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Laplace transforms for approximation of highly oscillatory Volterra integral equations of the first kind

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  • Xiang, Shuhuang

Abstract

This paper focuses on Laplace and inverse Laplace transforms for approximation of Volterra integral equations of the first kind with highly oscillatory Bessel kernels, where the explicit formulae for the solution of the first kind integral equations are derived, from which the integral equations can also be efficiently calculated by the Clenshaw–Curtis–Filon-type methods. Furthermore, by applying the asymptotics of the solution, some simpler formulas for approximating the solution for large values of the parameters are deduced. Preliminary numerical results are presented based on the approximate formulae and the explicit formulae, which are compared with the convolution quadrature and numerical inverse Laplace transform methods. All these methods share that the costs the same independent of large values of frequencies.

Suggested Citation

  • Xiang, Shuhuang, 2014. "Laplace transforms for approximation of highly oscillatory Volterra integral equations of the first kind," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 944-954.
    • Handle: RePEc:eee:apmaco:v:232:y:2014:i:c:p:944-954
    DOI: 10.1016/j.amc.2014.01.054
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    References listed on IDEAS

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    1. Joseph Abate & Ward Whitt, 2006. "A Unified Framework for Numerically Inverting Laplace Transforms," INFORMS Journal on Computing, INFORMS, vol. 18(4), pages 408-421, November.
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    Cited by:

    1. Casabán, M.-C. & Cortés, J.-C. & Jódar, L., 2015. "A random Laplace transform method for solving random mixed parabolic differential problems," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 654-667.
    2. Li, Bin & Kang, Hongchao & Chen, Songliang & Ren, Shanjing, 2023. "On the approximation of highly oscillatory Volterra integral equations of the first kind via Laplace transform," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 92-113.
    3. Ma, Junjie, 2017. "Oscillation-free solutions to Volterra integral and integro-differential equations with periodic force terms," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 294-298.

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