Quasi Wavelet For The Time-variable Order Fractional Partial Differential Equation

In the research of fluid mechanics, biology, finance, chemical process, s-tochastic process, material science and so on, the fractional order partial dif-ferential equation appears frequently. In recent years, with the development of research problems, the fractional order partial differential equations have been extended to the variable order fractional partial differential equations by many experts and scholars. But because of the variable order fractional partial differential equation,the numerical solution is more difficult to solve, so the study of the numerical solution of the differential equation with variable order fractional is less. At present, main domestic and international researcher-s on this problem are as follows:Chen,C.M.,F.Liu[1,2,3], Shen,S.J.[8], Soon, C.M.,F.M.Conbra, M.H.Kobayashi[ll], Combra, F.M.[12], Zhuang, P., Liu,F.,V.Anh[13], Sun,H.W.,W.Chen,Y.Chen[14]. In these papers, hardly anyone use the quasi wavelet method to solve the variable order fractional differential equations.This paper mainly discusses a class of time-variable order fractional par-tial differential equations by using the quasi wavelet method. This thesis is divided into six chapters. The first and the two chapter mainly introduce the development of the variable order fractional partial differential equations, the research status at home and abroad as well as the definition of some vari-able fractional order derivatives. The third chapter studies the Fully discrete schemes of the Euler-compact difference of the time-variable order fractional cable equation and its stability and convergence. The fourth chapter gives a brief introdution to the theory of the quasi wavelet function and the deriva-tive approximation. The fifth chapter illustrates the Euler time semi discrete scheme of the time-variable order fractional cable equation and the Euler quasi wavelet Fully discrete scheme. The sixth chapter verifies the validity and reliability of the proposed method to solve the time-variable order frac-tional partial differential equations by numerical examples.