arXiv subjects

Mahmoud A. Zaky

Publications and source records attributed to Mahmoud A. Zaky.

8 recordsLinked to original sources

arXiv2026

An optimal order fractional backward collocation method for adjoint Volterra integro-differential equations

This paper develops and analyzes an optimal-order fractional backward collocation method for adjoint Volterra integro-differential equations with weakly singular kernels. The backward Volterra structure, together with the weakly singular kernel, induces fractional-power singularities at the terminal endpoint, thereby reducing the classical regularity of the exact solution and causing order deterioration in standard polynomial collocation methods. We first establish a regularity result that characterizes the terminal singular behavior of the solution, showing that the solution is continuously differentiable, whereas its second derivative may exhibit a weak singularity at the terminal endpoint. Motivated by this regularity structure, we introduce a terminally graded mesh and construct a fractional backward collocation scheme whose local approximation space is adapted to the endpoint singularity. Rigorous convergence and superconvergence estimates are derived for both the solution and its derivative. With suitable choices of the fractional parameter and the mesh-grading exponent, the proposed method attains the optimal convergence orders dictated by the local approximation degree. Numerical experiments confirm the theoretical predictions and demonstrate the accuracy and effectiveness of the method for adjoint weakly singular Volterra integro-differential equations.

math.NA↗

arXiv2026

Backward log orthogonal functions and their approximation theory

We introduce a new class of backward logarithmic orthogonal functions and generalized backward logarithmic orthogonal functions, constructed by applying a terminal-endpoint logarithmic mapping to generalized Laguerre polynomials. These functions are designed for backward spectral approximations of problems whose solutions exhibit weak singularities at the terminal endpoint. The proposed basis functions generate non-polynomial weighted approximation spaces with nodes naturally clustered near the singular endpoint, and therefore provide an effective framework for resolving algebraic and logarithmic endpoint singularities. We develop the basic approximation theory for these backward logarithmic orthogonal functions, including recurrence relations, derivative formulas, orthogonality, Sturm--Liouville characterization, mapped Laguerre--Gauss quadrature rules, weighted projection estimates, backward Lagrange interpolation estimates, inverse inequalities, and stability properties in weighted Sobolev-type spaces defined through a terminal logarithmic pseudo-derivative. A generalized version of the basis is also introduced by incorporating an algebraic scaling parameter, which improves the flexibility of the approximation space and allows singular factors to be represented more effectively. The error analysis and numerical results show that the proposed backward logarithmic basis is particularly suitable for weakly regular functions with terminal-endpoint singularities and can recover exponential or high-order convergence rates that are typically lost when usual polynomial approximations are applied directly.

math.NA↗

arXiv2026

Fractional backward spectral approximation theory for weakly singular adjoint integral equations

We introduce a new class of fractional backward orthogonal functions designed for the spectral approximation of weakly singular adjoint Volterra integral equations. These basis functions generate an approximation space that naturally reflects the terminal-endpoint singular behaviour produced by weakly singular kernels. We develop the basic approximation theory for the proposed backward orthogonal basis, including weighted projection estimates, Gauss-type interpolation estimates, inverse inequalities, and stability bounds for the associated weakly singular adjoint integral operator. The error analysis and numerical results show that the proposed backward Jacobi method is particularly suitable for solutions with terminal-endpoint weak singularities and can recover high-order convergence rates that are typically lost when usual polynomial approximations are applied directly to such weakly regular solutions.

math.NA↗

arXiv2026

Generalized fractional Laguerre orthogonal functions: projection and interpolation estimates

Classical Laguerre spectral approximations are highly effective on the half-line when the target function is smooth in the usual polynomial scale. However, their accuracy deteriorates for nonsmooth functions. Such behavior appears naturally in fractional models, weakly singular integral equations, and semi-infinite-domain approximations with limited regularity near the origin. The main contribution of this work is the construction and analysis of a fractional Laguerre approximation framework tailored to nonsmooth functions on the half-line. We establish projection and interpolation error estimates in nonuniformly weighted Sobolev space. These estimates clarify how the fractional parameter adapts the approximation space to the regularity of nonsmooth functions and improves the resulting convergence behavior. We further introduce a generalized fractional Laguerre family with an additional algebraic parameter, which gives greater flexibility in controlling both the approximation space and the underlying weight. Numerical experiments confirm the theoretical estimates and demonstrate the advantage of the proposed functions over standard Laguerre-type approximations.

math.NA↗

arXiv2025

Reconstruction of a potential parameter in subdiffusion via a Kohn--Vogelius type functional: Theory and computation

This work considers the reconstruction of a space-dependent potential from boundary observations in subdiffusion by a stable and robust recovery method. Specifically, we develop an algorithm to minimize the Kohn-Vogelius cost function, which measures the difference between the solutions of two excitations. The inverse potential problem is recast into an optimization problem, where the objective is to minimize a Kohn-Vogelius-type functional within a set of admissible potentials. We establish the well-posedness of this optimization problem by proving the existence and uniqueness of a minimizer and demonstrating its stability with respect to perturbations in the boundary data. Furthermore, we analyze the Fréchet differentiability of the Kohn-Vogelius functional and prove the Lipschitz continuity of its gradient. These theoretical results enable the development of a convergent conjugate gradient algorithm for numerical reconstruction. The effectiveness and robustness of the proposed method are confirmed through several numerical examples in both one and two dimensions, including cases with noisy data.

math.NA↗

arXiv2024

A robust alternating direction numerical scheme in a shape optimization setting for solving geometric inverse problems

The alternating direction method of multipliers within a shape optimization framework is developed for solving geometric inverse problems, focusing on a cavity identification problem from the perspective of non-destructive testing and evaluation techniques. The rationale behind this method is to achieve more accurate detection of unknown inclusions with pronounced concavities, emphasizing the aspect of shape optimization. Several numerical results to illustrate the applicability and efficiency of the method are presented for various shape detection problems. These numerical experiments are conducted in both two- and three-dimensional settings, with a focus on cases involving noise-contaminated data. The main finding of the study is that the proposed method significantly outperforms conventional shape optimization methods in reconstructing unknown cavity shapes.

math.OC↗

arXiv2023

On the Rothe-Galerkin spectral discretisation for a class of variable fractional-order nonlinear wave equations

In this contribution, a wave equation with a time-dependent variable-order fractional damping term and a nonlinear source is considered. Avoiding the circumstances of expressing the nonlinear variable-order fractional wave equations via closed-form expressions in terms of special functions, we investigate the existence and uniqueness of this problem with Rothe's method. First, the weak formulation for the considered wave problem is proposed. Then, the uniqueness of a solution is established by employing Grönwall's lemma. The Rothe scheme's basic idea is to use Rothe functions to extend the solutions on single-time steps over the entire time frame. Inspired by that, we next introduce a uniform mesh time-discrete scheme based on a discrete convolution approximation in the backward sense. By applying some reasonable assumptions to the given data, we can predict a priori estimates for the time-discrete solution. Employing these estimates side by side with Rothe functions leads to proof of the solution's existence over the whole time interval. Finally, the full discretisation of the problem is introduced by invoking Galerkin spectral techniques in the spatial direction, and numerical examples are given.

math.NA↗

arXiv2022

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel

In this paper, a two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel is proposed to reduce the computation time and improve the accuracy of the scheme developed by Xu et al. (Applied Numerical Mathematics 152 (2020) 169-184). The proposed scheme consists of three steps: First, a small nonlinear system is solved on the coarse grid using fix-point iteration. Second, the Lagrange's linear interpolation formula is used to arrive at some auxiliary values for analysis of the fine grid. Finally, a linearized Crank-Nicolson finite difference system is solved on the fine grid. Moreover, the algorithm uses a central difference approximation for the spatial derivatives. In the time direction, the time derivative and integral term are approximated by Crank-Nicolson technique and product integral rule, respectively. With the help of the discrete energy method, the stability and space-time second-order convergence of the proposed approach are obtained in $L^2$-norm. Finally, the numerical results agree with the theoretical analysis and verify the effectiveness of the algorithm.

math.NA↗