This study developed a finite difference method (FDM) for a time-fractional inverse problem associated with Schrödinger partial differential equations. The main objective of the inverse problem is the simultaneous identification of the unknown source function $p(x)$ and the state variable $w(t,x)$. The mathematical model involves the Caputo fractional order derivative (CFOD) of order $0<\alpha \le 1$ and incorporates a spatially variable diffusion coefficient $a(x)$, which significantly increases the complexity of the problem compared with constant-coefficient models. Homogeneous Dirichlet boundary value conditions (DBVCs) were imposed on the spatial domain. For the numerical discretization, the time-CFOD was approximated using a consistent L1-type scheme, while the spatial derivatives were discretized by second-order central finite difference schemes (FDSs). Stability estimates and convergence properties of the proposed numerical scheme are rigorously established using discrete energy techniques. The analysis shows that the method achieves a convergence order of $O({\tau }^{2-\alpha }+h^2)$. To validate the theoretical results, numerical experiments were performed for two benchmark problems with diffusion coefficients $a(x)=x^2+1$ and $a(x)=x^3+1$. The obtained numerical results confirm the effectiveness and robustness of the proposed approach. Graphical comparisons illustrate the behavior of solutions for different fractional orders as time evolves, while error tables demonstrate that the fractional-order solutions provide more accurate approximations to the exact solution than the corresponding integer-order case.

This paper presents the inverse problem (IP) for the fractional order two-dimensional parabolic diffusion equation (FOTDPDE) formulated to depend on a initial-boundary value problem (IBVP) with homogeneous Dirichlet boundary conditions (DBC). The model involves a fractional-order Caputo derivative (FOCD) and an inverse time-dependent source term. A Crank-Nicholson finite difference scheme (CN-FDS) is constructed, and stability inequalities and a theorem in the discrete $L^2$ norm are proved to ensure unconditional stability of the proposed scheme. Results calculated by using finite difference methods (FDM) have a temporal convergence rate of $O({\tau }^{2-\alpha })$ and second-order spatial accuracy. Numerical examples are tested to confirm the theoretical stability results and to represent the effectiveness and the accuracy of the method for solving IP for FOTDPDE depending on BVP.

In this paper, we considered an inverse problem of recovering the time dependent potential and force coefficients in the third order pseudoparabolic equation from nonlocal integral observations. Existence and uniqueness of the solution are proved by means of the contraction principle on a small time interval. The stability results for the inverse problem is presented. The unique solvability theorem for this inverse problem is proved. However, since the governing equation is yet ill-posed (very slight errors in the integral input may cause relatively significant errors in the output potential and heat source terms), we need to regularize the solution. Therefore, to get a stable solution, a regularized cost function is to be minimized for retrieval of the unknown terms. The third order pseudoparabolic problem is discretized using the cubic B-spline (CB-spline) collocation technique and reshaped as nonlinear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine $lsqnonlin$ tool. Both analytical and perturbed data are inverted. Numerical outcomes for two benchmark test examples are reported and discussed. In addition, the von Neumann stability analysis for the proposed numerical approach has also been discussed.

In this study, we investigate the numerical solutions of the time-fractional Fisher equation by employing the collocation finite element method (FEM). The fractional derivative is considered in the Caputo sense, which provides a suitable framework for modeling memory-dependent diffusion-reaction processes. The temporal discretization is carried out using the $L1$ algorithm, while the spatial discretization is carried out using the unified hyperbolic polynomial B-spline basis within a collocation finite element framework. Two test problems are presented to demonstrate the efficiency of the proposed method, and the obtained numerical results are compared with available exact and reference solutions. The results confirm that the collocation finite element approach is a reliable and effective technique for solving fractional partial differential equations (FPDEs) arising in nonlinear diffusion-reaction models.

This work introduces an integrated numerical framework for obtaining numerical approximations to the extended Fisher-Kolmogorov (EFK) equation model. The methodology entails a two-stage process: initial linearization of the governing equation through a Taylor series approach, followed by the application of a spectral collocation method utilizing Bell polynomials to resolve the resultant linear system. Comprehensive theoretical considerations, including a detailed error estimate in the weighted $L^2$-norm, are established. The numerical investigation, comprising three illustrative test cases, confirms the scheme's computational efficiency and demonstrates a marked improvement in accuracy when compared to current state-of-the-art results and exact solutions.

This paper is devoted to investigating the well-posedness, as well as performing the numerical analysis, of an inverse source problem for linear pseudoparabolic equations with a memory term. The investigated inverse problem involves determining a right-hand side that depends on the spatial variable under the given observation at a final time along with the solution function. Under suitable assumptions on the problem data, the existence, uniqueness and stability of a strong generalized solution of the studied inverse problem are obtained. In addition, the pseudoparabolic problem is discretized using extended cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this problem is effectively solved using the MATLAB subroutine lsqnonlin. Both exact and noisy data are inverted. Numerical results for a benchmark test example are presented and discussed. Moreover, the von Neumann stability analysis is also discussed.

Two inverse source problems for a space-time fractional differential equation involving bi-fractional Laplacian operators in the spatial variable and Caputo time-fractional derivatives of different orders between 1 and 2 are studied. In the first inverse source problem, the space-dependent term along with the diffusion concentration is recovered, while in the second inverse source problem, the time-dependent term along with the diffusion concentration is identified. Both inverse source problems are ill-posed in the sense of Hadamard. The existence and uniqueness of solutions for both inverse source problems are investigated. Finally, several examples are presented to illustrate the obtained results for the inverse source problems.