Journal of Integral Equations and Applications

Numerical approximation of first kind Volterra convolution integral equations with discontinuous kernels

Penny J. Davies and Dugald B. Duncan

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The cubic ``convolution spline'' method for first kind Volterra convolution integral equations was introduced in P.J. Davies and D.B. Duncan, $\mathit {Convolution\ spline\ approximations\ of\ Volterra\ integral\ equations}$, Journal of Integral Equations and Applications \textbf {26} (2014), 369--410. 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.

Article information

J. Integral Equations Applications, Volume 29, Number 1 (2017), 41-73.

First available in Project Euclid: 27 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65M12: Stability and convergence of numerical methods 65R20: Integral equations

Volterra integral equations discontinuous kernel time delay


Davies, Penny J.; Duncan, Dugald B. Numerical approximation of first kind Volterra convolution integral equations with discontinuous kernels. J. Integral Equations Applications 29 (2017), no. 1, 41--73. doi:10.1216/JIE-2017-29-1-41.

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