Wavelet Galerkin Method For Fractional Elliptic Equations

Fractional derivatives are generalizations of the classical integer-order coun-terparts. Fractional operators can be used more accurately to describe the proba-bility distribution of particles in space and in time. In recent years, the fractional calculus and fractional order differential equations have been widely applied in the physical and biological engineering. At the same time, it has become the most effective tools to describe the abnormal diffusion problems. The modeling progress on using fractional differential equations has led to increasing interest in developing numerical schemes for their solutions. We present a novel solving method, called wavelets Galerkin method, for the numerical solutions of frac-tional elliptic equations, which are based on the B-spline functions to construct Riesz bases, in order to decrease the condition number of corresponding stiffness matrix of the variational formulation. There are many methods to solve fraction-al differential equation, such as the finite element method, the finite difference method and the spectral method. We primarily compare with the finite element method to indicate the effectiveness of our wavelets Galerkin method. Although, the finite element method could get the solutions of the fractional differential equation, the condition number of the corresponding stiffness matrix of the vari-ational formulation is a little more big when the order of the matrix increasing. Since we want to find a new method, which can decrease the corresponding con-dition number of the stiffness matrix. The detailed method is performed, and the extensive numerical experiments confirm the theoretical results and show the robustness of this method.