Linearized novel operational matrices-based scheme for classes of nonlinear time-space fractional unsteady problems in 2D

Abstract

Finding analytical and semi-analytical solutions of two-dimensional nonlinear fractional-order problems arising in mathematical physics is a challenging task for research community. In this work, an innovative scheme is proposed based on the shifted Gegenbauer wavelets. For this purpose first, we introduce shifted Gegenbauer polynomials via suitable transformation. Then, we present three-dimensional shifted Gegenbauer wavelets using shifted Gegenbauer polynomials. We illustrate function approximation of three variables, for example, u(x,y,t) through shifted Gegenbauer wavelets. To compute the novel operational matrices of positive integer and non-integer order derivative of shifted Gegenbauer wavelets vector in one, two and three dimensionals piecewise functions are utilized. Moreover, we describe associated theorems to validate our newly proposed scheme mathematically. The proposed algorithm is innovative because of the incorporation of Picard iterative scheme to tackle highly nonlinear problems of fractional-order. The current computational scheme coverts a mathematical model to a system of linear algebraic equations that are easier to solve. To validate the accuracy, credibility, and reliability of the present method, we analyze various fractional-order problems, including Bloch–Torrey, Burgers, Schrödinger, Rayleigh–Stokes and sine-Gordon. We also conduct a detailed comparative study, which demonstrates that the proposed computational scheme is effective to find the analytical and semi-analytical solutions of the aforementioned problems. Moreover, the proposed computational method can be utilized to analyze the solutions of other higher dimensional nonlinear fractional or variable order problems of physical nature.