Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels

• Published in: Symmetry (Basel) 2020-12, Vol.12 (12), p.2105
• Main Authors: Hashemizadeh, Elham, Ebadi, Mohammad Ali, Noeiaghdam, Samad
• Format: Article
• Language: English
• Subjects: Abel’s integral equations; Approximation; Fluid dynamics; Integral equations; Kernels; Mathematical analysis; Nonlinear equations; nonlinear Volterra integral equation; Numerical analysis; operational matrix; Ordinary differential equations; Partial differential equations; Polynomials; Spectral methods; Symmetry; the Genocchi polynomials; Volterra integral equations; weakly singular kernels
• ISSN: 2073-8994; 2073-8994
• DOI: 10.3390/sym12122105

In this study, we present a spectral method for solving nonlinear Volterra integral equations with weakly singular kernels based on the Genocchi polynomials. Many other interesting results concerning nonlinear equations with discontinuous symmetric kernels with application of group symmetry have remained beyond this paper. In the proposed approach, relying on the useful properties of Genocchi polynomials, we produce an operational matrix and a related coefficient matrix to convert nonlinear Volterra integral equations with weakly singular kernels into a system of algebraic equations. This method is very fast and gives high-precision answers with good accuracy in a low number of repetitions compared to other methods that are available. The error boundaries for this method are also presented. Some illustrative examples are provided to demonstrate the capability of the proposed method. Also, the results derived from the new method are compared to Euler’s method to show the superiority of the proposed method.