The approximate solutions of Fredholm–Volterra integro-differential equations of multi-fractional order within the Caputo sense (F-VIFDEs) under mixed conditions are presented in this article apply a collocation points technique based completely on Bessel polynomials of the first kind. This new approach depends particularly on transforming the linear equation and conditions into the matrix relations (some time symmetry matrix), which results in resolving a linear algebraic equation with unknown generalized Bessel coefficients. Numerical examples are given to show the technique’s validity and application, and comparisons are made with existing results by applying this process in order to express these solutions, most general programs are written in Python V.3.8.8 (2021).

The well-known central finite difference approximation was combined with the trapezoid quadrature method in this study to provide a numerical solution of the linear system of Volterra integro-fractional differential equations (LSVI-FDEs) of arbitrary orders, where the fractional derivative is described in the Caputo sense and the orders are between zero and one. The method works by first using the central finite difference approximation to approximate the Caputo derivative at any fixed point and then using the trapezoidal rule to obtain a finite difference expression for our fractional equation, while recalling the linear spline approximation for the first steps. This new, more efficient method involves converting sets of equations and conditions into matrix relationships, from which symmetry matrices can be created in some cases. We also present a new approach for error analysis of the discrete numerical scheme and the explicit numerical technique for LSVI-FDEs. The multi-level explicit finite difference approximation’s stability and convergence were explored, and a MatLab application was created to explain the results. Finally, several numerical examples are offered to demonstrate the technique's application.

In this article, the numerical solution of the mixed Volterra–Fredholm integro-differential equations of multi-fractional order less than or equal to one in the Caputo sense (V-FIFDEs) under the initial conditions is presented with powerful algorithms. The method is based upon the quadrature rule with the aid of finite difference approximation to Caputo derivative usage collocation points. For treatments, our technique converts the V-FIFDEs into algebraic equations with operational matrices, some of which have the symmetry property, which is simple for evaluating. Furthermore, numerical examples are presented to show the technique’s validity and usefulness as well comparisons with previous results. The majority of programs are performed using MATLAB v. 9.7.

In this paper a new method for solving the integral equation in space-time which arises in mathematical physics, mechanics and the heat conduction theory is presented. The central idea is: First, differentiating both sides of two-dimensions Fredholm integral Equations ntimes with respect to both variables x and t, second, substituting the Taylor series of two variables for the unknown function, and third, obtain a system of linear equations which can be solved by a suitable truncation scheme .Lastly,the double-quadrature rules are used to calculate the required integrals in this procedure. The presented algorithm is illustrated by some numerical examples with comparison tables.

This study uses for the first time a Laplace Adomian decomposition with a modified technique for solving linear system of Volterra integro-fractional delay differential equations. The fractional part is a Caputo type, and the delay part is a retarded delay time lag. This strategy is essentially based on an elegant combination of the Laplace transform strategy, the series expansion strategy, and the modified Adomian polynomial. This technique computes analytical work using multiple sorts of kernels, which have qualities such as difference kernels and simple degenerate kernels. This method is good for solving the problem like that and getting a good result. These are illustrated by assessing the efficiency and accuracy of the cause technique and solving specific examples.