Study On Two Kinds Of Spectral Distribution Method For Caputo Type Fractional Differential Equations

Fractional differential equations are more suitable in describing physical phenomena, engineering process, biological system and financial problem. So it attract more and more scholars’ attention. In this thesis, we introduce integral basis functions constructed by Jacobi polynomials. Then we derive the first order pseudospectral differential matrix. Next, we expand Lagrange interpolating polynomial with Jacobi polynomial. By using some special properties of fractional derivative of Jacobi polynomial, we construct the pseudospectral differential matrix of arbitrary order Caputo fractional derivative. We adjust the distribution of collocation points by the compressing mapping. So we can get better simulation of the singularity of fractional differential operator. Based on this, we derive pseudospectral differential matrices of left and right Caputotype derivatives. We can compute these differential matrices efficiently by downwind scheme.Numerical experiments show that our new methods can give sharp numerical results.