Numerical approximation of first kind Volterra convolution integral equations with discontinuous kernels. (English) Zbl 1361.65101

Davies, Penny J.; Duncan, Dugald B.

J. Integral Equations Appl. 29, No. 1, 41-73 (2017).

Summary: The cubic “convolution spline” method for first kind Volterra convolution integral equations was introduced in [P. J. Davies and D. B. Duncan, J. Integral Equations Appl. 26, No. 3, 369–410 (2014; Zbl 1307.65127)]. Here, we analyze its stability and convergence for a broad class of piecewise smooth kernel functions and show it is stable and fourth order accurate even when the kernel function is discontinuous. Key tools include a new discrete Gronwall inequality which provides a stability bound when there are jumps in the kernel function and a new error bound obtained from a particular B-spline quasi-interpolant.

MSC:

65R20	Numerical methods for integral equations
45D05	Volterra integral equations
45E10	Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)

Keywords:

Volterra integral equations; discontinuous kernel; time delay; convolution spline method; stability; convergence

Citations:

Zbl 1307.65127