An Efficient Spectral Approximation for Solving Several Types of Parabolic PDEs with Nonlocal Boundary Conditions

Abstract

The problem of solving several types of one-dimensional parabolic partial differential equations (PDEs) subject to the given initial and nonlocal boundary conditions is considered. The main idea is based on direct collocation and transforming the considered PDEs into their associated algebraic equations. After approximating the solution in the Legendre matrix form, we use Legendre operational matrix of differentiation for representing the mentioned algebraic equations clearly. Three numerical illustrations are provided to show the accuracy of the presented scheme. High accurate results with respect to the Bernstein Tau technique and Sinc collocation method confirm this accuracy.

A Bivariate Chebyshev Spectral Collocation Quasilinearization Method for Nonlinear Evolution Parabolic Equations

“…Darvishi et al [ 29 , 30 ] solved the KdV and the Burgers-Huxley equations using a combination of the Chebyshev spectral collocation method and Darvishi's preconditioning. Jacobs and Harley [ 31 ] and Tohidi and Kilicman [ 32 ] used spectral collocation directly for solving linear partial differential equations. Accuracy will be compromised if they implement their approach in solving nonlinear partial differential equations since they use Kronecker multiplication.…”

Solving parabolic integro-differential equations with purely nonlocal conditions by using the operational matrices of Bernstein polynomials

“…Similar problem can be found in [14]. Recently, various types of partial differential equations (PDEs) with nonlocal conditions have been studied in [2-4, 10, 11, 18, 19] among others, and the use of nonlocal conditions was extended to cover a wide variety of PDEs, and integro-differential equations; see, e.g., [13,15,16,22]. This paper is organized as follows.…”

Parameter estimation in uncertain differential equations based on the solution

“…Singh et al 22 presented an effective matrix method for solving nonlinear Volterra singular partial integro‐differential equations based on shifted Legendre polynomials. Tohidi and Kilicman 23 studied the solution of several one‐dimensional parabolic partial differential equations under given initial and nonlocal boundary conditions. These important achievements provide new directions and broaden our research ideas for the future study of uncertainty theory.…”