An effective scheme for solving system of fractional Volterra–Fredholm integro-differential equations based on the Müntz–Legendre wavelets

The present paper is devoted to the application of local polynomial and non-polynomial
interpolation splines of the third order of approximation for the numerical solution of the Volterra integral
equation of the second kind. Computational schemes based on the use of the splines include the ability to
calculate the integrals over the kernel multiplied by the basis function which are present in the computational
methods. The application of polynomial and nonpolynomial splines to the solution of nonlinear Volterra
integral equations is also discussed. The results of the numerical experiments are presented.

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Application Local Polynomial and Non-polynomial Splines of the Third Order of Approximation for the Construction of the Numerical Solution of the Volterra Integral Equation of the Second Kind

In this study, a wavelet method is developed to solve a system of nonlinear variable-order (V-O) fractional integral equations using the Chebyshev wavelets (CWs) and the Galerkin method. For this purpose, we derive a V-O fractional integration operational matrix (OM) for CWs and use it in our method. In the established scheme, we approximate the unknown functions by CWs with unknown coefficients and reduce the problem to an algebraic system. In this way, we simplify the computation of nonlinear terms by obtaining some new results for CWs. Finally, we demonstrate the applicability of the presented algorithm by solving a few numerical examples.

/pdf/chebyshev-wavelets-operational-matrices-for-solving-i5fy08916w.pdf

Chebyshev wavelets operational matrices for solving nonlinear variable-order fractional integral equations

Robust H∞ output feedback control for type-2 Takagi-Sugeno fuzzy systems with multiple time-delays and disturbances: A descriptor redundancy approach

This study is focused on the numerical solutions of the nonlinear Volterra-Fredholm integral equations (NV-FIEs) of the second kind, which have several applications in physical mathematics and contact problems. Herein, we develop a new technique that combines the modified Adomian decomposition method and the quadrature (trapezoidal and Weddle) rules that used when the definite integral could be extremely difficult, for approximating the solutions of the NV-FIEs of second kind with a phase lag. Foremost, Picard's method and Banach's fixed point theorem are implemented to discuss the existence and uniqueness of the solution. Furthermore, numerical examples are presented to highlight the proposed method's effectiveness, wherein the results are displayed in group of tables and figures to illustrate the applicability of the theoretical results.

Mingxu Yi

Papers

Numerical research of nonlinear system of fractional Volterra–Fredholm integral–differential equations via Block-Pulse functions and error analysis