Solving partial and fraction differential equations using corrected fourier series method - Zainal, Nor Hafizah (2014)
Partial differential equations (PDE) are often used to construct models of the most basic theories in physics and engineering. Our goal here is to solve the PDEs problem by using Fourier series method that always been used. However, the truncated Fourier series will cause the Gibbs phenomenon. To eliminate this phenomenon, the corrected Fourier series which consists of a uniformly convergent Fourier series and a correction function will be used. The correction function here is referred to the algebraic polynomials and Heaviside step functions. The Fourier series remains uniformly convergent until its -th derivative without Gibbs oscillation if the order of polynomial in correction functions not exceed -th order which the Gibbs oscillation of the Fourier series will be terminated until its -th derivative. In this study, we use the corrected Fourier series to solve partial differential equations and fractional partial differential equations. The theory of derivatives and integrals of fractional (non-integer) order was started over 300 years ago. In recent years, fractional calculus have been attract in various research due to its extensive application in engineering and science. We solve this problem by using corrected Fourier series method with modified Riemann-Liouville derivatives. The fractional derivatives are described in Riemann-Liouville sense. For the case PDEs, we compared the result with classical Fourier series method and exact solution. There is some case that classical Fourier series method cannot solve at a certain point. Meanwhile, corrected Fourier series method gives the solution at that point. For the case that cannot solve by using classical Fourier series method, we can solve by using corrected Fourier series method. For the fractional PDEs, there is no exact solution for order α as a non-integer number. Thus, we compared the result with others method which is variational iteration method (VIM) and homotopy method. The Maple software is used for all calculation in this study.