The ability of structured light to reconstruct after partial obstruction, known as the self-healing phenomenon, has attracted considerable attention for applications in imaging, optical communication, and quantum technologies. While Bessel and Airy beams have been extensively studied, the self-healing behavior of Hermite–Gauss (HG) beams remains less well quantified. In this work, we systematically investigate the self-healing properties of six HG modes (HG01, HG10, HG11, HG12, HG21, HG22) under three classes of obstructions: circular disks, vertical strips, and rectangular fringes. Both intensity- and amplitude-based similarity measures are employed within the normalized self-healing degree (SHD) framework, providing a rigorous and comparable metric of recovery efficiency. Numerical simulations reveal that higher-order modes, particularly HG22, demonstrate superior resilience, achieving SHD values above 1.1 even for moderate obstructions, whereas lower-order modes exhibit incomplete recovery. Geometry–mode alignment effects are also observed, with HG11 showing enhanced robustness under strip obstructions. Importantly, amplitude-based SHD is consistently lower than intensity-based SHD, confirming that apparent intensity recovery can overestimate robustness when phase fidelity is critical. These findings establish a comprehensive quantitative picture of HG self-healing and offer practical insights for designing resilient, structured beams in realistic optical environments.

In this paper, a mixed high order finite difference scheme-Padé approximation method is applied to obtain numerical solution of the Riesz fractional advection-dispersion equation. This method is based on the high order finite difference scheme that derived from fractional centered difference and Padé approximation method for space and time integration, respectively. The stability analysis of the proposed method is discussed via theoretical matrix analysis. Numerical experiments are carried out to confirm the theoretical results of the proposed method.

A new operational matrix based approach is studied for numerical solution of 2D-integro-differential equations with non-local (integral) boundary conditions whose arise in some physical problems. Some important theoretical results are presented to reduce complexity and computational costs of the proposed method. We also give an error estimation which will be useful in estimating the error of approximate solution for the problems that we do not have any information about their exact solution. Illustrative numerical examples are also given to clarify the performance and accuracy of the new method.

In this paper, a Laguerre collocation method is presented in order to obtain numerical solutions for linear and nonlinear Lane-Emden type equations and their initial conditions. The basis of the present method is operational matrices with respect to modified generalized Laguerre polynomials(MGLPs) that transforms the solution of main equation and its initial conditions to the solution of a matrix equation corresponding to the system of algebraic equations with the unknown Laguerre coefficients. By solving this system, coefficients of approximate solution of the main problem will be determined. Implementation of the method is easy and has more accurate results in comparison with results of other methods.

In this paper, by combining of fractional centered difference approach with alternating direction implicit method, we introduce a mixed difference method for solving two-dimensional Riesz space fractional advection-dispersion equation. The proposed method is a fourth order centered difference operator in spatial directions and second order Crank-Nicolson method in temporal direction. By reviewing the consistency and stability of the method, the convergence of the proposed method is achieved. Several numerical examples are considered aiming to demonstrate the validity and applicability of the proposed technique.

In this paper, a mixed matrix transform method with fractional centered difference scheme for solving fractional diffusion equation with Riesz fractional derivative was examined. It was obtained that the numerical scheme was unconditionally stable and feasible using the matrix analysis method. Numerical experiments were, then, carried out to support the theoretical predictions.

In this paper, a compact alternating direction implicit (ADI) method has been developed for solving two-dimensional Riesz space fractional diffusion equation. The precision of the discretization method used in spatial directions is twice the order of the corresponding fractional derivatives. It is proved that the proposed method is unconditionally stable via the matrix analysis method and the maximum error in achieving convergence is discussed. Numerical example is considered aiming to demonstrate the validity and applicability of the proposed technique.

Recently, there has been significant development in the existence of mild solutions for fractional semilinear integro-differential equations but optimal control is not provided. The aim of this paper is studying optimal feedback control for fractional semilinear integro-differential equations in an arbitrary Banach space associated with operators generating compact semigroup on the Banach space.

Natural frequencies and free vibration are important characteristics of beams with non-uniform cross section. Hence, the solution for free vibrations of non-uniform beams is presented using a Laguerre collocation method. The elastically restrained beam model is based on the Euler–Bernoulli theory. Also, the non-uniform beam is rested on a non-uniform foundation (Winkler type). The Laguerre collocation method is introduced for solving the differential equation. This approach reduces the governing differential equation to a system of algebraic equations, and finally, the problem is greatly simplified. Properties of Laguerre polynomials and the operational matrix of derivation are first presented. Eventually, the proposed method is applied for solving the governing differential equation subject to initial conditions, and the results are compared with other results from the literature.

The superposition of the Bessel and Mirrored Bessel beams, and their self-healing characteristic, using the defined similarity function have been investigated, quantitatively. We use the Huygens convolution method to propagate these beams to study their self-healing behavior. Numerical simulations and observational results show that the self-healing property of the superposed beams is better than the self-healing property of the Bessel beams.

We use a matrix formulated algorithm to approximate solutions of a class of nonlinear reaction-diffusion equations with nonlocal boundary conditions. Some theoretical results are presented to simplify application of operational matrix formulation and reduce the computational cost. Convergence analysis and error estimation of the method are also investigated. Finally, some numerical examples are given to demonstrate accuracy and efficiency of the proposed method.

In this paper, a numerical method is developed for solving linear and nonlinear integro-partial differential equations in terms of the two variables Jacobi polynomials. First, some properties of these polynomials and several theorems are presented then a generalized approach implementing a collocation method in combination with two dimensional operational matrices of Jacobi polynomials is introduced to approximate the solution of some integro-partial differential equations with initial or boundary conditions. Also, it is shown that the resulted approximate solution is the best approximation for the considered problem. The main advantage is to derive the Jacobi operational matrices of integration and product to achieve the best approximation of the two dimensional integro-differential equations. Numerical results are given to confirm the reliability of the proposed method for solving these equations.

This article introduces the matrix differential transform method (MDTM) to apply to matrix partial differential equations (MPDEs) and employs it for solving matrix Fisher equations, matrix Burgers equations and matrix KdV equations. We show how the MDTM applies to the linear part and nonlinear part of any MPDE and give various examples of MPDEs to illustrate the efficiency of the method. The results obtained are in excellent agreement with the exact solution and show that the proposed method is powerful, accurate, and easy.

Anomalous diffusion and non-exponential relaxation patterns can be described by a space - time fractional diffusion equation. This paper aims to present a Pade approximation for Mittag-Leffler function mixed finite difference method to develop a numerical method to obtain an approximate solution for the space and time fractional diffusion equation. The truncation error of the method is theoretically analyzed. It is proved that the numerical proposed method is unconditionally stable from the matrix analysis point of view. Finally, some numerical results are given, which demonstrate the efficiency of the approximate scheme.

This paper aims to construct a general formulation for the shifted Jacobi operational matrices of integration and product. The main aim is to generalize the Jacobi integral and product operational matrices to the solving system of Fredholm and Volterra equations. These matrices together with the collocation method are applied to reduce the solution of these problems to the solution of a system of algebraic equations. The method is applied to solve system of linear and nonlinear Fredholm and Volterra equations. Illustrative examples are included to demonstrate the validity and efficiency of the presented method. Also, several theorems, which are related to the convergence of the proposed method, will be presented.

In this paper, we implement the shifted Jacobi operational matrix of derivative with spectral tau method and collocation method for numerical solution for the systems of linear and non-linear ordinary differential equations subject to initial or boundary conditions. By means of this approach, such problems are reduced for solving a system of algebraic equations and are greatly simplified the problems. We compare the obtained numerical results with the exact solutions. Also, we present and prove several theorems, which are related to the convergence of the proposed methods. Finally, some numerical test examples are presented to illustrate the validity and the great potential of the proposed technique.

In this paper, a class of linear and nonlinear functional integro-differential equations are considered that can be found in the various fields of sciences such as: stress–strain states of materials, motion of rigid bodies and models of polymer crystallization. The operational collocation method with shifted Jacobi polynomial bases is applied to approximate the solution of these equations. In addition, some theoretical results are given to simplify and reduce the computational costs. Finally, some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

In this paper, the numerical solution of two dimensional Fredholm and Volte rra integral equations will be investigated. For this order, two dimensional collocation method is applied to solve system o f two dimensional linear and nonlinear Fredholm and Volterra integral equations. Using the Jacobi polynomials, two dimensiona l integral equations reduce to a system of algebraic equations. The main aim is the developing the Jacobi operational matrices of integration and product for the solving system of two dimensional Fredholm and Volterra integral equations. These matrices together with the collocation method are applied to reduce the solution of these problems to the solution of a system of algebraic equations. The nume rical examples illustrate the efficiency and accuracy of this method.

In this paper, non-linear parabolic-hyperbolic partial differential equations are solved by means of Variational iteration method (VIM) and the results are compared with those of Homotopy perturbation method (HPM), and lesser computations will be expected. Parabolic-hyperbolic partial differential equations appear in mathematical modeling of many phenomena. Also, the convergence of the method is addressed briefly. For this purpose, Banach's fixed point theorem has been used. Some examples are presented to illustrate the ability and the simplicity of the method. The results reveal that this method is very effective and simple and can be applied for other problems in different fields of mathematics.

In this study a fractional Poisson equation is scrutinized through finite difference using the shifted Grunwald estimate. A novel method is proposed numerically. The existence and uniqueness of the solution for the fractional Poisson equation is proved. Exact and numerical solutions are constructed and compared. Then numerical results show the efficiency of the proposed method.