Bessel Collocation Method for Solving Fredholm–Volterra Integro-Fractional Differential Equations of Multi-High Order in the Caputo Sense

Abstract

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).

Solving a System of Fractional-Order Volterra Integro-Differential Equations Based on the Explicit Finite Difference Approximation via the Trapezoid Method with Error Analysis

“…. , N. From Equations ( 9) and (10), and Equations ( 18) and (19), respectively, where n = 3 and m = 2 with σ 13 = 0.9, σ 12 = 0.6, σ 11 = 0.3, σ 10 = 0, σ 23 = 0.8, σ 22 = 0.5, σ 21 = 0.4,…”

“…. , N. From Equations ( 9), ( 10), (18), and (19), respectively, where n = 2 and m = 2 with σ 12 = 0.6, σ 11 = 0.2, σ 10 = 0,…”

“…From Equations ( 9), ( 10), (18), and (19), respectively, where n = 1 and m = 3 with σ 11 = 0.9, σ 10 = 0, σ 21 = 0.7, σ 20 = 0, σ 31 = 0. We can solve it by any numerical way, obtaining the L 1 = [1.100025247 1.001854479 0.799941862].…”

“…. Furthermore, Y r is defined in Equation (12), while H σ i n and A σ i n ( ) are defined in Equations ( 9) and (10), respectively, and L H σ i n,1 and L A σ i n,1 ( ) are defined in Equations ( 18) and (19), respectively. Finally, here K is the kernel matrix at any point (t r , s ) in the basic domain.…”

Numerical Computation of Mixed Volterra–Fredholm Integro-Fractional Differential Equations by Using Newton-Cotes Methods

“…The approximate methods have gained importance to prevent this difficulty. Many methods evaluate the approximate solution of integro-fractional differential equations; see references [9][10][11][12][13][14][15][16][17][18][19]. In [11], the author utilized modified Navot-Simpson's quadrature for solving second-kind Volterra integral equations of singular type.…”