Explicit scheme for solving variable-order time-fractional initial boundary value problems

The creation of an explicit finite difference scheme with the express purpose of resolving initial boundary value issues with linear and semi-linear variable-order temporal fractional properties is presented in this study. The rationale behind the utilization of the Caputo derivative in this scheme stems from its known importance in fractional calculus, an area of study that has attracted significant interest in the mathematical sciences and physics. Because of its special capacity to accurately represent physical memory and inheritance, the Caputo derivative is a relevant and appropriate option for representing the fractional features present in the issues this study attempts to address. Moreover, a detailed Fourier analysis of the explicit finite difference scheme’s stability is shown, demonstrating its conditional stability. Finally, certain numerical example solutions are reviewed and MATLAB-based graphic presentations are made.


Methodology
In this section, we discuss the explicit finite difference scheme for solving linear or semi-linear variable-order time fractional differential equations.In the end of this section, we discuss the stability of the proposed scheme.

Explicit finite difference scheme
Consider the variable-order time fractional semi-linear differential equation where or The function f (ψ) is nonlinear.Without f (ψ) function, the Eq. ( 1) becomes linear.

Discretization
Let [0, L x ] be the domain of interest, first, we discretize this domain.To do this, let us define x i = ih , where 0 ≤ i ≤ M , Mh = L x , t j = jk , 0 ≤ j ≤ N , Nk = T , where h is the space step length and k is time step size.Sup- pose that ψ j i be the numerical approximation of ψ(x i , t j ) and f j i (ψ j i ) = f (x i , t j , ψ j i ) .Further, suppose that the non-linear function f j i (ψ j i ) satisfies the Lipschitz condition.

Scientific
The comparison of the proposed method with previous techniques is given in the Table 1 below.
The following section evaluates the stability of the discrete equation scheme (5, 6, 7).

Stability analysis
The most common technique for stability analysis of the explicit finite difference scheme is the Von Neumann stability analysis, which involves linearizing the problem around a steady state and studying the eigenvalues of the resulting linear system 43 .It is also possible to use other stability analysis techniques such as Fourier analysis 44 , Lax-Richtmyer 45 , and Von Neumann Courant-Friedrichs-Lewy (CFL) conditions 46 .
To evaluate the stability of the scheme, we consider ρ , where ϕ j i is the accurate solution at (x i , t j ) and apply the Fourier method.The discrete function ρ j (x * i ) is formulated as: In Fourier series, the function ( 8) can be expanded where Properties of the coefficient It is easy to prove property (2).

Stability of the scheme
The stability of the scheme is analyzed in this section.By inserting (5, 6)   Evaluating sum for l = j , we get Simplification yields us (8) Table 1.Proposed method comparison with the previous methods.

Strategies Methodology Benefits and shortcomings
Alia et al. 41 proposed a new group of iterative techniques to address the numerical solution of a two-dimensional subdiffusion equation that involves fractional derivatives and specific boundary conditions These iterative schemes are designed to provide a robust and efficient means of solving two-dimensional subdiffusion equations This method is computationally efficient, but no stability analysis has been performed Oderinu et al. 42 proposed a method that focuses on finding approximate solutions to linear time fractional differential equations under specific boundary conditions It investigates a numerical approach for the solution of linear time fractional differential equations of the Caputo type.The results of the research culminated in the establishment of a theorem that showcases the Kamal transform of the nth-order Caputo derivatives The proposed numerical scheme provides highly accurate solutions for linear time fractional differential equations.However, no stability analysis of the scheme has been performed.This method is limited in scope and can only be applied to linear time fractional differential equations

Proposed
The proposed method uses the central finite difference method for approximating the second-order spatial derivative and forward difference for approximating the Caputo derivative of time.This combination of techniques allows for an efficient and accurate numerical approximation of the solutions to linear/semi-linear time fractional differential equations This numerical scheme is versatile and can be applied to both linear and semi-linear equations, providing a flexible solution for a range of problems.The stability of the method has been rigorously verified, ensuring reliable results for a wide range of parameters.Additionally, the method is not limited by specific boundary conditions, making it suitable for a wide range of applications Vol:.( 1234567890) where Let solution at grid points be of the form Substituting (13) in Eq. ( 12) Simplifying and re-arranging the terms Using identity e ix = cosx + isinx , and re-arranging the terms or ( 10) www.nature.com/scientificreports/ The following lemma provides a framework for evaluating the stability of the scheme.
Lemma Suppose that ξ j be the solution of ( 14) and ∀(i, j), r .Taking modulus on both sides where For j > 0 , the Eq. ( 14) can be written as Let us now assume that the given result holds for j and prove it for j + 1 , i.e., it holds |ξ j | ≤ C|ξ 0 | and we are to show that |ξ j+1 | ≤ C * |ξ 0 | .Taking modulus on both sides of (15), i.e., We know that, |ξ j | ≤ C * |ξ 0 | for all k > 1.
Proof To prove that the explicit finite difference schemes given in Eqs.(5, 6) to (7) are stable under the condition r j i ≤ 1 2 sin 2 h 2 for all (i, j), where i = 1, 2, . . ., M and j = 0, 1, . . ., N , we can proceed as follows: Using Eq. ( 10) and the given lemma, we have: To establish stability, we need to show that |ρ j | 2 remains bounded for all j = 1, 2, . . ., N , given the stability condition on the coefficients r j i .First, let's define E j = |ρ j | 2 for convenience.Our goal is to show that E j is bounded for all j.Using Eq. ( 7) and the definition of | • |2 norm, we can rewrite E j as: Now, we can analyze the behavior of E j in terms of E j−1 .By substituting the expression for E j and E j−1 into equation ( 56), we have: is a valid norm, we have the triangle inequality: Therefore, we can bound E j as follows: Thus, we have shown that E j is bounded for all j = 1, 2, . . ., N.
Therefore, the explicit finite difference schemes (56) to (7) are stable under the given condition.This proves the stability of the scheme.

Numerical experiments
In this section, we present a numerical solution for variable-order time fractional linear initial boundary value problems using an explicit finite difference scheme.We investigate the influence of the fractional order γ by solv- ing problems for different values of γ ranging from 0 to 1.To perform the numerical calculation, we discretize the spatial domain into N = 10 equal intervals, each with a step size of h.The final solution is obtained at a specified final time T and stored in a matrix for each γ value.Finally, to visualize the results, we plot the solution against the spatial variable x, with each line representing the solution for a different fractional order value.
All the tests are performed on a Windows 10 Pro operating system using Matlab version R2016b on a computer equipped with an Intel(R) Core(TM) i5-7200U CPU running at 2.5 GHz and with 8GB of RAM.

Solution:
To obtain the discrete form of Eq. ( 16), the time fractional approximation (3) must be used for the time derivative and the central difference approximation for the space derivative.
The numerical solution for different values of γ at the final time T is depicted in 3D Fig. 1 and also for 2D Fig. 2.

Example II
Subject to the conditions:

Solution:
Using time fractional approximation (3) the ( 17) can be written as Re-arranging the terms . The numerical solution for different values of γ at the final time T is shown in 3D Fig. 3 and 2D Fig. 4.

Example III
Subject to the conditions:

Conclusion and future work
This paper presents a novel explicit finite difference scheme specifically designed for solving initial boundary value problems with linear and semi-linear variable-order time fractional characteristics.The choice of employing the Caputo derivative in this scheme is motivated by its well-established significance in fractional calculus, enabling effective modeling of physical memory and inheritance.The thorough stability analysis using the Fourier method confirms the conditional stability of the proposed scheme.Numerical examples demonstrate the efficacy of the method, with graphical representations using MATLAB showcasing solution curves for different fractional orders.Moving forward, future research directions could include extending the scheme to more complex nonlinear problems, investigating adaptive mesh refinement techniques, exploring the application of the method to other scientific and engineering domains, and considering parallel computing techniques to enhance computational efficiency.This work contributes to the advancement of fractional partial differential equations solving methods and provides a foundation for further exploration and refinement of this approach.

Figure
Figure 4. 2D Solution curves for Example II.
4. 2D Solution curves for Example II.paper, we explore three examples of initial boundary value problems involving fractional partial differential equations.In Example I, we consider a problem governed by a time-fractional partial differential equation with a variable-order fractional derivative.The equation exhibits a linear and semi-linear variable-order time fractional characteristic.The Caputo derivative is employed to model physical memory and inheritance, and the problem is solved subject to specified initial and boundary conditions.Example II presents a different equation, also with a variable-order fractional derivative, but involving a different spatial derivative term.The equation includes a term that accounts for the singular behavior near the origin (at x=0) and requires additional boundary conditions at the right boundary (at x=1).Finally, in Example III, we explore a problem described by a time-fractional partial differential equation with a variable-order fractional derivative and a spatial derivative term.This example includes a non-homogeneous initial condition and satisfies zero-flux boundary conditions.By considering these three examples, we demonstrate the versatility and applicability of the explicit finite difference scheme in solving initial boundary value problems with various fractional characteristics, showcasing the effectiveness of the proposed approach.