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).

Background In this paper, we propose two new techniques called the Sawi transformations and Sequential approximations method, which are applied to solve multi-higher order linear fractional differential equations with constant coefficients. In which Riemann-Liouville and Caputo define the fractional derivatives and fractional integral, fractional formula for all types have been derived, we first developed the Sawi transform of foundational mathematical functions for this purpose and then described the important characteristics of the Sawi transform, which may be applied to solve ordinary differential equations and fractional differential equations. Following that, the authors found an exact solution to a particular example of fractional differential equations. Materials and Methods With these methods good exact and approximate solutions can be obtained with only a few iterations for the Sequential approximations method, and the approximate solutions guarantee the desired accuracy. For more validation of the methods, and fractional formula of sawi transformation method work such as other transformation. Results: The exact and approximation solutions for some fractional differential equations are obtained, and several examples are explained to demonstrate the efficiency and implementation of the proposed methods. Conclusions It is clear from reading the literature on fractional differential equations (FDEs) that the Sawi transformation method is a workable solution to these problems. To use this approach, which also completely resolved fractional differential equations, the FDEs must be simplified.

Background: This article presents a method for finding numerical solutions to Fredholm integro-differential equations (FIFDEs) with multi-fractional orders of one or less, using a useful algorithm. Materials and Methods: A finite difference approximation to Caputo's derivative using collocation points is used to build the midpoint method for the quadrature rule, which forms the basis of the approach. Results: Our method simplifies the evaluation of treatments by transforming the FIFDEs into algebraic equations with operational matrices. After calculating the Caputo derivative at a specific point using the finite difference method, we use the quadrature method, which includes the midpoint rule, to create a finite difference formula for our fractional equation. Conclusions: Additionally, numerical examples are provided to demonstrate the validity and use of the approach as well as comparisons with earlier findings. The results are expressed using a program created in MATLAB.

This paper describes a novel integral transform called the "Dukani Transform," which we can employ to precisely (analytically) solve the Volterra integral equations (VIE) of the first sort. To do this, first the Dukani transform of transcendental and elementary algebraic mathematical functions is derived by the authors. Next, the basic properties of the Dukani transform were discussed. Then, using a range of differential equations, including integral equations (IEs), ordinary differential equations (ODEs), and integro differential equations (IDEs), the exact (analytic) solution for first-class generic VIE was found by the authors. Numerical problems have been thoroughly examined and solved, in a step wise manner, demonstrating the applicability of the Dukani transform. The findings demonstrate that, without need for time-consuming calculations, the suggested "Dukani Transform" new integral transform produces accurate (VIEs.) of first-kind and second kind.

This paper introduces a new computational method for solving high fractional-order multi-term linear Fredholm–Volterra integro-differential equations (F-VIFDEs) with mixed boundary conditions. The approach utilizes the Caputo fractional derivative and shifted Lucas polynomial collocation technique. The core of the method involves transforming the F-VIFDE and its constraints into a system of algebraic equations by computing generalized Lucas coefficients. This matrix-based formulation provides an efficient framework for numerical solutions. The accuracy of the method is validated through numerical examples, which compare the approximate solutions with exact solutions. A least-squares error minimization approach across the domain further ensures precision. The presented method is computationally robust and reproducible, with implementations available for broader research use. The results demonstrate its effectiveness and potential as a reliable tool for theoretical and applied problems in fields such as physics, engineering, and mathematical modeling.