Compact Finite Difference Schemes And Fast Solution Techniques For A Class Of Fractional Partial Differential Equations

In this paper we introduce the compact difference scheme for the following quasilinear fractal mobile/immobile transport model and the fast procedure for new definition.The nonlinear reaction term f(u)satisfies the assumed condition:A1:|f(u)|?C|u|,A2:the first-order derivative of f(u)with respect to u is bounded,that is to say,there exists a positive constant C such that |f'(u)|?C.The definition of fractional derivative of this model has been proposed by Caputo and Fabrizia.Definition:If u(·,t)?H1(a,b),b>a,??(0,1),then the new Caputo derivative of fractional order is defined as:where M(?)is a normalization function such that M(0)=M(1)=1.By changing the(t-s)-? with the function exp[-?t-s/1-a],the new definition of fractional derivative without singular kernel does not have singularity for s=t.The new definition of fractional derivative can portray substance hetero-geneities and configurations with different scales,which the original definition description is not good at describing.Our target is to get a second order finite difference method to solve a quasilinear fractal mobile/immobile transport model.In order to get a higher precision,at the same time of solving the equation,an effective compact differ-ence scheme is considered in space.And we analysis the stability and optimal error estimate.Some a priori estimates of discrete errors with optimal order of convergence rate is O(?2 + h4).We get a fast procedure for the new definition.The article is mainly to study the new definition of fractional derivative without singular kernel.The paper is organized as follow four section:Chapter 1.The research background and the model described as above and domestic and foreign research literature review are introducedChapter 2.Introduce some necessary theories including the new definition of fractional derivative without singular kernel and the compact difference and numerical scheme and get a fast solution technique for the new definition.Chapter 3.Prove the stability and optimal error estimate about compact difference scheme and get discrete errors with optimal order of convergence rate O(?2 + h4).Chapter 4.The numerical experiment is given to show that the conver-gence rates are in agreement with the theoretical analysis.