Operational matrix of two dimensional Chebyshev wavelets and its applications in solving nonlinear partial integro-differential equations

Abstract

Purpose This paper aims to present a new method for the approximate solution of two-dimensional nonlinear Volterra–Fredholm partial integro-differential equations with boundary conditions using two-dimensional Chebyshev wavelets. Design/methodology/approach For this purpose, an operational matrix of product and integration of the cross-product and differentiation are introduced that essentially of Chebyshev wavelets. The use of these operational matrices simplifies considerably the structure of the computation used for a set of the algebraic system has been obtained by using the collocation points and solved. Findings Theorem for convergence analysis and some illustrative examples of using the presented method to show the validity, efficiency, high accuracy and applicability of the proposed technique. Some figures are plotted to demonstrate the error analysis of the proposed scheme. Originality/value This paper uses operational matrices of two-dimensional Chebyshev wavelets and helps to obtain high accuracy of the method.

Crank Nicholson scheme to examine the fractional-order unsteady nanofluid flow of free convection of viscous fluids

“…Here h and τ indicate the space and time step length, M and N are represents the number of grids point. Fractional order derivate can discretize as [ 34 ]: and the second-order derivative using Crank-Nicholson idea can be discretized as under: …”

“…In above and b n +1 are represents the block matrices which are defined as follow: where the matrices and present in [ 33 , 34 ] and u and T are given as: …”

“…This research looked at the physical elements of the issue of fractional-order derivatives between certain domains. The fractional calculus makes visible contributions to various technical and scientific circumstances, including neurology, capacitor theory, viscoelasticity, electro-analytical chemistry, and electrical circuits [ 33 , 34 ]. Several authors [ 35 – 39 ] suggested several techniques for dealing with the nonlinearity of fractional differential equations.…”

A generalized Chebyshev operational method for Volterra integro-partial differential equations with weakly singular kernels

“…Moreover, Rostami and Maleknejad introduced a novel method utilizing 2D hybrid Taylor polynomials and Block-Pulse functions for nonlinear mixed Volterra-Fredholm partial integro-differential equations [25] , [26] . In [27] , 2D Chebyshev wavelets were applied to approximate solutions for 2D Volterra-Fredholm partial integro-differential equations with boundary conditions. Hermite wavelets, in conjunction with the Galerkin method, were dedicated to numerically solving Volterra integro-differential equations [28] .…”

A novel approach to solving system of integral partial differential equations based on hybrid modified block‐pulse functions

“…Therefore, it simplifies problems and reduces the complexity of computing PIDEs. Due to this fact, some operational matrices made of different functions to find numerical solutions to problems are Legendre polynomial solving PDE [12], shifted Legendre polynomial in solving fractional partial differential equations [13], Legendre wavelet in solving fractional partial differential equations [14], Chebyshev wavelet in PIDE [15], Bernoulli polynomials solving PDE [16], Bernoulli wavelet in PDE [17], Müntz–Legendre polynomials in the fractional optimal control problem [18], and so on.…”