Maha M. Hamood

This paper presents a comprehensive numerical study on the efficacy of a Hermite polynomial-based least-squares method for solving Volterra–Fredholm fractional integro-differential equations (V-FFIDEs). In our approach, we construct an approximate solution as a finite expansion of Hermite polynomials. This trial solution is systematically substituted into the governing V-FFIDE. Following the analytical evaluation of the fractional and integral operators, we formulate a residual function. The core of our method involves minimizing the squared norm of this residual over the problem domain, a process that transforms the original problem into a well-defined system of linear algebraic equations. To validate our methodology, we conducted a series of numerical experiments on a collection of representative examples. The results of our study, presented through detailed tables of numerical outcomes and comparative graphical illustrations, conclusively demonstrate the high accuracy, computational efficiency, and robust convergence of the proposed technique.

This paper establishes the existence and uniqueness of solutions for a class of implicit fractional Volterra–Fredholm integrodifferential equations with finite delay and instantaneous impulses. By employing the Banach contraction principle and Schaefer’s fixed‐point theorem, we provide a rigorous analytical foundation for our results. Furthermore, through illustrative examples and numerical simulations, we validate our theoretical findings, which provide a deeper understanding of the problem’s behavior for researchers. Our work not only advances the theoretical framework of impulsive fractional dynamic systems but also provides practical tools for further exploration in applied mathematics and related fields.

In this study, we explore the solutions of fractional differential equations (FDEs) involving the Atangana–Baleanu–Caputo derivative, subject to both integral and impulsive implicit boundary conditions. To establish the existence and uniqueness of solutions, we utilize the Banach Contraction Mapping Principle and Krasnoselskiis fixed point theorem. The Banach principle helps demonstrate the uniqueness by showing the mapping is a contraction under suitable conditions, while Krasnoselskiis theorem is applied to confirm the existence of solutions without requiring strict contraction. An illustrative example is provided to support the theoretical results, demonstrating how the applied methods effectively guarantee the solvability of the given fractional boundary value problem. The example also highlights the practical relevance of the theoretical framework developed. This approach contributes to the broader understanding of FDEs, especially those modeled with nonlocal and impulsive implicit conditions, which are increasingly important in various scientific and engineering applications.

The purpose of this paper is to investigate, the existence and uniqueness of solutions for nonlinear fractional impulsive fractional Volterra–Fredholm integro-differential equations of the Caputo–Hadamard type, with a new modeling integral boundary value problem. The Krasnoselskii fixed-point theorem, Schaefer’s fixed point theorem, and the Banach contraction principle serve as the basis of this unique strategy and are used to achieve the desired results. An example illustrates the theoretical findings.

This research will integrate the Laplace transform method with the Adomian Decomposition Method to semi‐analytically treat nonlinear integro‐fractional differential equations of the Volterra–Fredholm–Hammerstein type. The higher‐order fractional derivative will be expressed in the Caputo sense, and the first‐order simple degenerate and the difference kernel will be used. With this approach, the inverse Laplace transform is applied, and the solution of the equation is viewed as the sum of an endless series of components that usually converge to the solution. Numerical applications frequently employ a shortened number of terms when a closed‐form solution is not possible. Lastly, a diagram displaying the arrived at and discussed solutions was shown along with illustrative examples.