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

Abstract

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.

Computational Algorithm for Approximating Fractional Derivatives of Functions

“…Therefore, we can observe that when ℎ (step size) is reduced by a factor of 1/2 the successive errors are diminished by approximately 1/4 this confirms the order is O(ℎ 2 ) and consistent with the error analysis presented above. This method is effective and consistent especially when compared with other methods and solvers [38]- [42].…”

“…Trapezoidal rule is an effective tool for approximation of derivatives and integral of arbitrary order particularly when combined with the finite difference scheme [38]. Engineers and scientist find it useful especially in dealing with problems that are either difficult or cannot be solved analytically, this approach is not only unique but limited in literature and efficient in practice especially when numerical solution is sought, [38], [42], [45]. .…”

“…Therefore, identifying not only obstacles, but potentials and also indicating a route for the future could have a big impact, as a result numerous strategies and tactics have been put forth [15], [20]- [37]. Recent study on this approach can be found, for example, in [38]- [43]. This article shares the author's perspective on the major, and rapidly developing topic of fractional calculus and propose and algorithm for easy computation of functions with the aid of finite difference scheme and the trapezoidal rule, with this method the problems are resolved due to the methods high adaptability and applications are made easier while maintaining efficiency.…”

On study the existence and uniqueness of the solution of the Caputo–Fabrizio coupled system of nonlocal fractional q‐integro differential equations

“…The authors of [19] used a technique known as the trapezoidal method to solve the Volterra integro-differential equations numerically. The numerical solution of a system of Volterra integro-differential equations of fractional order is obtained using a mix of the trapezoidal method and the finite difference method [20]. Now, we discuss the following coupled system:…”

On the analytical and numerical study for fractional q-integrodifferential equations

“…Garba and Bichi used the finite-difference-composite Simpson's method to find the numerical solution of the first-order Fredholm integrodifferential equation [11]. Pandey in [12], Saadati et al in [13] and Ahmed in [14] used the finite-difference-Trapezoidal method to find the solutions to various integrodifferential equations. Mittal and Jain in [15], Gholamian, Nadjaf in [16], Hamzah in [17], and Mirzaee, and Alipour in [18] used the cubic B-spline to solve the different types of integrodifferential equations.…”