Theoretical and numerical analysis of third-kind auto-convolution Volterra integral equations

In this paper, we deal with the implicit Euler method on uniform meshes for third-kind auto-convolution Volterra integral equations (AVIEs). The existence, uniqueness, and boundedness of the exact solution are presented by a novel weighted exponential norm and the smoothness is discussed based on linear cordial Volterra integral equations (CVIEs). The solvability of the implicit Euler method is discussed by the step-by-step approach and the uniform boundedness of the numerical solution is provided by an analogous discrete weighted exponential norm. The attainable convergence order is investigated theoretically with the help of the error analysis for linear CVIEs. Finally, some numerical examples, including an application to the theory of viscoelasticity, are presented to verify our theoretical results.