Compact Finite Difference Method For Solving The Fractional Delay Partial Differential Equations

In this paper, the time fractional neutral delay parabolic differential e-quations with initial condition and Dirichlet boundary condition is considered. We establish an implicit compact finite difference scheme for solving the one-dimension and two-dimension time fractional neutral delay parabolic differen-tial equations respectively and present their theoretical analysis. Numerical results are provided to verify the accuracy and efficiency of the proposed algo-rithm.Firstly, the research background,research meaning and research status of the fractional partial differential equations as well as some elementary knowl-edge of fractional differential and fractional calculus are introduced. After having a preliminary understanding of fractional partial differential equations, we give an introduction to the research background,research meaning and re-search status of the fractional delay differential equations.Secondly, the main idea about the equations which we study is:using numerical methods to obtain a implicit compact finite difference scheme of our equation model. After approximating the second-order derivative with respect to space by the compact finite difference method, we use the L1 algorithm to discrete The Caputo fractional derivative to obtain a fully discrete implicit scheme,then we analyze the local truncation error of our scheme. In the end, we prove the stability and convergence by the Fourier Methods.We discuss one-dimensional problem of the time fractional neutral delay parabolic differential equations as well as its theoretical analysis in second chapter, then we give a further study of two-dimensional case of the time frac-tional neutral delay parabolic differential equations and provide its theoretical analysis in third chapter.In the end, a numerical example has demonstrated the effectiveness of our numerical methods, we also give a brief summary of our numerical methods and theoretical analysis.