1. Introduction
Over the last decade, there has been considerable interest in fractional partial differential equations (PDEs), making it a vibrant research field for scientists and engineers. PDEs possess the capability to describe numerous complex phenomena across diverse fields, including biology, fluid mechanics, plasma physics, optics, acoustics, financial mathematics, climate modeling, materials science, and electromagnetics [
1,
2]. Fractional calculus, which deals with derivatives and integrals of non-integer order, finds broad applications in science and technology. In physics, it models complex systems with non-local behavior, such as anomalous diffusion in fluid dynamics and memory effects in viscoelastic materials [
3]. Engineering benefits from fractional calculus are found in signal processing, control theory, and optimization, offering robustness and efficiency in dynamic systems. Telecommunications utilize it for modeling wireless communication channels and improving channel estimation algorithms. Fractional calculus enables advancements in understanding and optimizing real-world processes and systems across various domains [
4]. The fractional spatio-temporal PDE models are crucial in describing complex phenomena [
5], whereas the two-scale fractal PDE models are essential for simulating intricate structures in materials science and physics, offering insights into multiscale behaviors and enhancing our understanding of fractal geometries in natural and engineered systems [
6]. In particular, the Sobolev model equation is integral across various scientific fields, providing a robust framework for investigating phenomena with spatial variations. Its relevance extends across disciplines, encompassing physics, engineering, biology, and finance. In physics, Sobolev PDEs are commonly utilized to model phenomena such as heat conduction, fluid dynamics, and quantum mechanics [
7,
8]. Engineers employ them to analyze structural mechanics, electromagnetism, and signal-processing problems [
7]. Furthermore, Sobolev PDEs find applications in image processing, medical imaging, and geophysical exploration. Their adaptability lies in their capacity to handle irregular domains and boundary conditions while providing solutions that demonstrate smoothness properties.
Numerous studies explored the applications and theoretical foundations of Sobolev PDEs, underscoring their significance in both theoretical and applied contexts. However, numerous researchers faced challenges in deriving and formulating various complex phenomena within nonlinear PDEs with integer orders [
9]. In response, fractional calculus is regarded as a viable solution to this issue, as it incorporates a nonlocal property that is absent in nonlinear PDEs with integer orders [
10]. Observations indicate that multi-term time-fractional PDEs were proposed to enhance the modeling accuracy in depicting anomalous diffusion processes, capturing various types of viscoelastic damping, accurately representing power-law frequency dependence, and simulating the flow of a fractional Maxwell fluid [
11]. This study focused on the two-term time-fractional Sobolev equation in two and three dimensions, defined as follows:
and
with the following initial and boundary conditions:
where the constants
,
,
, and
have known values. Moreover,
and
denote the Caputo derivatives in accordance with Caputo [
12] and applied to
, where
.
Several works delved into the existence and uniqueness of solutions for Sobolev equations, as documented in [
13]. Ewing [
14] applied the finite difference method to tackle 2D Sobolev equations. The study [
15] used a computationally appealing and precise local meshless technique to provide numerical solutions for three-dimensional two- and three-term time-fractional PDE models. Various test problems were employed to assess the accuracy and reliability of the proposed approach. The paper by Luo et al. [
16] introduces a reduced-order extrapolated finite difference iterative scheme for 2D Sobolev equations, employing proper orthogonal decomposition to construct a reduced-order solution space model, thus reducing the computational costs. Numerical experiments validated the scheme’s efficiency and accuracy, indicating its potential for efficient Sobolev equation solving. An accurate and efficient local meshless technique is used in the article [
17] to explore numerical solutions for two-term time-fractional Sobolev models. The method approximates the solution on a uniform or scattered set of nodes, yielding sparse and well-conditioned coefficient matrices. Another paper by Luo et al. [
18] introduces a novel method that employs reduced-order techniques and extrapolation within a Crank–Nicolson finite volume framework for 2D Sobolev equations. It aims to improve the computational efficiency and accuracy, showcasing promising outcomes via numerical simulations. The paper by Li et al. [
19] introduces an extended mixed finite element approach for solving 2D Sobolev equations, integrating a novel formulation with stable discretization for accurate solutions and computational efficiency. Numerical experiments highlighted its effectiveness and superiority over existing methods, especially in maintaining stability and convergence properties. Heydari et al. [
20] introduced a novel method that employs orthonormal Bernoulli polynomials to solve distributed-order time-fractional 2D Sobolev equations, which demonstrated accuracy through four test problems. Gao et al. [
21] used the local discontinuous Galerkin finite element method to solve a particular class of 2D Sobolev equations, while another work [
22] applied the weak Galerkin finite element method, providing an error estimate. Abu et al. [
23] presented a method for solving time- and space-fractional Sobolev equations with Caputo fractional derivatives in n-dimensional space, utilizing the reproducing kernel Hilbert space method, which is particularly effective for Caputo class derivatives. A meshless RBFs technique for solving 2D time-fractional Sobolev equations was presented by Hussain et al. [
24]. This method uses a finite difference formula for time-fractional derivative approximation and RBFs for spatial operator approximation. Using a Liouville–Caputo derivative technique for the time derivative, Ahmad et al. [
25] suggested an efficient meshless method for estimating the numerical solution of 3D time-fractional Sobolev equations. Furthermore, Zhang et al. [
26] used a characteristic splitting mixed finite element approach to solve convection-dominated Sobolev equations.
In response to its extensive applications, researchers explored a multitude of numerical and analytical methodologies [
27,
28] for addressing complex PDEs. These methods encompass various techniques, such as the alternating direction implicit method [
29], the homotopy perturbation method [
30], the Laplace transform technique [
31], the variational approach [
32,
33], finite element methods (FEMs) [
34], finite difference methods (FDMs) [
16,
35], gradient descent iterative method [
36], spectral methods [
37], the exp-function method [
38], the alternating direction method [
39], and meshless methods [
40]. Notably, the straightforward nature of the FDM, FEM, and meshless techniques is noteworthy. Recently, hybrid methodologies have emerged to enhance the efficiency and accuracy of numerical solutions, including meshless methods, the method of lines utilizing Fibonacci polynomials, and combinations of the FDM and FEM.
This investigation focused on computing numerical solutions for the suggested model using a hybrid methodology based on the Caputo derivative. This approach integrates Fibonacci polynomials with the established Caputo derivative concept, exploiting the relationship between Fibonacci and Lucas polynomials. This integration provides a significant advantage in the straightforward implementation of higher-order derivatives. Moreover, the proposed approach reduces computational costs by enhancing the accuracy, even with a limited number of collocation sites. Remarkably, these polynomials find diverse real-world applications in the realm of differential equations.
Addressing boundary value problems accurately involves exploring the interplay between Chebyshev and Lucas polynomials, as demonstrated in previous studies [
41,
42]. For instance, the Lucas sequence was utilized to approximate integro-differential equations [
43], while Lucas polynomials were applied to solve higher-order differential equations [
44]. Additionally, the efficacy of a Fibonacci polynomial methodology in resolving Volterra–Fredholm integral differential equations was demonstrated [
45]. Furthermore, a hybrid Taylor–Lucas polynomial approach was introduced for addressing delay difference equations [
46]. Notably, novel methodologies for solving time-dependent PDEs were proposed by combining hybrid Fibonacci and Lucas polynomial schemes [
47,
48]. Moreover, researchers employed finite differences and Lucas polynomials to achieve effective numerical solutions for various PDE models [
49,
50].
Motivation
The primary objective of this research was to introduce a relatively new numerical approach designed to solve two-term time-fractional PDE models in both two and three dimensions. This method combines the Liouville–Caputo fractional derivative scheme with the Strang splitting algorithm for the temporal component and utilizes a meshless technique for spatial derivatives, incorporating Lucas and Fibonacci polynomials. Below are some of the highlighted key features of the proposed study:
The complex characteristics of fractional nonlinear PDEs make calculating analytical solutions challenging, driving ongoing research efforts to develop accurate and efficient numerical methodologies, with the two-term fractional order Sobolev model equation in both two and three dimensions holding significant importance across multiple scientific domains.
This study aimed to introduce an efficient numerical framework specifically designed for solving PDEs with temporal fractions.
The proposed methodology adopts a hybrid approach, integrating Fibonacci and Lucas polynomials with finite difference techniques, while also addressing the temporal direction through the utilization of the Liouville–Caputo fractional derivative in conjunction with a splitting mechanism.
Lucas and Fibonacci polynomials, unlike orthogonal counterparts, like Chebyshev polynomials, are non-orthogonal, eliminating the need for interval transformations. Additionally, they facilitate the straightforward approximation of higher-order derivatives for unknown functions.
Furthermore, the approach is characterized by its simplicity and ability to enhance the accuracy, even in scenarios involving fewer nodal points, with the aim to provide a robust and effective numerical solution to the intricate challenges posed by nonlinear PDEs.
This paper is organized as follows:
Section 2 provides an overview of the fundamental terms and concepts. The proposed technique for the underlying model equations is discussed in
Section 3, while theoretical results regarding stability and error analysis are presented in
Section 4.
Section 5 utilizes numerical experiments to validate the method’s efficacy, and finally,
Section 6 summarizes the outcomes and presents concluding remarks to finalize the work.