Mathematical Analysis and the Local Discontinuous Galerkin Method for Caputo-Hadamard Fractional Partial Differential Equation

In this paper, we study the Caputo-Hadamard fractional partial differential equation where the time derivative is the Caputo-Hadamard fractional derivative and the space derivative is the integer-order one. We first introduce a modified Laplace transform. Then using the newly defined Laplace transform and the well-known finite Fourier sine transform, we obtain the analytical solution to this kind of linear equation. Furthermore, we study the regularity and logarithmic decay of its solution. Since the equation has a time fractional derivative, its solution behaves a certain weak regularity at the initial time. We use the finite difference scheme on non-uniform meshes to approximate the time fractional derivative in order to guarantee the accuracy and use the local discontinuous Galerkin method (LDG) to approximate the spacial derivative. The fully discrete scheme is established and analyzed. A numerical example is displayed which support the theoretical analysis.