A robust scheme based on novel‐operational matrices for some classes of time‐fractional nonlinear problems arising in mechanics and mathematical physics. Issue 6 (25th June 2020) - Usman, Muhammad; Hamid, Muhammad; Khalid, Muhammad Saif Ullah; Haq, Rizwan Ul; Liu, Moubin
Abstract: In this paper, we present a novel approach based on shifted Gegenbauer wavelets to attain approximate solutions of some classed of time‐fractional nonlinear problems. First, we present the approximation of a function of two variables u ( x, t ) with help of shifted Gegenbauer wavelets and then some novel operational matrices are proposed with the help of piecewise functions to investigate the positive integer derivative (D x and D t ), fractional‐order derivative ( P x α and P t β ), fractional‐order integration ( J x α and J t β ) and delay terms ( D b a and D d c ) of approximated function u ( x, t ). In order to transform the discussed nonlinear problem into linear problem Picard iterative scheme has been adopt. The current scheme converts the discussed highly nonlinear time‐fractional problem into system of linear algebraic equation the help of developed operational matrices and Picard idea. Analysis on the error bound and convergence to authenticate the mathematical formulation of the computational algorithm. We solve various test problems, such as the van der Pol oscillator model, generalized Burger–Huxley, neutral delay parabolic differential equations, sine‐Gordon, parabolic integro‐differential equation and nonlinear Schrödinger equations to show the efficiency and accuracy of linearized shifted Gegenbauer wavelets method. A comprehensive comparative examination shows the credibility, accuracy, and reliability of the presently proposed computational approach. Also, this scheme can be extended conveniently to other multi‐dimensional physical problems of highly nonlinear fractional or variable order of complex nature.