Designing a Matrix Collocation Method for Fractional Delay Integro-Differential Equations with Weakly Singular Kernels Based on Vieta–Fibonacci Polynomials

Khadijeh Sadri,Soheil Salahshour,Choonkil Park,Dumitru Baleanu,Kamyar Hosseini

doi:10.3390/fractalfract6010002

Abstract

In the present work, the numerical solution of fractional delay integro-differential equations (FDIDEs) with weakly singular kernels is addressed by designing a Vieta–Fibonacci collocation method. These equations play immense roles in scientific fields, such as astrophysics, economy, control, biology, and electro-dynamics. The emerged fractional derivative is in the Caputo sense. By resultant operational matrices related to the Vieta–Fibonacci polynomials (VFPs) for the first time accompanied by the collocation method, the problem taken into consideration is converted into a system of algebraic equations, the solving of which leads to an approximate solution to the main problem. The existence and uniqueness of the solution of this category of fractional delay singular integro-differential equations (FDSIDEs) are investigated and proved using Krasnoselskii’s fixed-point theorem. A new formula for extracting the VFPs and their derivatives is given, and the orthogonality of the derivatives of VFPs is easily proved via it. An error bound of the residual function is estimated in a Vieta–Fibonacci-weighted Sobolev space, which shows that by properly choosing the number of terms of the series solution, the approximation error tends to zero. Ultimately, the designed algorithm is examined on four FDIDEs, whose results display the simple implementation and accuracy of the proposed scheme, compared to ones obtained from previous methods. Furthermore, the orthogonality of the VFPs leads to having sparse operational matrices, which makes the execution of the presented method easy.

Highlights

The Vieta–Fibonacci polynomials (VFPs) are not widely taken into consideration as basis functions in the spectral methods. These reasons motivate the authors to employ the VFPs to proffer a numerical method with a smaller computational size to solve the fractional delay integro-differential equations (FDIDEs) with weakly singular kernels
Due to the need of the orthogonality of derivatives of the shifted VFPs in the subject of the approximation error, we show that the mth derivative of these polynomials is orthogonal with respect to the weight function wm (t) = tm+ 2 (1 − t)m+ 2, m = 1, 2
Operational matrices of the integration of fractional and integer orders, a pseudo-operational matrix for the integral parts with singular kernels, a delay operational matrix, and a product operational matrix were constructed, and the collocation method along with the obtained operational matrices were used for reducing the FDIDEs to an algebraic system

Summary

Introduction

Numerous works have been devoted to numerically solving fractional integro-differential equations with weakly singular kernels. These equations emerged in diverse fields of science, such as the heat conduction problem, radiative equilibrium, elasticity, and fracture mechanics [1,2,3,4]. Determining the analytic solutions of fractional integro-differential equations with weakly singular kernels is often complicated and even infeasible. Finding new numerical methods or developing existing methods to solve this class of equations is unavoidable. Different computing methods have been presented, for Fractal Fract.

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Conclusion