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An algorithm for the inversion of Laplace transforms using Puiseux expansions

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Abstract

This paper is devoted to designing a practical algorithm to invert the Laplace transform by assuming that the transform possesses the Puiseux expansion at infinity. First, the general asymptotic expansion of the inverse function at zero is derived, which can be used to approximate the inverse function when the variable is small. Second, an inversion algorithm is formulated by splitting the Bromwich integral into two parts. One is the main weakly oscillatory part, which is evaluated by a composite Gauss–Legendre rule and its Kronrod extension, and the other is the remaining strongly oscillatory part, which is integrated analytically using the Puiseux expansion of the transform at infinity. Finally, some typical tests show that the algorithm can be used to invert a wide range of Laplace transforms automatically with high accuracy and the output error estimator matches well with the true error.

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Acknowledgements

The authors would like to thank the anonymous referees for their useful suggestions and valuable remarks, which significantly improve the quality of the paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tianjin Normal University, Tianjin, 300387, China

    Tongke Wang

  2. College of Information Engineering, Henan Institute of Science and Technology, Xinxiang, Henan, 453003, China

    Yuesheng Gu

  3. Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Zhiyue Zhang

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  1. Tongke Wang
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  2. Yuesheng Gu
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  3. Zhiyue Zhang
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Correspondence to Tongke Wang.

Additional information

This project was partially supported by the National Natural Science Foundation of China (grant No. 11471166), Natural Science Foundation of Jiangsu Province (grant No. BK20141443) and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

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Wang, T., Gu, Y. & Zhang, Z. An algorithm for the inversion of Laplace transforms using Puiseux expansions. Numer Algor 78, 107–132 (2018). https://doi.org/10.1007/s11075-017-0369-y

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  • DOI: https://doi.org/10.1007/s11075-017-0369-y

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