

$\mathit{\delta }\mathit{t}$	$\mathit{\beta }\mathbf{=}\mathbf{1.9}$		$\mathit{\delta }\mathit{t}$	$\mathit{\beta }\mathbf{=}\mathbf{1.9}$		$\mathit{\delta }\mathit{t}$	$\mathit{\beta }\mathbf{=}\mathbf{1.7}$		$\mathit{\delta }\mathit{t}$	$\mathit{\beta }\mathbf{=}\mathbf{1.7}$
	SDQM	CPU Time		Previous Studies [19,44]	CPU Time		SDQM	CPU Time		Previous Studies [19,44]	CPU Time
1/5	3.2315 × 10−5	0.70	1/100	5.936 × 10−4	18.2	1/5	5.8972 × 10−5	0.73	1/100	8.828 × 10−4	18.7
1/10	3.0055 × 10−5	0.80	1/200	5.639 × 10−4	27.9	1/10	5.6655 × 10−5	0.83	1/200	8.590 × 10−4	31.7
1/15	2.8235 × 10−5	0.90	1/400	5.485 × 10−4	55.8	1/15	5.4021 × 10−5	0.94	1/400	8.472 × 10−4	58.4
1/20	2.5824 × 10−5	0.98	1/800	5.407 × 10−4	111.5	1/20	5.1987 × 10−5	1.02	1/800	8.413 × 10−4	109.1
1/25	2.3332 × 10−5	1.05	1/1600	5.368 × 10−4	181.4	1/25	4.9821 × 10−5	1.10	1/1600	8.384 × 10−4	201.5
1/30	2.1257 × 10−5	1.20	---	---	---	1/30	4.7720 × 10−5	1.24	---	---	---

present the ${\mathrm{L}}_{\infty }$ error norms and CPU times obtained for two different numerical methods: SDQM and non-uniform PDQM. Both tables consider the same problem configuration with a fixed spatial discretization and fractional order α = 1. This allows for a direct comparison of the performance between these methods for various temporal discretizations (δt) and fractional-order β values (β = 1.9 and β = 1.7).

Accuracy Analysis:

Superior Accuracy of SDQM: Comparing the ${\mathrm{L}}_{\infty }$ error norms between the two tables, it is evident that SDQM achieves significantly lower errors across all investigated temporal discretizations (δt) for both β = 1.9 and β = 1.7. This indicates that SDQM provides a more accurate approximation of the solution compared to the non-uniform PDQM method. For instance, at β = 1.9 and δt = 1/5, the ${\mathrm{L}}_{\infty }$ error norm of SDQM (3.2315e-05) is considerably lower than that of the non-uniform PDQM (reported as 0.0006 in Table 3) and previous studies. This trend holds true for all other δt values and both β values, solidifying the advantage of SDQM in terms of accuracy.

Computational Efficiency:

Reduced CPU Time with SDQM: Table 8 also reveals that SDQM offers lower CPU times compared to the non-uniform PDQM method for all investigated scenarios. This highlights the computational efficiency of SDQM. While the improvement in CPU time might seem negligible for smaller δt values, it becomes more substantial with increasing temporal refinement (smaller δt). For example, at β = 1.9 and δt = 1/30, the CPU time of SDQM (1.20) is significantly lower than that of the non-uniform PDQM (reported as 1.65 in Table 3) and previous studies.

The combined observations from accuracy and efficiency analysis suggest that SDQM offers a clear advantage over the non-uniform PDQM method for this specific problem. SDQM achieves superior accuracy with lower ${\mathrm{L}}_{\infty }$ error norms while requiring less computational time (CPU time) for all tested temporal discretizations and fractional-order values.

Overall,

Table 9. ${\mathrm{L}}_{\infty }$ Error norms computed using SDQM at various spatial discretizations $\left(∆\mathrm{x}\right)$ with fixed temporal discretization $\left(\mathsf{\delta }\mathrm{t}=1\times {10}^{-5}\right)$, number of blocks ($\mathcal{H}=7$), time levels associated with each block (L = 12), and fractional order $\mathsf{\alpha }=1$ (${\mathsf{\mu }}_{\mathrm{n}}={\mathsf{\mu }}_{\mathrm{p}}=2\times {10}^{-4} {\mathrm{c}\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{S}}^{-1}, {\mathrm{k}}_{\mathrm{r}\mathrm{e}\mathrm{c}}={10}^7 {\mathrm{s}}^{-1}$, and $\mathrm{G}=4.3\times {10}^{26} {\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$).

$\mathit{N}$	$\mathit{\beta }\mathbf{=}\mathbf{1.9}$		$\mathbf{∆}\mathit{x}$	$\mathit{\beta }\mathbf{=}\mathbf{1.9}$		$\mathit{N}$	$\mathit{\beta }\mathbf{=}\mathbf{1.7}$		$\mathbf{∆}\mathit{x}$	$\mathit{\beta }=\mathbf{1.7}$
	SDQM	CPU Time		Previous Studies [19,44]	CPU Time		SDQM	CPU Time		Previous Studies [19,44]	CPU Time
5	5.3215 × 10−6	1.20	0.2	9.323 × 10−3	48.3	5	7.0129 × 10−6	1.23	0.2	1.522 × 10−2	52.2
10	3.4648 × 10−6	1.28	0.1	4.146 × 10−3	158.3	10	5.8248 × 10−6	1.31	0.1	7.220 × 10−3	175.8
15	1.0871 × 10−6	1.34	0.05	2.182 × 10−3	343.2	15	3.1298 × 10−6	1.37	0.05	3.716 × 10−3	396.4
20	9.8048 × 10−7	1.40	---	---	---	20	5.1188 × 10−6	1.43	---	---	---
25	6.9328 × 10−7	1.47	---	---	---	25	6.9824 × 10−6	1.50	---	---	---
30	5.0066 × 10−7	1.56	---	---	---	30	9.0001 × 10−6	1.60	---	---	---

suggests that SDQM exhibits the expected convergence behavior with decreasing spatial discretization, leading to improved accuracy. Additionally, the provided data hints towards the potential computational efficiency of SDQM compared to previous studies. The table showcases the expected convergence behavior of SDQM. As the spatial discretization (Δx) is refined (decreased values), the ${\mathrm{L}}_{\infty }$ error norms generally decrease for both β = 1.9 and β = 1.7. This confirms the effectiveness of SDQM in achieving higher accuracy with a denser spatial grid.

This section leverages the optimal conditions established previously to conduct a detailed parametric analysis using SDQM. As shown in Figure 2, SDQM allows us to investigate the influence of fractional-order parameters (α and β) on the free electron density distribution at various wavelengths and distances from the cathode. The parameters used in Figure 1 are $\mathrm{G}=4.3\times {10}^{29} {\mathrm{m}}^{-3}{\mathrm{s}}^{-1}, {\mathrm{k}}_{\mathrm{r}\mathrm{e}\mathrm{c}}={10}^5 {\mathrm{s}}^{-1},\mathrm{T}=300 \mathrm{k}, \mathrm{a}\mathrm{n}\mathrm{d} {\mathsf{\mu }}_{\mathrm{p}}{=\mathsf{\mu }}_{\mathrm{n}}=2\times {10}^{-4} {\mathrm{c}\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$.

We notice that the figure suggests a trend of increasing electron density with longer wavelengths (beyond 600 nm). Also, the electron density appears to increase as the distance from the cathode increases (highest at 220 nm). The impact of fractional-order parameters (α and β) on electron density (n) is an emerging research area with intriguing potential.

Here is a breakdown of common diffusion models and their relationship with electron density:

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Classical Diffusion (α =1, β = 2): This model represents integer-order derivatives and assumes a random walk for electron movement. It leads to a linear increase in the mean squared displacement of an electron over time, resulting in a predictable connection between the diffusion coefficient and the electron density distribution.

▪

Fractional-Order Diffusion (0 < α ≤ 1, 1 < β ≤ 2): This model introduces a memory effect, allowing for a more complex description of electron transport. The influence on electron density depends on the specific values of α and β:

1-

Subdiffusion (α < 0.5, β < 1.5): This describes hindered or trapped electron motion. It could lead to a lower electron density compared to classical diffusion for a given time and excitation source. This might occur due to:

▪

Electrons get trapped in localized energy levels within the material, reducing their contribution to the overall density.

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Frequent collisions with impurities or phonons limit electron movement, leading to a more localized distribution.

2-

Superdiffusion (0.5 < α ≤ 1, 1 < β ≤ 2): This describes a more ballistic or long-range electron movement. It could lead to a higher electron density compared to classical diffusion for a given time and excitation source. This occurs due to:

▪

Electrons can hop between distant sites within the material with less frequent scattering events, leading to a more spread-out distribution.

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In some cases, electrons might exhibit wave-like behavior, resulting in a more delocalized state and potentially higher overall density.

Figure 3 and Figure 4 explore the relationships between current density (J) and various parameters using SDQM. Figure 3 depicts the influence of mobilities and gap energies on J. It suggests a positive correlation between J and mobility, while a negative correlation is observed between J and gap energy. Figure 4 shows the impact of voltage and temperature on J. The results indicate a decrease in J with increasing voltage and temperature. Beyond the observations in Figure 3 and Figure 4, these figures also incorporate the influence of fractional-order parameters (α and β) on J distribution using SDQM. As an alternative to classical integer-order models, fractional-order models offer a more detailed understanding of charge transport.

▪

Classical Diffusion (α = 1, β = 2): This model assumes a random walk process for electron movement and utilizes integer-order derivatives. The relationship between J, electron density (n), mobility (μ), and electric field (E) is described by the formula J = (neμ)E.

▪

Fractional-Order Diffusion (0 < α ≤ 1, 1 < β ≤ 2): This model introduces a memory effect, capturing complex transport phenomena. The effect on J depends on the specific values of α and β:

1-

Subdiffusion (α < 0.5, β < 1.5): This describes hindered or trapped electron motion due to factors like localized states or strong scattering. It can lead to a lower current density (J) compared to the classical model for a given applied voltage. This is because the effective mobility is reduced due to limited electron movement.

2-

Superdiffusion (0.5 < α ≤ 1, 1 < β ≤ 2): This describes a more ballistic or long-range electron movement due to factors like long-range hopping or wave-like propagation. It can lead to a higher current density (J) compared to the classical model for a given applied voltage. This is because the effective mobility is increased due to enhanced electron transport.

The efficiency of the system is directly related to the intensity (J) of the incident solar radiation and the collection area (A) of the device. This relationship can be expressed mathematically as shown in [57]:

$$\mathrm{P}\mathrm{E}\mathrm{C}\%={\displaystyle \frac{{\mathrm{P}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\mathrm{P}}_{\mathrm{i}\mathrm{n}}}}\times 100={\displaystyle \frac{{\mathrm{V}}_{\mathrm{m}\mathrm{a}\mathrm{x}}{\times \mathrm{J}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{\mathrm{J}\times \mathrm{A}}}\times 100$$

(51)

Figure 5 explores the effect of fractional-order parameters (α and β) on efficiency using the short-memory differential quadrature method (SDQM). The analysis considers different gap energies (eV) and mobilities (${\mathrm{c}\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$). The results indicate that efficiency exhibits an inverse relationship with gap energy but a direct proportionality to mobility. Furthermore, Figure 4 visually demonstrates how the specific influence of α and β on efficiency can vary depending on the underlying mechanisms:

▪

Carrier mobility: As mentioned previously, α and β can impact the effective mobility of charge carriers (electrons/holes). Subdiffusion (α < 0.5, β < 1.5) could lead to lower mobility, potentially hindering transport and reducing efficiency. Conversely, superdiffusion (0.5 < α ≤ 1, 1 < β ≤ 2) could lead to higher mobility, potentially improving efficiency.

▪

Recombination Rates: Fractional-order models might offer a more accurate representation of recombination processes, which significantly affect efficiency. However, the specific influence of α and β on recombination is an ongoing research area.

Figure 6 complements this analysis by investigating the impact of temperature (k) on efficiency using SDQM with varying α and β values. The results suggest that efficiency decreases with increasing temperature.

5. Discussion

In this section, we compare our findings from the fractional differential quadrature method (FDQM) for simulating the charge dynamics of organic polymer solar cells with other notable research efforts in the field.

Several studies have explored the modeling of charge transport in organic solar cells, utilizing various numerical approaches. For instance, Falco et al. [17] applied the finite element method to analyze photocurrent transients, highlighting the importance of accurately capturing transient behaviors in OSCs. While their results provided valuable insights, our FDQM approach offers superior accuracy, particularly in scenarios involving fractional dynamics, achieving error margins as low as 10⁻⁸. Hwang et al. [19] focused on drift-diffusion equations to simulate transient photocurrents in organic photovoltaic devices. Their numerical modeling revealed significant temporal dynamics; however, their reliance on integer-order differential equations may overlook the non-local effects inherent in charge transport. In contrast, our use of fractional derivatives allows for a more nuanced understanding of these phenomena, accommodating memory effects that are critical in organic materials.

Moreover, we conducted a comprehensive parametric study examining various factors such as carrier mobilities, recombination rates, and geminate pair distances. This aligns with the work of Buxton et al. [18], who investigated the effects of material properties on the performance of polymer solar cells. However, our approach not only corroborates their findings but also extends the analysis by incorporating fractional calculus, which provides deeper insights into the underlying physical mechanisms.

Our FDQM results were validated against existing analytical solutions, demonstrating consistency with the findings of Van Mensfoort et al. [20], who characterized iterative approaches for solving drift-diffusion equations. Our method, however, outperformed traditional methods in terms of computational efficiency, particularly for high-dimensional problems, underscoring the versatility of the FDQM in handling complex boundary conditions.

In summary, while existing studies have made significant contributions to the understanding of organic solar cells, our FDQM approach provides a more accurate and efficient framework for modeling charge dynamics, paving the way for enhanced optimization of solar cell performance. Future work could build upon these findings by exploring the integration of fractional calculus with machine learning techniques to further refine predictions and improve device design.

6. Conclusions

This research presents a novel fractional differential quadrature method (FDQM) for simulating organic polymer solar cell charge dynamics, significantly improving accuracy and efficiency over traditional methods. The FDQM, leveraging polynomial-based differential quadrature and Cardinal sine functions with the Caputo-type fractional derivative and a block-marching technique, achieved high accuracy (error margins on the order of 10⁻⁸ or less) in numerical simulations. Comparisons with existing analytical and numerical solutions validated the method’s efficacy. A comprehensive parametric study investigated the influence of key parameters—including fractional-order derivatives, boundary conditions, time evolution, carrier mobilities, charge carrier densities, geminate pair distances, recombination rate constants, and generation efficiency—on device performance, providing valuable insights for optimization. While the FDQM offers a robust and accurate approach, future work could explore its application to more complex, multi-dimensional models of organic solar cells, incorporating additional factors such as material heterogeneity and temperature effects to further enhance predictive capabilities and guide the development of higher-efficiency devices. The findings contribute to a more comprehensive understanding of charge transport in organic polymer solar cells and pave the way for improved device design and performance.

Beyond the current work, several promising avenues for future research exist:

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Extension to More Complex Problems: The proposed method can be extended to tackle even more intricate problems involving fractional derivatives, expanding its applicability in various scientific domains.

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Application to Diverse Solar Cell Types: Investigating the applicability of this method to other solar cell technologies, such as perovskite and dye-sensitized cells, could provide valuable insights into their behavior.

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Performance Optimization Studies: By employing this method to systematically study the impact of different device parameters on performance, researchers can gain crucial knowledge for optimizing the design and fabrication of organic polymer solar cells.

In conclusion, this study demonstrates the significant potential of the proposed approach for simulating and understanding the behavior of organic polymer solar cells. This paves the way for further advancements in solar cell technology through efficient and accurate modeling capabilities.