Finite integration method with chebyshev expansion for finding numerical solution of nonlinear and fractional order differential equations

Abstract

In this dissertation, we develop the finite integration method by using Chebyshev polynomial expansion (FIM-CPE) for solving one- and two-dimensional nonlinear differential equations. The developed FIM-CPE can be used on any domains. Then, we utilize our FIM-CPE to deal with the spatial variable and the forward difference quotient to handle the derivative involving temporal variable. Thus, the numerical algorithms based on this idea are devised to overcome three nonlinear problems including one-dimensional Burgers' equation with shock wave, time-fractional Benjamin-Bona-Mahony-Burgers' equation and two-dimensional nonlinear Poisson equation over irregular domains. Moreover, we examine our algorithms with several experimental examples by comparing the approximate results obtained by our methods and other methods with their analytical solutions. Those examples show that the proposed algorithms provide a significant improvement of the approximate solution in terms of accuracy with low computational cost.