This work investigates the inverse problem of identifying a time-dependent source term in a time-fractional semi-linear degenerate parabolic equation using integral measurement data. We establish the unique solvability of the inverse problem within a suitable functional framework. The proof methodology is based on the Rothe method, where the variational formulation is discretized in time, and a priori estimates for discrete solutions are derived. These estimates are then utilized to demonstrate the convergence of Rothe approximations to a unique weak solution. Additionally, we develop a numerical scheme based on the $L^1$-Galerkin finite element method, combined with iterative refinement, to reconstruct the unknown source term. The numerical performance of the proposed method is validated through a series of computational experiments, demonstrating its stability and robustness against noisy data.

An efficient Legendre-Galerkin spectral method and its error analysis for a one-dimensional parabolic equation with Dirichlet-type non-local boundary conditions are presented in this paper. The spatial discretization is based on Galerkin formulation and the Legendre orthogonal polynomials, while the time derivative is discretized by using the symmetric Euler finite difference schema. The stability and convergence of the semi-discrete spectral approximation are rigorously set up by following a novel approach to overcome difficulties caused by the non-locality of the boundary condition. Several numerical tests are included to confirm the efficacy of the proposed method and to support the theoretical results.

Many physical phenomena can be modelled through nonlocal boundary value problems whose boundary conditions involve integral terms. In this work we propose a numerical algorithm, by combining second-order Crank–Nicolson schema for the temporal discretization and Legendre–Chebyshev pseudo-spectral method (LC–PSM) for the space discretization, to solve a class of parabolic integrodifferential equations subject to nonlocal boundary conditions. The approach proposed in this paper is based on Galerkin formulation and Legendre polynomials. Results on stability and convergence are established. Numerical tests are presented to support theoretical results and to demonstrate the accuracy and effectiveness of the proposed method

A semilinear pseudoparabolic equation with nonlocal integral boundary conditions is studied in the present paper. Using Rothe method, which is based on backward Euler finitedifference schema, we designed a suitable semidiscretization in time to approximate the original problem by a sequence of standard elliptic problems. The questions of convergence of the approximation scheme as well as the existence and uniqueness of the solution are investigated. Moreover, the Legendre pseudospectral method is employed to discretize the time-discrete approximation scheme in the space direction. The main advantage of the proposed approach lies in the fact that the full-discretization schema leads to a symmetric linear algebraic system, which may be useful for theoretical and practical reasons. Finally, numerical experiments are included to illustrate the effectiveness and robustness of the presented algorithm.

The present study is devoted to investigating the inverse problem of simultaneously reconstructing two source terms that depend solely on time in a system of coupled fractional reaction–diffusion equations. Such coupled systems are fundamental for modelling multispecies anomalous diffusion processes, where the evolution of each state variable is governed by unknown time-varying sources. The inverse problem is tackled using supplementary measurements of the state variables over the spatial domain. The coupling between the two equations presents a distinct complexity, making the simultaneous reconstruction problem notably challenging and important. Results on the unique solvability, are established by employing the Rothe method, whereby the problem is first discretized in time and then stability results for the semi-discrete approximations are derived. These estimates, together with compactness arguments, are then used to rigorously prove the convergence of Rothe approximations to a unique weak solution. Finally, the proposed method's effectiveness and stability are further validated through a series of numerical simulations.

In the present work, we aim to study the inverse problem of recovering an unknown spatial source term in a multi-term time-fractional diffusion equation involving the fractional Laplacian. The forward problem is first analyzed in appropriate fractional Sobolev spaces, establishing the existence, uniqueness, and regularity of solutions. Exploiting the spectral representation of the solution and properties of multinomial Mittag–Leffler functions, we prove uniqueness and derive a stability estimate for the spatial source term from finaltime observations. The inverse problem is then formulated as a Tikhonov regularized optimization problem, for which existence, uniqueness, and strong convergence of the regularized minimizer are rigorously established. On the computational side, we propose an efficient reconstruction algorithm based on the conjugate gradient method, with temporal discretization via an L¹-type scheme for Caputo derivatives and spatial discretization using a Galerkin approach adapted to the nonlocal fractional Laplacian. Numerical experiments confirm the accuracy and robustness of the proposed method in reconstructing the unknown source term.