Hp-Discontinuous Galerkin Time-Stepping Method For Integro-Differential Equations

In this paper the hp-discontinuous Galerkin time-stepping method for quasi-linear ordinary integro-differential equations and then for partial integro-differential equations are considered. Firstly, we consider quasi-linear Volterra integro-differential equations with weakly singular kernels. The primary problems are linearized and the equivalence of the primary problems and the linearized ones are proved. And the error estimates in L₂ are derived. For quasi-linear parabolic integro-differential equations, similarly, using the linearization method and the equivalence of the primary problems and the linearized ones, under the hp-discontinuous Galerkin time-stepping method the existence and uniqueness of the numerical solution of quasi-linear parabolic integro-differential equations are proved. And the error estimates in L₂ are derived.