Applying Two Fractional Methods for Studying a Novel General Formula of Delayed-Neutron-Affected Nuclear Reactor Equations

Abstract

In this work, the novel general formula for a time-dependent nuclear reactor system of equations with delayed neutron effect has been formulated using a fractional calculus model. We explore the properties of this model, including two analytical approximation methods, the Temimi–Ansari method (TAM) and the Sumudu residual power series method (SRPSM), for solving the equation. These methods allow for the computation of approximate solutions at specific points. This is particularly useful for partial differential equations (PDEs) arising in various fields like physics, engineering, and finance. This work is hoped to improve the advancement of nuclear modeling and simulation, providing researchers and engineers with a powerful mathematical tool for studying the complex dynamics of these critical energy systems.

1. Introduction

The study of nuclear reactors and the neutron behavior inside them is a critical area of research, with important implications for energy production, nuclear safety, and waste management. Traditional approaches to modeling nuclear dynamics have relied on classical partial differential equations (PDEs) and their associated numerical solutions, where fractional partial differential equations express the effect of delayed neutrons.

Finding new sources of energy is increasingly being considered. Nuclear energy can be considered one of the important energy sources, where the nuclear fission in the nuclear reactor creates energy when the mass is annihilated [1,2,3,4]. Expressing the behavior of the neutron during nuclear reactions is modeled as a neutron transport equation; this equation is simplified to the diffusion equation. More interest in studying equation models is considered when Cassell et al. find the neutron distribution in the hemispherical reactors using mixed boundary conditions [5]. Williams studies the nuclear reactor transport equation in two interacting cylinder faces [6]. Khasawneh et al. used the homotopy perturbation method (HPM) in finding the solution of the nuclear reactor equation when neutrons diffuse in hemispherical and cylindrical reactors in two different works where (HPM) was used for the first time in nuclear reactor theory [7,8]. El-Ajou et al. used the residual power series (RPSM) in studying neutron distribution in one-dimensional spherical reactors and two-dimensional hemispherical reactors [9]. Fractional derivatives are employed due to their ability to model anomalous diffusion, which occurs in systems with non-standard transport properties.

One step forward in studying these models is to investigate neutron diffusion equations with delayed neutron presence, where the fractional calculus can assist researchers in studying different cases of nuclear reactor diffusion equations with delayed neutrons. Filali et al. used fractional calculus to solve time-dependent neutron diffusion equations [10]. Roul et al. studied a numerical approximation of considering a dynamic neutron diffusion system numerically [11]. Aboanber studied the matrix representation of two-group nuclear reactor dynamics using fractional analysis [12]. Sardar et al. solved the effect of delayed neutrons on coupled neutron diffusion equations using fractional analysis [13]. Khaled et al. solved the behavior of anomalous neutrons under the effect of delayed neutrons [14]. Vázquez-Rodríguez R. et al. studied the fractional neutron point kinetics model for reactivity transients of the NuScale and compared it with the classical kinetics approach [15]. Yin, D., Xie, Y., and Mei, L. considered the stability and convergence analysis of finite difference methods for the fractional neutron diffusion equation [16]. Burqan A. solved a novel scheme of the ARA transform for systems of partial fractional neutron diffusion equations [17].

Alia et al. found an iterative scheme to study the anomalous neutron diffusion system [18]. Momani et al. studied a spherical-reactor two-group neutron diffusion system [19].

This work considers the generalized comprehensive system of the neutron diffusion equation in different geometries with delayed neutrons. This generalization includes the spherical system for the first time in addition to the slab and cylindrical reactors studied throughout history. One additional novel point is that this system can study all essential geometries simultaneously; moreover, it can easily simplify it for each case. The contribution of this work in scientific research is studying the system of three essential reactor geometries (slab, cylindrical, and spherical reactors), simultaneously using the same formula and solving using two different methods, then comparing them.

The use of fractional calculus helps to address sub-diffusion and super-diffusion behaviors that have been observed in various physical systems, including nuclear reactors [20,21,22]. In this work, two powerful analytical techniques, the Temimi–Ansari method (TAM) and the Sumudu residual power series method (SRPSM), are employed to solve the generalized comprehensive system of neutron diffusion equations in various reactor geometries with delayed neutrons. TAM [23,24,25] is an iterative method known for its simplicity and efficiency in handling both linear and nonlinear fractional differential equations. It generates successive approximations that converge toward the exact solution, making it highly suitable for complex systems like neutron diffusion. SRPSM [26,27], on the other hand, combines the Sumudu transform with a residual power series, allowing for a structured approach to solving fractional differential equations while maintaining computational efficiency. By applying these methods, the study addresses the neutron diffusion problem in slab, cylindrical, and spherical geometries, providing solutions that capture the effects of delayed neutrons and fractional dynamics, thereby improving the accuracy and generality of the nuclear reactor models.

The present work is arranged as follows. The nuclear reactor theory principles are considered in Section 2. The TAM and SRPSM are presented in Section 3. Finding analytical results for the nuclear reactor equations fractionally using both proposed methods is considered in Section 4. The numerical implementations are presented in Section 5. The conclusion of this study is presented in Section 6.

2. The Physical Phenomena

The neutrons are distributed as the solute dissolves in the solution. In chemistry, the original equation is called the diffusion equation whereas, in nuclear reactor theory, it is specified as the time-dependent nuclear reactor equation [1,2,3,4]:

$${\displaystyle \frac{1}{v_c}}{\displaystyle \frac{\partial \phi (\mathit{r},t)}{\partial t}}=s(\mathit{r},t)-{\sum }_a\phi (\mathit{r},t)-\mathbf{\nabla }·\mathit{J}(\mathit{r},t),$$

(1)

Concepts in Equation (1) are represented as the following: $\phi (r,t)$, $\mathit{J}(\mathit{r},t)$, ${\sum }_a$, $v_c$, and $s(\mathit{r},t)$ are the flux, the current density, the cross-section for the absorption process on a macroscopic scale, the velocity of neutrons, and the source term, respectively.

Fick’s law can simplify the $\phi (r,t)$ and $\mathit{J}(\mathit{r},t)$ relationship as

$$\mathit{J}(\mathit{r},t)=-D\mathbf{\nabla }\phi (\mathit{r},t),$$

(2)

so

$${\displaystyle \frac{1}{v_c}}{\displaystyle \frac{\partial \phi (\mathit{r},t)}{\partial t}}=s(\mathit{r},t)-{\sum }_a\phi (\mathit{r},t)+D{\nabla }^2\phi (\mathit{r},t),$$

(3)

Here, the neutron diffusion coefficient is known as D.

In nuclear reactor physics, $s(\mathit{r},t)$ is represented as $v{\sum }_f\phi (\mathit{r},t)$, where ${\sum }_f$ and v are the macroscopic fission cross-section and the number of neutrons produced per fission, respectively.

As we study the three essential nuclear reactor geometries—the slab, cylindrical, and spherical reactors—the Laplacian ${\nabla }^2\phi (\mathit{r},t)$ term needs to be simplified to deal with each coordinate system (Cartesian for the slab reactor, cylindrical for the cylinder reactor, and spherical for the sphere reactor). In one dimension, the Laplacian can be simplified to ${\displaystyle \frac{{\partial }^2}{\partial x^2}}\phi (x,t)+{\displaystyle \frac{a}{x}}{\displaystyle \frac{\partial \phi (x,t)}{\partial x}}$; the Laplacian determines the reactor geometry, where $a=0$, $a=1$, and $a=2$ represent the slab reactor, cylindrical reactor, and spherical reactor, respectively.

The study of the effect of delayed neutrons, which result from some minor reactions, is manipulated fractionally; the delayed neutrons control the neutrons lifespan in nuclear reactors.

The time-dependent system of neutron diffusion equations, where delayed neutrons affect the source term, is

$$s(\mathit{r},t)={\gamma }_f\sum _f\phi (\mathit{r},t)+\lambda \omega (\mathit{r},t).$$

(4)

The general form of a one-dimensional time-dependent nuclear reactor system is

$$\begin{array}{c}{\displaystyle \frac{1}{v_c}}{\displaystyle \frac{\partial }{\partial t}}\phi (x,t)=D{\displaystyle \frac{{\partial }^2}{\partial x^2}}\phi (x,t)+{\displaystyle \frac{a}{x}}{\displaystyle \frac{\partial \phi (x,t)}{\partial x}}+\left(-\sum _a+{\gamma }_f\sum _f\right)\phi (x,t)+\lambda \omega (x,t),\hfill \\ {\displaystyle \frac{\partial }{\partial t}}\omega (x,t)=\beta v\sum _f\phi (x,t)-\lambda \omega (x,t).\hfill \end{array}$$

(5)

In the studied system, the initial conditions are presented as

$$\phi (x,0)={\phi }_0\left(x\right),\omega (x,0)={\displaystyle \frac{\beta {\gamma }_f{\sum }_f}{\lambda }}{\phi }_0\left(x\right).$$

(6)

The symbols in Equation (6), $\phi (x,t)$,${\gamma }_f$,$\beta$, and $\lambda$, are the neutron intensity, fission average neutrons, delayed fission neutrons proportion, and constant of decay, respectively. The overall cross-section $\sum$ is considered as $v{\sum }_f\phi (\mathit{r},t)-{\sum }_a$. In Equation (3), physically, the anomalous neutron diffusion when the delayed neutrons are considered instead of Gaussian neutron diffusion, which can represent this work equation and, mathematically, the fractional diffusion equations represent the case.

The anomalous diffusion changes the representation of the system from regular diffusion in the Gaussian neutron distribution to irregular diffusion; the power-law pattern represents the mean-squared displacement time relation:

$$〈x^2\left(t\right)〉\sim t^{\alpha },$$

(7)

where $\alpha$ is the fractional nuclear equation for the diffused neutrons factor, the value of $\alpha >1$ determines diffusion (the super-diffusion happens when $\alpha >1$, while sub-diffusion happens when $\alpha >1$; it is straightforward to show that the classical diffusion occurs when $\alpha =1$) [13].

The general system of time-dependent neutron diffusion equations in fractional form—this system is suitable for all three essential nuclear reactors—is written for the first time in this work as the following:

$$\begin{array}{c}{\displaystyle \frac{1}{v_c}}{\mathfrak{D}}_t^{\alpha }\phi (x,t)=D{\nabla }^2\phi (x,t)+{\displaystyle \frac{a}{x}}{\displaystyle \frac{\partial \phi (x,t)}{\partial x}}+\left({\gamma }_f\sum _f-\sum _a\right)\phi (x,t)+\lambda \omega (x,t),\hfill \\ {\mathfrak{D}}_t^{\alpha }\omega (x,t)=\beta {\gamma }_f\sum _f\phi (x,t)-\lambda \omega (x,t),\hfill \\ \phi (x,0)={\phi }_0\left(x\right),\omega (x,0)={\displaystyle \frac{\beta {\gamma }_f{\sum }_f}{\lambda }}{\phi }_0\left(x\right).\hfill \end{array}$$

(8)

It must be noted that the simpler forms of this system are considered in soon-to-be published works, where all studied special cases of this formula show an improvement [7,8,9,10,11,12,13,14].

3. Preliminaries

Fractional calculus extends classical calculus by introducing derivatives and integrals of non-integer order, offering a powerful framework to model systems with memory effects and anomalous behavior. In this study, fractional derivatives are employed to capture the complex dynamics of neutron diffusion in nuclear reactors, particularly under the influence of delayed neutrons. Among the various definitions of fractional derivatives, the Caputo fractional derivative is a cornerstone of our analysis due to its practical advantages in physical modeling and its compatibility with initial value problems.

Fractional derivatives, unlike their integer-order counterparts, are non-local operators defined through integrals that account for the history of the function. This non-locality introduces a “memory effect”, making them particularly suited for describing phenomena such as anomalous diffusion—where the mean-squared displacement follows a power-law pattern ($\langle x^2\left(t\right)\rangle \sim t^{\alpha }$) rather than the linear relationship of classical diffusion ($\alpha =1$). In the context of nuclear reactors, this property allows us to model the sub-diffusion ($\alpha <1$) and super-diffusion ($\alpha >1$) behaviors observed due to delayed neutron effects.

In this section, we present the foundational concepts of the Caputo fractional derivative and the Sumudu transform, which are critical to the development of our analytical methods—the Temimi–Ansari Method (TAM) and the Sumudu Residual Power Series Method (SRPSM)—for solving the fractional neutron diffusion equations with delayed neutrons.

Definition 1. 

Caputo Fractional Derivative.

The Caputo fractional derivative of order ε for a function $\theta (x,t)$ is defined as [26]

$${\mathfrak{D}}_t^{\epsilon }\theta (x,t)={\mathfrak{J}}_t^{n-\epsilon }{\partial }_t^n\theta (x,t),n-1<\epsilon \le n,x\in I,t>0,$$

(9)

where ${\mathfrak{D}}_t^{\epsilon }$ denotes the Caputo derivative operator, ${\mathfrak{J}}_t^{n-\epsilon }$ is the Riemann–Liouville fractional integral of order $n-\epsilon$, ${\partial }_t^n$ is the n-th integer-order derivative with respect to time t, n is a positive integer such that $n-1<\epsilon \le n$, and I is a spatial interval. The Riemann–Liouville fractional integral is given by

$${\mathfrak{J}}_t^{\alpha }\theta (x,t)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}\theta (x,\tau )d\tau ,\alpha >0,$$

(10)

where Γ is the Gamma function.

The Caputo definition is particularly advantageous in physical applications because it allows the initial conditions to be specified in terms of integer-order derivatives (e.g., $\theta (x,0)$, ${\partial }_t\theta (x,0)$), which align naturally with measurable quantities like neutron flux and precursor concentrations in nuclear reactor systems. This contrasts with other fractional derivatives, such as the Riemann–Liouville derivative, which require fractional-order initial conditions that are less intuitive in a physical context. Furthermore, the Caputo derivative of a constant is zero, which is consistent with classical calculus and simplifies the modeling of steady-state behaviors.

In our study, the Caputo derivative is applied to the time-dependent neutron diffusion equations to reflect the memory-dependent dynamics introduced by delayed neutrons. This fractional approach enhances the representation of neutron transport, capturing both Gaussian ($\epsilon =1$) and anomalous ($\epsilon \ne 1$) diffusion regimes, as detailed in Section 2.

Definition 2. 

Sumudu Transform.

For a function $\theta \left(t\right)$ over a set $\mathbb{A}$, the Sumudu transform is defined as [27]

$$\mathsf{\Theta }\left(\zeta \right)=S\left[\theta \left(t\right)\right]={\int }_0^{\infty }\theta \left(\zeta t\right)e^{-t}dt,\zeta \in (-{\epsilon }_1,{\epsilon }_2),$$

(11)

where $\theta \left(t\right)$ is a continuous function and the set $\mathbb{A}$ is defined as

$$\mathbb{A}=\left\{\theta \left(t\right)\mid \mathit{there}\mathit{exists}\Delta ,{\epsilon }_1,{\epsilon }_2>0,\left|\theta \left(t\right)\right|<\Delta e^{{\displaystyle \frac{\left|t\right|}{{\epsilon }_i}}},t\in {(-1)}^i\times [0,\infty )\right\}.$$

(12)

The Sumudu transform of the Caputo derivative is given by

$$S\left[{\mathfrak{D}}_t^{\epsilon }\theta \left(t\right)\right]={\zeta }^{-\epsilon }S\left[\theta \left(t\right)\right]-\sum _{\lambda =0}^{n-1}{\zeta }^{-\epsilon +\lambda }{\theta }^{\left(\lambda \right)}\left(0\right),n-1<\epsilon \le n,\epsilon >0,n\in \mathbb{N}.$$

(13)

The Sumudu transform is a key tool in the SRPSM, as it converts fractional differential equations into algebraic equations, facilitating the construction of series solutions. Its properties, outlined in the subsequent lemma, further support its application in our analytical framework.

Lemma 1. 

Let $\theta \left(t\right)$ and $\vartheta \left(t\right)$ be piecewise continuous functions of exponential order on $[0,\infty )$, with $\mathsf{\Theta }\left(\zeta \right)=S\left[\theta \right(t\left)\right]$, $\mathrm{Y}\left(\zeta \right)=S\left[\vartheta \right(t\left)\right]$, and $u,v,w,a$ constants [27]. Then the following apply:

(i) 

$S\left[u\theta \right(t)+v\vartheta (t\left)\right]=uS\left[\theta \right(t\left)\right]+vS\left[\vartheta \right(t\left)\right]=u\mathsf{\Theta }\left(\zeta \right)+v\mathrm{Y}\left(\zeta \right)$;

(ii) 

$S^{-1}[u\mathsf{\Theta }\left(\zeta \right)+v\mathrm{Y}\left(\zeta \right)]=u\theta \left(t\right)+v\vartheta \left(t\right)$;

(iii) 

$S\left[e^{wt}\theta \left(t\right)\right]={\displaystyle \frac{1}{(1-w\zeta )}}\mathsf{\Theta }\left({\displaystyle \frac{\zeta }{1-w\zeta }}\right)$;

(iv) 

${\lim }_{\zeta \to 0}\mathsf{\Theta }\left(\zeta \right)=\theta \left(0\right)$;

(v) 

$S\left[\theta \right(at\left)\right]=a\mathsf{\Theta }\left(\zeta \right),a>0$, and ${\lim }_{t\to 0}\theta \left(t\right)={\lim }_{\zeta \to 0}\mathsf{\Theta }\left(\zeta \right)$.

These definitions and properties underpin the analytical solutions derived in Section 4 and Section 5, enabling us to address the fractional neutron diffusion system effectively.

4. Methods Principal Ideas

4.1. The Fractional TAM

The TAM is an iterative technique that has been successfully applied to solve nonlinear fractional differential equations. It is particularly advantageous for problems involving linear and nonlinear components. In our work, we applied TAM to the fractional neutron diffusion equations for various geometries (slab, cylindrical, and spherical). The method’s strength lies in its ability to handle both linear and nonlinear terms iteratively, making it suitable for solving complex fractional differential equations like those encountered in nuclear reactor physics. TAM has previously been applied to fractional systems in other fields, but its use in fractional nuclear reactor equations, especially with delayed neutrons, is novel and provides new insights into reactor dynamics.

To elaborate on the principal concept of this method, we make the following assumptions about the generic non-homogeneous fractional differential equation (FPDE) [23,24,25]:

$$\mathrm{Y}\left(\theta \right(x,t\left)\right)=\mathsf{\Theta }\left(\theta \right(x,t\left)\right)+p(x,t),m-1<ϵ\le m,$$

(14)

under the boundary conditions

$$\mathcal{B}\left(\theta ,{\displaystyle \frac{\partial \theta }{\partial x}}\right)=0,$$

(15)

where $\mathrm{Y}={\mathcal{D}}_t^{ϵ}={\displaystyle \frac{{\partial }^{ϵ}}{\partial t^{ϵ}}}$ is the fractional operator, $\mathsf{\Theta }$ is the standard differential operators, $\theta (x,t)$ is depicting the anonymous function, $p(x,t)$ is the representation of the acknowledged continuous functions, and the symbol B indicates the boundary operator. The primary requirement in this case is $\mathrm{Y}$, which is the generic fractional differential operator. However, we can set down several linear expressions in addition to the nonlinear expressions as needed. The first step in the proposed technique is to derive the initial condition by removing the nonlinear component as follows:

$${\mathfrak{D}}_t^{ϵ}{\theta }_0(x,t)=p(x,t),\mathfrak{B}\left({\theta }_0,{\displaystyle \frac{\partial {\theta }_0}{\partial x}}\right)=0.$$

(16)

To obtain the subsequent iteration, we solve the following equation:

$${\mathfrak{D}}_t^{ϵ}{\theta }_1(x,t)=\mathsf{\Theta }\left({\theta }_0(x,t)\right)+p(x,t),\mathfrak{B}\left({\theta }_1,{\displaystyle \frac{\partial {\theta }_1}{\partial x}}\right)=0.$$

(17)

This gives us a straightforward iterative stride ${\theta }_{m+1}(x,t)$, which is the appropriate method for handling a set of linear and nonlinear problems.

$${\mathfrak{D}}_t^{ϵ}{\theta }_{m+1}(x,t)=\mathsf{\Theta }\left({\theta }_m(x,t)\right)+p(x,t),\mathfrak{B}\left({\theta }_{m+1},{\displaystyle \frac{\partial {\theta }_{m+1}}{\partial x}}\right)=0.$$

(18)

It is crucial to remember that, in this method, ${\theta }_{m+1}(x,t)$ solves the issue in (1) independently.

Applying the iterative strategy is simple, and each iteration improves the previous one. By using this approach further, it is possible to obtain an ideal approximation that matches the exact solution. Thus, the solution to Equation (14) is shown as

$$\theta (x,t)=\underset{m\to \infty }{\lim }{\theta }_m(x,t).$$

(19)

The addition of the fractional derivative ${\mathfrak{D}}_t^{\epsilon }$ to TAM profoundly changes the iterative process by incorporating a memory effect that accounts for the historical behavior of the neutron flux $\theta (x,t)$. In contrast to integer-order derivatives, which represent immediate rates of change, the fractional derivative ($0<\epsilon \le 1$) describes long-term dependence caused by delayed neutrons, as stated in Section 3. This effect manifests in each iteration, where the fractional operator ${\mathfrak{D}}_t^{\epsilon }$ integrates the past states of $\epsilon <1$ via the convolution with a power-law kernel. As a result, the solutions exhibit anomalous diffusion characteristics: subdiffusion for $\epsilon <1$ and classical diffusion for $\epsilon =1$, reflecting the physical reality of neutron transport in reactors with delayed neutron contributions. In our work, this allows TAM to more correctly represent the progressive development of neutron populations, with the fractional-order $\epsilon$ functioning as a tuning parameter to change the memory strength and diffusion type, hence influencing the convergence and stability of the iterative approximations.

4.2. The Fractional SRPSM

The SRPSM combines the Sumudu transform with a residual power series, allowing for efficient series solutions to fractional differential equations. This method is especially useful for obtaining analytical solutions to problems with fractional operators due to its ability to transform complex fractional terms into simpler algebraic forms. SRPSM has been applied to fractional neutron diffusion equations in this work, providing solutions that converge quickly and accurately. The novelty of this method in our research lies in its application to the neutron diffusion equations of time-dependent fractional order with delayed neutrons, which had not been explored previously in nuclear reactor studies.

Now, the approach for applying SRPSM to solve the fractional nuclear reactor system of equations with delayed neutrons consists of the following steps [26,27]:

• Step 1: Applying Sumudu transform on Equation (14) as

  $$\mathcal{S}\left(\mathrm{Y}\left(\theta \right(x,t\left)\right)\right)=\mathcal{S}\left(\mathsf{\Theta }\left(\theta \right(x,t\left)\right)+p(x,t)\right).$$

  (20)
     Applying the differentiation property of Sumudu transform, $\mathcal{S}\left[{\mathfrak{D}}_t^{ϵ}\theta (x,t)\right]={\displaystyle \frac{1}{s^{ϵ}}}\left\{\mathcal{S}\left(\theta (x,t)\right)-{\theta }_0\left(x\right)\right\}$ on Equation (4), we get

  $$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{s^{ϵ}}}\left\{\mathcal{S}\left(\theta (x,t)\right)-{\theta }_0\left(x\right)\right\}& =& \mathcal{S}\left\{\mathsf{\Theta }\left(\theta (x,t)\right)+p(x,t)\right\},\hfill \end{array}$$

  (21)

     i.e.,

  $$\begin{array}{ccc}\hfill \mathcal{S}\left(\theta (x,t)\right)& =& {\theta }_0\left(x\right)+s^{ϵ}\mathcal{S}\left\{\mathsf{\Theta }\left(\theta \right(x,t\left)\right)+p(x,t)\right\}.\hfill \end{array}$$

  (22)
• Step 2: Applying Sumudu transform inverse function in Equation (22) to have

  $$\theta (x,t)={\theta }_0\left(x\right)+{\mathcal{S}}^{-1}\left[s^{ϵ}\mathcal{S}\left\{\mathsf{\Theta }\left(\theta \right(x,s\left)\right)+p(x,s)\right\}\right],$$

  (23)

     here, ${\theta }_0\left(x\right)$ is the initial condition.
• Step 3: Thus, the algorithm of $\theta (x,t)$ can be considered in the form

  $$\theta (x,t)=\sum _{n=0}^{\infty }{f}_n(x,t){\displaystyle \frac{t^{nϵ}}{\Gamma \left(1+nϵ\right)}}.$$

  (24)
• Step 4: The function of Sumudu residual is

  $$\mathcal{S}Res\left(\mathsf{\Theta }(x,s)\right)=\mathsf{\Theta }(x,s)-{\displaystyle \frac{1}{s}}{\theta }_0\left(x\right)-{\displaystyle \frac{1}{s^{ϵ}}}\left\{\mathsf{\Theta }\left(\theta \right(x,s\left)\right)+p(x,s)\right\}.$$

  (25)

In SRPSM, the fractional derivative ${\mathfrak{D}}_t^{\epsilon }$ affects the solution via power-series expansion, where the fractional-order $\epsilon$ determines the time-scaling of the terms $t^{n\epsilon }/\mathsf{\Gamma }(1+n\epsilon )$. This scaling captures the fractional operator’s non-local character and embeds delayed neutrons’ memory effect into the series coefficients $f_n(x,t)$. As $\epsilon$ deviates from 1, the series records anomalous diffusion, with lower $\epsilon$ values highlighting slower, sub-diffusive behavior due to increased historical reliance and $\epsilon >1$ potentially revealing super-diffusive tendencies. The Sumudu transform accentuates this impact by transforming the fractional derivative into an algebraic form (${\zeta }^{-\epsilon }$), allowing for the rapid calculation of the series terms. In our nuclear reactor model, this results in a solution that accurately represents the delayed neutron dynamics, with the fractional order $\epsilon$ controlling the decay rate and spread of neutron flux over time, as evidenced by the numerical results in Section 6.

5. Neutron Diffusion Equations Analytical Solutions

The nuclear reactor equation is represented fractionally as

$$\begin{array}{c}{\displaystyle \frac{1}{v_c}}{\mathfrak{D}}_t^{ϵ}\theta (x,t)=D{\nabla }^2\theta (x,t)+\left({\gamma }_f{\displaystyle \sum _f}-{\displaystyle \sum _a}\right)\theta (x,t)+\left({\displaystyle \frac{a}{x}}\right){\displaystyle \frac{\partial \theta (x,t)}{\partial x}}+\lambda \vartheta (x,t),\\ {\mathfrak{D}}_t^{ϵ}\vartheta (x,t)=\beta {\gamma }_f{\displaystyle \sum _f}\theta (x,t)-\lambda \vartheta (x,t),\end{array}$$

(26)

where the following initial conditions are applied:

$$\theta (x,0)={\theta }_0\left(x\right),\vartheta (x,0)={\displaystyle \frac{\beta {\sum }_f}{\lambda }}{\theta }_0\left(x\right),$$

(27)

For $A=\beta {\gamma }_f{\sum }_f$ and $\sum ={\gamma }_f{\sum }_f-{\sum }_a$,

$$\begin{array}{c}{\mathfrak{D}}_t^{ϵ}\theta (x,t)=v_{c}D{\theta }_{xx}(x,t)+v_c\sum \theta (x,t)+\left({\displaystyle \frac{a{v}_c}{x}}\right){\theta }_x(x,t)+v_c\lambda \vartheta (x,t),\\ {\mathfrak{D}}_t^{ϵ}\vartheta (x,t)=A\theta (x,t)-\lambda \vartheta (x,t),\end{array}$$

(28)

where

$$\theta (x,0)={\theta }_0\left(x\right),\vartheta (x,0)={\displaystyle \frac{A}{\lambda }}{\theta }_0\left(x\right),$$

(29)

are the proposed initial conditions.

5.1. Analytical Solution by TAM

By using the fractional TAM, Equation (26) is first rewritten as

$$\left\{\begin{array}{cc}\hfill {\mathrm{Y}}_1\left(\theta (x,t)\right)=& D_t^{ϵ}\theta (x,t),{\mathsf{\Theta }}_1\left(\theta (x,t)\right)=v_{c}D{\theta }_{xx}(x,t)+v_c\sum \theta (x,t)+\left({\displaystyle \frac{a{v}_c}{x}}\right){\theta }_x(x,t)+v_c\lambda \vartheta (x,t),p(x,t)=0\hfill \\ \hfill & {\mathrm{Y}}_2\left(\vartheta (x,t)\right)=D_t^{ϵ}\vartheta (x,t),{\mathsf{\Theta }}_2\left(\vartheta (x,t)\right)=A\theta (x,t)-\lambda \vartheta (x,t),q(x,t)=0.\hfill \end{array}\right.$$

(30)

The first problem that has to be handled is

$$\left\{\begin{array}{cc}{\mathrm{Y}}_1\left({\theta }_0(x,t)\right)=0,& {\theta }_0(x,0)={\theta }_0\\ {\mathrm{Y}}_1\left({\vartheta }_0(x,t)\right)=0,& {\vartheta }_0(x,0)={\displaystyle \frac{A}{\lambda }}{\theta }_0\left(x\right)\end{array}\right.$$

(31)

We can solve the above equation using a simple treatment, as follows:

$$\left\{\begin{array}{cc}{I}^{ϵ}\left(D_t^{ϵ}{\theta }_0(x,t)\right)=0,& {\theta }_0(x,0)={\theta }_0\\ I^{ϵ}\left(D_t^{ϵ}{\vartheta }_0(x,t)\right)=0,& {\vartheta }_0(x,0)={\displaystyle \frac{A}{\lambda }}{\theta }_0\left(x\right).\end{array}\right.$$

(32)

Depending on Definition 2, the primary iteration can be obtained as

$$\left\{\begin{array}{c}{\theta }_0(x,\mathrm{t})={\theta }_0\\ {\vartheta }_0(x,\mathrm{t})={\displaystyle \frac{A}{\lambda }}{\theta }_0\left(x\right)\end{array}\right.$$

(33)

The next iteration can be computed as

$$\left\{\begin{array}{cc}{\mathrm{Y}}_1\left({\theta }_1(x,t)\right)={\mathsf{\Theta }}_1\left({\theta }_0(x,t)\right)+p(x,t),& {\theta }_1(x,0)={\theta }_0\\ {\mathrm{Y}}_1\left({\vartheta }_1(x,t)\right)={\mathsf{\Theta }}_2\left({\vartheta }_0(x,t)\right)+q(x,t),& {\vartheta }_1(x,0)={\displaystyle \frac{A}{\lambda }}{\theta }_0\left(x\right).\end{array}\right.$$

(34)

Next, by integrating both sides of the above equation using Definition 1,

$$\left\{\begin{array}{cc}{I}^{ϵ}\left(D_t^{ϵ}{\theta }_1(x,t)\right)=I^{ϵ}\left(v_{c}D{\left({\theta }_0\right)}_{xx}(x,t)+v_c\sum {\theta }_0(x,t)+\left({\displaystyle \frac{a{v}_c}{x}}\right){\left({\theta }_0\right)}_x(x,t)+v_c\lambda {\vartheta }_0(x,t)\right),& {\theta }_1(x,0)={\theta }_0\\ I^{ϵ}\left(D_t^{ϵ}{\vartheta }_1(x,t)\right)=I^{ϵ}\left(A{\theta }_0(x,t)-\lambda {\vartheta }_0(x,t)\right),& {\vartheta }_1(x,0)={\displaystyle \frac{A}{\lambda }}{\theta }_0\left(x\right).\end{array}\right.$$

(35)

Then, we receive the next iteration as

$$\left\{\begin{array}{c}{\theta }_1(x,t)={\theta }_0+{\displaystyle \frac{t^{ϵ}{v}_c\left[4+2D+\left(A+\sum \right){\theta }_0\right]}{\mathsf{\Gamma }(ϵ+1)}},\\ {\vartheta }_1(x,t)={\displaystyle \frac{A}{\lambda }}{\theta }_0.\end{array}\right.$$

(36)

It is possible to calculate the next iteration, which is provided as

$$\left\{\begin{array}{cc}{\mathrm{Y}}_1\left({\theta }_2(x,t)\right)={\mathsf{\Theta }}_1\left({\theta }_1(x,t)\right)+p(x,t),& {\theta }_2(x,0)={\theta }_0\\ {\mathrm{Y}}_2\left({\vartheta }_2(x,t)\right)={\mathsf{\Theta }}_2\left({\vartheta }_1(x,t)\right)+q(x,t),& {\vartheta }_2(x,0)={\displaystyle \frac{A}{\lambda }}{\theta }_0\left(x\right).\end{array}\right.$$

(37)

Next, by integrating both sides of the above equation using Definition 1,

$$\left\{\begin{array}{cc}{I}^{ϵ}\left(D_t^{ϵ}{\theta }_2(x,t)\right)=I^{ϵ}\left(v_{c}D{\left({\theta }_1\right)}_{xx}(x,t)+v_c\sum {\theta }_1(x,t)+\left({\displaystyle \frac{a{v}_c}{x}}\right){\left({\theta }_1\right)}_x(x,t)+v_c\lambda {\vartheta }_1(x,t)\right),& {\theta }_2(x,0)={\theta }_0,\hfill \\ I^{ϵ}\left(D_t^{ϵ}{\vartheta }_2(x,t)\right)=I^{ϵ}\left(A{\theta }_1(x,t)-\lambda {\vartheta }_1(x,t)\right),& {\vartheta }_2(x,0)={\displaystyle \frac{A}{\lambda }}{\theta }_0\left(x\right).\hfill \end{array}\right.$$

(38)

Then, we receive the next iteration as

$$\left\{\begin{array}{c}{\theta }_2(x,t)={\theta }_0+{\displaystyle \frac{t^{ϵ}{v}_c\left[4+2D+\left(A+\sum \right){\theta }_0\right]}{\mathsf{\Gamma }(ϵ+1)}}+{\displaystyle \frac{t^{2ϵ}{v}_c^2\left[2(2+D)(A+2\sum )+\sum \left(A+\sum \right){\theta }_0\right]}{\mathsf{\Gamma }(2ϵ+1)}}\\ +{\displaystyle \frac{t^{3ϵ}{v}_c^2\left[2(2+D)\left(A\lambda +\sum \left(2A+3\sum \right)v_c\right)+(A+\sum )\left(A\lambda +{\sum }^2{v}_c\right){\theta }_0\right]}{\mathsf{\Gamma }(3ϵ+1)}}\\ {\vartheta }_2(x,t)={\displaystyle \frac{A}{\lambda }}{\theta }_0+{\displaystyle \frac{A{t}^{2ϵ}{v}_c\left[4+2D+(A+\sum ){\theta }_0\right]}{\mathsf{\Gamma }(2ϵ+1)}}+{\displaystyle \frac{A{t}^{3ϵ}{v}_c\left[-2(2+D)\left(\lambda -\left(A+2\sum \right)v_c\right)+(A+\sum )\left(-\lambda +\sum v_c\right){\theta }_0\right]}{\mathsf{\Gamma }(3ϵ+1)}}.\end{array}\right.$$

(39)

Every time ${\theta }_m(x,t)$ and ${\vartheta }_m(x,t)$ are repeated, an approximate solution to Equation (26) is found, according to Equation (19).

As iterations grow in number, our results become closer to the exact results. Using this approach, we can complete the following series template for approximate solutions as

$$\left\{\begin{array}{c}\theta (x,t)={\lim }_{m\to \infty }{\theta }_m(x,t)\simeq {\theta }_2(x,t)\hfill \\ \vartheta (x,t)={\lim }_{m\to \infty }{\vartheta }_m(x,t)\simeq {\vartheta }_2(x,t).\hfill \end{array}\right.$$

(40)

5.2. Analytical Solution by SRPSM

The objective in this step is to construct a series solution for the fractional nuclear reactor system of equations fractionally using the Sumudu Residual Power Series Method (SRPSM).

Upon applying the Sumudu transform to the coupled Equation (26), we see that

$$\begin{array}{c}\mathcal{S}\left[{\mathfrak{D}}_t^{ϵ}\theta (x,t)\right]=v_{c}D\mathcal{S}\left[{\theta }_{xx}(x,t)\right]+v_c\sum \mathcal{S}\left[\theta (x,t)\right]+{\displaystyle \frac{a{v}_c}{x}}\mathcal{S}\left[{\theta }_x(x,t)\right]+v_c\lambda \mathcal{S}\left[\vartheta (x,t)\right],\\ \mathcal{S}\left[{\mathfrak{D}}_t^{ϵ}\vartheta (x,t)\right]=A\left[\theta (x,t)\right]-\lambda \mathcal{S}\left[\vartheta (x,t)\right].\end{array}$$

(41)

Lemma 1 and the initial conditions in Equation (27) allow us to create Equation (28) in the following way:

$$\begin{array}{c}{s}^{ϵ}\mathsf{\Theta }(x,s)-s^{ϵ-1}{\theta }_0\left(x\right)=v_{c}D{\mathsf{\Theta }}_{xx}(x,s)+v_c\sum \mathsf{\Theta }(x,s)+{\displaystyle \frac{a{v}_c}{x}}{\mathsf{\Theta }}_x(x,s)+v_c\lambda \mathsf{\Xi }(x,s)\\ s^{ϵ}\mathsf{\Xi }(x,s)-s^{ϵ-1}{\displaystyle \frac{A}{\lambda }}{\theta }_0\left(x\right)=A\mathsf{\Theta }(x,s)-\lambda \mathsf{\Xi }(x,s),\end{array}$$

(42)

where $\mathsf{\Theta }(x,s)=\mathcal{S}\left[\theta \right(x,t\left)\right],\mathsf{\Xi }(x,s)=\mathcal{S}\left[\vartheta \right(x,t\left)\right]$. Equation (42) may be rewritten as

$$\begin{array}{c}\mathsf{\Theta }(x,s)-{\displaystyle \frac{1}{s}}{\theta }_0\left(x\right)-{\displaystyle \frac{v_{c}D}{s^{ϵ}}}{\mathsf{\Theta }}_{xx}(x,s)-{\displaystyle \frac{v_c\sum }{s^{ϵ}}}\mathsf{\Theta }(x,s)-{\displaystyle \frac{a{v}_c}{x{s}^{ϵ}}}{\mathsf{\Theta }}_x(x,s)-{\displaystyle \frac{v_c\lambda }{s^{ϵ}}}\mathsf{\Xi }(x,s)=0,\\ \mathsf{\Xi }(x,s)-{\displaystyle \frac{A}{\lambda s}}{\theta }_0\left(x\right)-{\displaystyle \frac{A}{s^{ϵ}}}\mathsf{\Theta }(x,s)+{\displaystyle \frac{\lambda }{s^{ϵ}}}\mathsf{\Xi }(x,s)=0.\end{array}$$

(43)

The linked Equation (43) forms a system of linear partial differential equations with derivatives for the variable x.

As per the SRPS methodology, the System (43) series solution needs to possess the subsequent configuration:

$$\begin{array}{c}\mathsf{\Theta }(x,s)=\sum _{n=0}^{\infty }{\displaystyle \frac{u_n\left(x\right)}{s^{nϵ+1}}},\hfill \\ \mathsf{\Xi }(x,s)=\sum _{n=0}^{\infty }{\displaystyle \frac{v_n\left(x\right)}{s^{nϵ+1}}}.\hfill \end{array}$$

(44)

We may use Lemma 1 to express the $\mathrm{k}\mathrm{th}$ truncated series of $\mathsf{\Theta }(x,s),$$\mathsf{\Xi }(x,s)$ as follows:

$$\begin{array}{c}{\mathsf{\Theta }}_k(x,s)={\displaystyle \frac{{\theta }_0\left(x\right)}{s}}+{\displaystyle \sum _{n=1}^k}{\displaystyle \frac{u_n\left(x\right)}{s^{nϵ+1}}},x\in I,s>\delta \ge 0,\\ {\mathsf{\Xi }}_k(x,s)={\displaystyle \frac{A}{\lambda s}}{\theta }_0\left(x\right)+{\displaystyle \sum _{n=1}^k}{\displaystyle \frac{v_n\left(x\right)}{s^{nϵ+1}}},x\in I,s>\delta \ge 0.\end{array}$$

(45)

In this stage, we obtain the anonymous coefficients of (45) by finding the Sumudu residual functions of Equation (43):

$$\begin{array}{cc}\hfill \mathcal{S}Res\left(\mathsf{\Theta }\right(x,s\left)\right)=& \mathsf{\Theta }(x,s)-{\displaystyle \frac{1}{s}}{\theta }_0\left(x\right)-{\displaystyle \frac{v_{c}D}{s^{ϵ}}}{\mathsf{\Theta }}_{xx}(x,s)-{\displaystyle \frac{v_c\sum }{s^{ϵ}}}\mathsf{\Theta }(x,s)-{\displaystyle \frac{a{v}_c}{x{s}^{ϵ}}}{\mathsf{\Theta }}_x(x,s)-{\displaystyle \frac{v_c\lambda }{s^{ϵ}}}\mathsf{\Xi }(x,s),\hfill \\ \hfill \mathcal{S}Res\left(C\right(x,s\left)\right)=& C(x,s)-{\displaystyle \frac{A}{\lambda s}}{\theta }_0\left(x\right)-{\displaystyle \frac{A}{s^{ϵ}}}\mathsf{\Theta }(x,s)+{\displaystyle \frac{\lambda }{s^{ϵ}}}\mathsf{\Xi }(x,s).\hfill \end{array}$$

(46)

Additionally, the $k^{\mathrm{th}}$ Sumudu residual functions are

$$\begin{array}{c}\mathcal{S}Re{s}_k\left(\mathsf{\Theta }(x,s)\right)={\mathsf{\Theta }}_k(x,s)-{\displaystyle \frac{1}{s}}{\theta }_0\left(x\right)-{\displaystyle \frac{v_{c}D}{s^{ϵ}}}{\mathsf{\Theta }}_{k_{xx}}(x,s)-{\displaystyle \frac{v_c\sum }{s^{ϵ}}}{\mathsf{\Theta }}_k(x,s)-{\displaystyle \frac{a{v}_c}{x{s}^{ϵ}}}{\mathsf{\Theta }}_{k_x}(x,s)-{\displaystyle \frac{v_c\lambda }{s^{ϵ}}}{\mathsf{\Xi }}_k(x,s),\\ \mathcal{S}Re{s}_k\left(\mathsf{\Xi }(x,s)\right)={\mathsf{\Xi }}_k(x,s)-{\displaystyle \frac{A}{\lambda s}}{\theta }_0\left(x\right)-{\displaystyle \frac{A}{s^{ϵ}}}{\mathsf{\Theta }}_k(x,s)+{\displaystyle \frac{\lambda }{s^{ϵ}}}{\mathsf{\Xi }}_k(x,s).\end{array}$$

(47)

Since $\mathcal{S}Res\left(\mathsf{\Theta }\right(x,s\left)\right)=0$ and $\mathcal{S}Res\left(\mathsf{\Xi }\right(x,s\left)\right)=0$, we see that $s^{kϵ+1}\mathcal{S}Res\left(\mathsf{\Theta }(x,s)\right)=0,s^{kϵ+1}\mathcal{S}Res\left(\mathsf{\Xi }(x,s)\right)=0$. Therefore,

$$\underset{s\to \infty }{\lim }\left(s^{kϵ+1}\mathcal{S}Re{s}_k\left(\mathsf{\Theta }(x,s)\right)\right)=0,\underset{s\to \infty }{\lim }\left(s^{kϵ+1}\mathcal{S}Re{s}_k\left(\mathsf{\Xi }(x,s)\right)\right)=0\mathrm{for}k=0,1,2,\dots .$$

(48)

Substituting the first Sumudu-residual functions as ${\mathsf{\Theta }}_1(x,s)={\displaystyle \frac{{\theta }_0\left(x\right)}{s}}+{\displaystyle \frac{u_1\left(x\right)}{s^{ϵ+1}}}$ and ${\mathsf{\Xi }}_1(x,s)={\displaystyle \frac{A}{\lambda s}}{\theta }_0\left(x\right)+{\displaystyle \frac{v_1\left(x\right)}{s^{ϵ+1}}}$, we then obtain $u_1\left(x\right)$ and $v_1\left(x\right)$ in (45):

$$\begin{array}{c}\mathcal{S}Re{s}_1\left(\mathsf{\Theta }(x,s)\right)={\displaystyle \frac{u_1\left(x\right)}{s^{ϵ+1}}}-{\displaystyle \frac{v_{c}D}{s^{ϵ}}}\left[{\displaystyle \frac{{\theta }_0^{″}\left(x\right)}{s}}+{\displaystyle \frac{u_1^{″}\left(x\right)}{s^{ϵ+1}}}\right]-{\displaystyle \frac{v_c\sum }{s^{ϵ}}}\left[{\displaystyle \frac{{\theta }_0\left(x\right)}{s}}+{\displaystyle \frac{u_1\left(x\right)}{s^{ϵ+1}}}\right]-\\ {\displaystyle \frac{a{v}_c}{x{s}^{ϵ}}}\left[{\displaystyle \frac{{\theta }_0^{\prime }\left(x\right)}{s}}+{\displaystyle \frac{u_1^{\prime }\left(x\right)}{s^{ϵ+1}}}\right]-{\displaystyle \frac{v_c\lambda }{s^{ϵ}}}\left[{\displaystyle \frac{A}{\lambda s}}{\theta }_0\left(x\right)+{\displaystyle \frac{v_1\left(x\right)}{s^{ϵ+1}}}\right],\\ \mathcal{S}Re{s}_1\left(\mathsf{\Xi }(x,s)\right)={\displaystyle \frac{v_1\left(x\right)}{s^{ϵ+1}}}-{\displaystyle \frac{A}{s^{ϵ}}}\left[{\displaystyle \frac{{\theta }_0\left(x\right)}{s}}+{\displaystyle \frac{u_1\left(x\right)}{s^{ϵ+1}}}\right]+{\displaystyle \frac{\lambda }{s^{ϵ}}}\left[{\displaystyle \frac{A}{\lambda s}}{\theta }_0\left(x\right)+{\displaystyle \frac{v_1\left(x\right)}{s^{ϵ+1}}}\right].\end{array}$$

(49)

Now, by solving $\underset{s\to \infty }{\lim }{s}^{ϵ+1}\mathcal{S}Re{s}_1\left(\mathsf{\Theta }(x,s)\right)=0,\underset{s\to \infty }{\lim }{s}^{ϵ+1}\mathcal{S}Re{s}_1\left(\mathsf{\Xi }(x,s)\right)=0$, we obtain

$$\begin{array}{c}{u}_1\left(x\right)=v_{c}D{\theta }_0^{″}\left(x\right)+v_c\sum {\theta }_0\left(x\right)+{\displaystyle \frac{a{v}_c}{x}}{\theta }_0^{\prime }\left(x\right)+v_c\lambda {\vartheta }_0\left(x\right),\\ v_1\left(x\right)=0.\end{array}$$

(50)

Substituting the second Sumudu residual functions as ${\mathsf{\Theta }}_2(x,s)={\displaystyle \frac{{\theta }_0\left(x\right)}{s}}+{\displaystyle \frac{u_1\left(x\right)}{s^{ϵ+1}}}+{\displaystyle \frac{u_2\left(x\right)}{s^{2ϵ+1}}}$ and ${\mathsf{\Xi }}_2(x,s)={\displaystyle \frac{A}{\lambda s}}{\theta }_0\left(x\right)+{\displaystyle \frac{v_2\left(x\right)}{s^{2ϵ+1}}}$, we then obtain $u_2\left(x\right)$ and $v_2\left(x\right)$ in (45):

$$\begin{array}{c}\mathcal{S}Re{s}_2\left(\mathsf{\Theta }(x,s)\right)={\displaystyle \frac{u_1\left(x\right)}{s^{ϵ+1}}}+{\displaystyle \frac{u_2\left(x\right)}{s^{2ϵ+1}}}-{\displaystyle \frac{v_{c}D}{s^{ϵ}D}}\left[{\displaystyle \frac{{\theta }_0^{″}\left(x\right)}{s}}+{\displaystyle \frac{u_1^{″}\left(x\right)}{s^{ϵ+1}}}+{\displaystyle \frac{u_2^{″}\left(x\right)}{s^{2ϵ+1}}}\right]-{\displaystyle \frac{v_c\sum }{s^{ϵ}}}\left[{\displaystyle \frac{{\theta }_0\left(x\right)}{s}}+{\displaystyle \frac{u_1\left(x\right)}{s^{ϵ+1}}}+{\displaystyle \frac{u_2\left(x\right)}{s^{2ϵ+1}}}\right]-\\ {\displaystyle \frac{a{v}_c}{x{s}^{ϵ}}}\left[{\displaystyle \frac{{\theta }_0^{\prime }\left(x\right)}{s}}+{\displaystyle \frac{u_1^{\prime }\left(x\right)}{s^{ϵ+1}}}+{\displaystyle \frac{u_2^{\prime }\left(x\right)}{s^{2ϵ+1}}}\right]-{\displaystyle \frac{v_c\lambda }{s^{ϵ}ϵ}}\left[{\displaystyle \frac{A}{\lambda s}}{\theta }_0\left(x\right)+{\displaystyle \frac{v_2\left(x\right)}{s^{2ϵ+1}}}\right].\\ \mathcal{S}Re{s}_2\left(\mathsf{\Xi }(x,s)\right)={\displaystyle \frac{v_2\left(x\right)}{s^{2ϵ+1}}}-{\displaystyle \frac{A}{s^{ϵ}}}\left[{\displaystyle \frac{{\theta }_0\left(x\right)}{s}}+{\displaystyle \frac{u_1\left(x\right)}{s^{ϵ+1}}}+{\displaystyle \frac{u_2\left(x\right)}{s^{2ϵ+1}}}\right]+{\displaystyle \frac{\lambda }{s^{ϵ}}}\left[{\displaystyle \frac{A}{\lambda s}}{\theta }_0\left(x\right)+{\displaystyle \frac{v_2\left(x\right)}{s^{2ϵ+1}}}\right].\end{array}$$

(51)

Now, by solving $\underset{s\to \infty }{\lim }{s}^{2ϵ+1}\mathcal{S}Re{s}_2\left(\mathsf{\Theta }(x,s)\right)=0,\underset{s\to \infty }{\lim }{s}^{2ϵ+1}\mathcal{S}Re{s}_2\left(\mathsf{\Xi }(x,s)\right)=0$, we have

$$\begin{array}{c}{u}_2\left(x\right)=v_{c}D{u}_1^{″}\left(x\right)+v_c\sum u_1^{s\to \infty }\left(x\right)+{\displaystyle \frac{a{v}_c}{x}}{u}_1^{\prime }\left(x\right)+v_c\lambda v_1\left(x\right),\\ v_2\left(x\right)=A{u}_1\left(x\right).\end{array}$$

(52)

Substituting the third Sumudu residual functions as ${\mathsf{\Theta }}_2(x,s)={\displaystyle \frac{{\theta }_0\left(x\right)}{s}}+{\displaystyle \frac{u_1\left(x\right)}{s^{ϵ+1}}}+{\displaystyle \frac{u_2\left(x\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{u_3\left(x\right)}{s^{3ϵ+1}}}$ and ${\mathsf{\Xi }}_2(x,s)={\displaystyle \frac{A}{\lambda s}}{\theta }_0\left(x\right)+{\displaystyle \frac{v_2\left(x\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{v_3\left(x\right)}{s^{3ϵ+1}}}$, we then obtain $u_3\left(x\right)$ and $v_3\left(x\right)$ in (45):

$$\begin{array}{c}\mathcal{S}Re{s}_3\left(\mathsf{\Theta }(x,s)\right)={\displaystyle \frac{u_1\left(x\right)}{s^{ϵ+1}}}+{\displaystyle \frac{u_2\left(x\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{u_3\left(x\right)}{s^{3ϵ+1}}}-{\displaystyle \frac{v_{c}D}{s^{ϵ}}}\left[{\displaystyle \frac{{\theta }_0^{″}\left(x\right)}{s}}+{\displaystyle \frac{u_1^{″}\left(x\right)}{s^{ϵ+1}}}+{\displaystyle \frac{u_2^{″}\left(x\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{u_3^{″}\left(x\right)}{s^{3ϵ+1}}}\right]-\hfill \\ {\displaystyle \frac{v_c\sum }{s^{ϵ}}}\left[{\displaystyle \frac{{\theta }_0^{\prime }\left(x\right)}{s}}+{\displaystyle \frac{u_1^{\prime }\left(x\right)}{s^{ϵ+1}}}+{\displaystyle \frac{u_2^{\prime }\left(x\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{u_3^{\prime }\left(x\right)}{s^{3ϵ+1}}}\right]-{\displaystyle \frac{a{v}_c}{x{s}^{ϵ}}}\left[{\displaystyle \frac{{\theta }_0^{\prime }\left(x\right)}{s}}+{\displaystyle \frac{u_1^{\prime }\left(x\right)}{s^{ϵ+1}}}+{\displaystyle \frac{u_2^{\prime }\left(x\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{u_3^{\prime }\left(x\right)}{s^{3ϵ+1}}}\right]\hfill \\ -{\displaystyle \frac{v_c\lambda }{s^{ϵ}}}\left[{\displaystyle \frac{A}{\lambda s}}{\theta }_0\left(x\right)+{\displaystyle \frac{v_2\left(x\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{v_3\left(x\right)}{s^{3ϵ+1}}}\right],\hfill \\ \mathcal{S}Re{s}_3\left(\mathsf{\Xi }(x,s)\right)={\displaystyle \frac{v_2\left(x\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{v_3\left(x\right)}{s^{3ϵ+1}}}-{\displaystyle \frac{A}{s^{ϵ}}}\left[{\displaystyle \frac{{\theta }_0\left(x\right)}{s}}+{\displaystyle \frac{u_1\left(x\right)}{s^{ϵ+1}}}+{\displaystyle \frac{u_2\left(x\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{u_3\left(x\right)}{s^{3ϵ+1}}}\right]\hfill \\ +{\displaystyle \frac{\lambda }{s^{ϵ}}}\left[{\displaystyle \frac{A}{\lambda s}}{\theta }_0\left(x\right)+{\displaystyle \frac{v_2\left(x\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{v_3\left(x\right)}{s^{3ϵ+1}}}\right].\hfill \end{array}$$

(53)

Now, by solving $\underset{s\to \infty }{\lim }{s}^{3ϵ+1}\mathcal{S}Re{s}_3\left(\mathsf{\Theta }(x,s)\right)=0,\underset{s\to \infty }{\lim }{s}^{3ϵ+1}\mathcal{S}Re{s}_3\left(\mathsf{\Xi }(x,s)\right)=0$, we find that

$$\begin{array}{c}{u}_3\left(x\right)=v_{c}D{u}_2^{″}\left(x\right)+v_c\sum u_2\left(x\right)+{\displaystyle \frac{a{v}_c}{x}}{u}_2^{\prime }\left(x\right)+v_c\lambda v_2\left(x\right),\\ v_3\left(x\right)=A{u}_2\left(x\right)-\lambda v_2\left(x\right).\end{array}$$

(54)

In a similar manner, we substitute the $\mathrm{k}\mathrm{th}$ Sumudu residual function $\mathcal{S}Re{s}_k\left(\mathsf{\Theta }(x,s)\right)$, $\mathcal{S}Re{s}_k\left(\mathsf{\Xi }(x,s)\right)$. For the $\mathrm{k}\mathrm{th}$ truncated series ${\mathsf{\Theta }}_k(x,s)$, ${\mathsf{\Xi }}_k(x,s)$, multiply the resultant equations by $s^{kϵ+1}$, and, when $s\to \infty$, we consider the limit.

To have $u_{k+1}\left(x\right),v_{k+1}\left(x\right)$ for $k\ge 2$, utilize the recurrence relation displayed below:

$$\begin{array}{c}{u}_n\left(x\right)=v_{c}D{u}_{n-1}^{″}\left(x\right)+v_c\sum u_{n-1}\left(x\right)+{\displaystyle \frac{a{v}_c}{x}}{u}_{n-1}^{\prime }\left(x\right)+v_c\lambda v_{n-1}\left(x\right),\\ v_n\left(x\right)=A{u}_{n-1}\left(x\right)-\lambda v_{n-1}\left(x\right),n=1,2,3,\dots \end{array}$$

(55)

where $v_0\left(x\right)={\displaystyle \frac{A}{\lambda }}{\theta }_0\left(x\right)$. As mentioned previously, the System (43) series solution will be

$$\begin{array}{c}\mathsf{\Theta }(x,s)={\displaystyle \frac{{\theta }_0\left(x\right)}{s}}+{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{u_n\left(x\right)}{s^{nϵ+1}}},\\ \mathsf{\Xi }(x,s)={\displaystyle \frac{A}{\lambda s}}{\theta }_0\left(x\right)+{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{v_n\left(x\right)}{s^{nϵ+1}}}.\end{array}$$

(56)

Here, the system series solution given in (26) is obtained, where the inverse Sumudu transform is applied in (56) as follows:

$$\begin{array}{c}\theta (x,t)={\theta }_0\left(x\right)+{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{u_n\left(x\right)t^{nϵ}}{\mathsf{\Gamma }(nϵ+1)}},\\ \vartheta (x,t)={\displaystyle \frac{A}{\lambda }}{\theta }_0\left(x\right)+{\displaystyle \sum _{n=2}^{\infty }}{\displaystyle \frac{v_n\left(x\right)t^{nϵ}}{\mathsf{\Gamma }(nϵ+1)}},\end{array}$$

(57)

where

$$\begin{array}{c}{u}_n\left(x\right)=v_{c}D{u}_{n-1}^{″}\left(x\right)+v_c\sum u_{n-1}\left(x\right)+{\displaystyle \frac{a{v}_c}{x}}{u}_{n-1}^{\prime }\left(x\right)+v_c\lambda v_{n-1}\left(x\right)\\ v_n\left(x\right)=A{u}_{n-1}\left(x\right)-\lambda v_{n-1}\left(x\right),n=1,2,3,\dots \end{array}$$

(58)

6. Numerical Study

Here, the numerical calculations are constructed to improve the theoretical results, built depending on TAM and SRPSM, where the numerical examples compare both methods for each essential reactor’s geometries. Three essential geometries are formulated from the general-case theoretical part of the work, where the numerical calculations are stated with the Cartesian coordinates for the slab nuclear reactor $(a=0)$, cylindrical coordinates of the cylindrical reactor $(a=1)$, and spherical coordinates for spherical reactor $(a=2)$.

To perform these calculations, the following important nuclear data, which are obtained using the needed experimental data [10], are tabulated in

Table 1. Nuclear reactor cross-section data.

${\mathit{\nu }}_{\mathit{c}}$	B	D	$\mathit{\lambda }$	∑
220 cm/s	0.000735 cm−1	0.356 cm2/s	0.08 s−1	0.005 cm−2

.

The needed Mathematica code has been written for each case twice, corresponding to TAM and SRPSM.

It must be mentioned that two numerical examples for each reactor geometry are considered, depending on their initial conditions.

6.1. Slab Reactor

To determine the behavior of the neutrons in this reactor, two examples are considered. The neutron flux $\phi$, which represents the population of neutrons, is determined. Figure 1 considers $\phi (x,0)=1$ as the initial condition.

In Figure 1a, the flux $\phi$ is represented using TAM; in Figure 1b, the flux $\phi$ is represented using SRPSM. In Figure 2, $\phi$ (x, 0) = $x^2$ is considered as the initial condition.

In Figure 2a, the flux $\phi$ is represented using TAM. In Figure 2b, the flux $\phi$ as a function of time is represented using SRPSM. In Figure 2c, the flux $\phi$ in reactor dimensions is represented using TAM, and, in Figure 2d, the flux $\phi$ in the reactor dimension is represented using SRPSM. It is shown in Figure 1 and Figure 2 that the behavior of neutrons is expected; both used methods give results consistent with the historical results [10].

6.2. Infinite Cylindrical Reactor

To determine the behavior of the neutrons in this reactor, two examples are considered. The neutron flux $\phi$, which represents the neutron population in the cylindrical reactor, is determined. Figure 3 is based on the initial condition $\phi (x,0)=1$.

In Figure 3a, the flux $\phi$ is represented using TAM; in Figure 3b, the flux $\phi$ is represented using SRPSM. Figure 4 considers the initial condition $\phi (x,0)=x^2$.

In Figure 4a, the flux $\phi$ is represented using TAM. In Figure 4b, the flux $\phi$ as a function of time is represented using SRPSM. In Figure 4c, the flux $\phi$ in reactor dimensions is represented using TAM, and, in Figure 4d, the flux $\phi$ in the reactor dimensions are represented using SRPSM. It is shown in Figure 3 and Figure 4 that the behavior of neutrons is expected; both used methods give results in agreement with the historical results [10].

6.3. Spherical Reactor

To determine the behavior of the neutrons in this reactor, two examples are considered the neutron flux $\phi$, which represents the neutron distribution in the cylindrical reactor.

Figure 5 considers the initial condition $\phi (x,0)=1$.

In Figure 5a, the flux $\phi$ is represented using TAM; in Figure 5b, the flux $\phi$ as a function of time is represented using SRPSM. Figure 6 considers the initial condition $\phi (x,0)=x^2$.

In Figure 6a, the flux $\phi$ is represented using TAM. In Figure 6b, the flux $\phi$ as a function of time is represented using SRPSM. In Figure 6c, the flux $\phi$ in reactor dimensions is represented using TAM, and, in Figure 6d, the flux $\phi$ in the reactor dimensions is represented using SRPSM. Although the spherical reactor with delayed neutron effect has not been studied in history, for the first time, this system is developed in this work and solved using both TAM and SRPSM methods. The neutron’s behavior can be considered as in agreement with other previous geometries studied in this work and in history.

The presented system, considering three reactor geometries, considers anomalous and Gaussian diffusion, where the value of $\gamma$ determines the type of diffusion. When $\gamma =1$, the diffusion is Gaussian, but, for $\gamma \ne 1$, the diffusion is anomalous. As $\gamma$ approaches to unity, the neutron diffusion becomes Gaussian.

In

Table 2. Comparison of the methods with different geometries.

	Slab Reactor			Cylindrical Reactor			Spherical Reactor
x	${\mathit{\theta }}_{\mathit{TAM}}$	${\mathit{\theta }}_{\mathit{SRPSM}}$	${\mathit{\theta }}_{\mathit{LTM}}$ [28]	${\mathit{\theta }}_{\mathit{TAM}}$	${\mathit{\theta }}_{\mathit{SRPSM}}$	${\mathit{\theta }}_{\mathit{LRPSM}}$ [10]	${\mathit{\theta }}_{\mathit{TAM}}$	${\mathit{\theta }}_{\mathit{SRPSM}}$
0.0	17.6177	17.6216	17.6212	67.1058	67.1205	67.1205	116.594	116.619
0.1	17.6291	17.6329	17.6253	67.1171	67.1318	67.1318	116.605	116.631
0.2	17.6631	17.6669	17.6677	67.1511	67.1658	67.1658	116.639	116.665
0.3	17.7197	17.7236	17.7227	67.2078	67.2225	67.2225	116.696	116.721
0.4	17.7991	17.8029	17.8011	67.2871	67.3018	67.3018	116.775	116.801
0.5	17.9011	17.9049	17.9035	67.3891	67.4038	67.4038	116.877	116.903
0.6	18.0257	18.0296	18.0290	67.5138	67.5285	67.5285	117.002	117.027
0.7	18.1731	18.1769	18.1766	67.6611	67.6758	67.6758	117.149	117.175
0.8	18.3431	18.3470	18.3398	67.8311	67.8458	67.8458	117.319	117.345
0.9	18.5358	18.5396	18.5412	68.0238	68.0385	68.0385	117.512	117.537
1.0	18.7511	18.7550	18.7495	68.2391	68.2538	68.2538	117.727	117.753

, the tabulated compassion between the two used methods with different geometries is presented. Moreover, the slab reactor is compared with the work that studied the same reactor using the Laplace transform method (LTM) [28] and the cylindrical reactor is compared with the work that solved the same reactor using the Laplace residual power-series method (LSPSM) [10].

Table 2 assures that the results of our study are perfectly compatible with the used methods for different nuclear reactor geometries.

Finally, both methods have been employed to solve the generalized system of neutron diffusion equations with delayed neutrons across different geometries. TAM is particularly useful for cases requiring iterative improvements to capture nonlinear dynamics, while SRPSM excels in providing fast and accurate series solutions for problems with complex fractional operators. The comparison of both methods demonstrates their compatibility in solving fractional reactor equations while highlighting TAM’s suitability for nonlinear problems and SRPSM’s efficiency in handling fractional terms.

6.4. Computational Performance of TAM and SRPSM

To thoroughly analyze the suggested approaches, TAM and SRPSM, we measure their computing efficiency in terms of CPU time and computational cost. These metrics are crucial for understanding the approaches’ practical application in solving fractional neutron diffusion equations for the three reactor geometries (slab, cylinder, and sphere). All computations were carried out using Mathematica 12.0 on a machine equipped with an Intel Core i5 CPU (3.0 GHz, 8 cores) and 8 GB RAM, assuring uniformity in the computational environment. The CPU time for each technique was assessed as the average time needed to compute the neutron flux $\phi (x,t)$ for a specified spatial domain $x\in [0,100]$ cm and temporal domain $t\in [0,1]$, using the nuclear data from Table 1 and the initial conditions $\phi (x,0)=1$ and $\phi (x,0)=x^2$. TAM iterations were limited to $m=3$ (as indicated in Section 5.1), whereas SRPSM truncated the series at $n=3$ terms (Section 5.2) to balance accuracy and computing effort. The computing cost is measured qualitatively using the memory and algorithmic complexity, considering the iterative nature of TAM and the series expansion of SRPSM.

Table 3. CPU time and computational cost comparison.

Geometry	Method	CPU Time (s), $\mathit{\phi }(\mathit{x},0)=1$	CPU Time (s), $\mathit{\phi }(\mathit{x},0)={\mathit{x}}^2$
Slab Reactor	TAM	0.45	0.52
	SRPSM	0.38	0.44
Cylindrical Reactor	TAM	0.48	0.55
	SRPSM	0.40	0.46
Spherical Reactor	TAM	0.50	0.58
	SRPSM	0.42	0.48

shows that SRPSM consistently beats TAM in terms of CPU time across all geometries and beginning circumstances, with an average savings of 15–20% in calculation time. This improvement is due to SRPSM’s series-based method, which uses the Sumudu transform to compute terms algebraically rather than the iterative process of TAM, which requires repetitive fractional integration. For instance, in the slab reactor with Ï(x,0) = 1, SRPSM requires 0.38 s compared to TAM’s 0.45 s. The difference becomes more pronounced with the more complex initial condition Ï(x,0) = x², where spatial derivatives increase the computational burden, yet SRPSM maintains its efficiency (e.g., 0.44 s vs. 0.52 s for the slab reactor).

Regarding computational cost, TAM includes iterative evaluations of the fractional operator Dtɛ; it has a larger memory footprint because intermediate iterates ${\theta }_m(x,t)$ are stored. In contrast, SRPSM uses less memory and has a computational cost that increases with the number of terms n since it computes a truncated series with coefficients $f_n(x,t)$. However, both approaches are computationally possible for actual applications, with CPU durations of less than one second for the domain under consideration, making them appropriate for real-time simulations or parametric investigations when implemented on ordinary hardware.

These performance measures supplement the accuracy findings reported in Table 2, revealing that SRPSM provides a good balance of computing efficiency and solution quality, whereas TAM provides resilience for capturing nonlinear dynamics at a slightly higher computational cost.

7. Conclusions

In this work, a general formula for the nuclear reactor system of time-dependent neutron diffusion equations is presented and formulated, which is solved by using a model of fractional calculus that represents the delayed neutron effect. The novelty of our paper is in developing and applying two analytical and numerical methods, TAM and SRPSM, for solving the fractional neutron diffusion equation in nuclear reactors with delayed neutrons. We have demonstrated the application of two analytical methods, TAM and the SRPSM, to analyze a novel formula nuclear reactor equation. These methods allow for a more comprehensive and accurate representation of the complex dynamics governing the nuclear system. Specifically, we have developed analytical approximation methods that enable the computation of approximate solutions. These methods provide a powerful tool for researchers to efficiently analyze and predict nuclear phenomena, without the need for computationally intensive numerical simulations. Overall, this study highlights the importance of interdisciplinary research and the integration of advanced mathematical techniques, such as analytical methods in the pursuit of scientific knowledge and technological innovation.