On Caputo-Type Cable Equation: Analysis and Computation

In this paper, a special case of nonlinear time fractional cable equation is studied. For the equation defined on a bounded domain, the existence, uniqueness, and regularity of the solution are firstly studied. Furthermore, it is numerically studied via the weighted and shifted Grunwald difference (WSGD) methods/the local discontinuous Galerkin (LDG) finite element methods. The derived numerical scheme has been proved to be stable and convergent with order O(Delta t(2) + h(k+1)), where Delta t, h, k are the time stepsize, the spatial stepsize, and the degree of piecewise polynomials, respectively. Finally, a numerical experiment is presented to verify the theoretical analysis.