Numerical Inverse Laplace Transform Using Chebyshev Polynomial

Authors: Vinod Mishra, Dimple Rani

Abstract:

In this paper, numerical approximate Laplace transform inversion algorithm based on Chebyshev polynomial of second kind is developed using odd cosine series. The technique has been tested for three different functions to work efficiently. The illustrations show that the new developed numerical inverse Laplace transform is very much close to the classical analytic inverse Laplace transform.

Keywords: Chebyshev polynomial, Numerical inverse Laplace transform, Odd cosine series.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1124103

References:

[1] A. M. Cohen, Numerical Methods for Laplace Transform Inversion, Springer, 2007.
[2] R. E. Bellman, H. H. Kagiwada and R. E. Kalba, “Numerical Inversion of Laplace Transforms and Some Inverse Problems in Radiative Transfer,” Journal of Atmospheric Sciences, vol. 23, pp. 555-559, 1966.
[3] L. D’Amore, G. Laccetti and A. Murli, “An Implementation of a Fourier Series Method for the Numerical Inversion of the Laplace Transform,” ACM Transactions on Mathematical Software, vol. 25, No. 3, pp. 279–305, 1999.
[4] B. Davies and B. Martin, “Numerical Inversion of Laplace Transform: A Survey and Comparison of Methods,” Journal of Computational Physics, vol. 33, pp. 1-32, 1979.
[5] A. Papoulis, “A New Method of Inversion of the Laplace Transform,” Quart. Appl. Math., vol. 14, pp. 405-414, 1956.
[6] V. Mishra, “Review of Numerical Inverse of Laplace transforms using Fourier Analysis, Fast Fourier transform and Orthogonal polynomials,” Mathematics in Engineering Science and Aerospace, vol. 5, pp. 239-261, 2014.
[7] J. L. Schiff, The Laplace Transform: Theory and Applications, Springer, 2005.
[8] F. A. Uribe, J. L. Naredo, P. Moreno and L. Guardado, “Electromagnetic Transients in Underground Transmission Systems through the Numerical Laplace Transform,” International Journal of Electrical Power and Energy Systems, vol. 24, pp. 215-221, 2002.
[9] V. Mishra and D. Rani, “Chebyshev Polynomial Based Numerical Inverse Laplace Transform Solutions of Linear Volterra Integral and Integro-differential Equations,” American Research Journal of Mathematics, vol. 1, pp. 22-32, 2015.
[10] R. E. Bellman, R.E. Kalaba and J.A. Lockett, Numerical Inversions Laplace Transforms with Applications, American Elsevier Publishing Co., New York, 1966, p. 255.
[11] J. Glyn, Advanced Modern Engineering Mathematics, Pearson Education, 2004 (Indian reprint).
[12] H. Sheng, Y. Li and Y. Q. Chen, “Application of Numerical Inverse Laplace Transform Algorithms in Fractional Calculus,” in Proceedings of FDA’10, Article No. 108, pp. 1-6.
[13] L. C. Andrews and B. K. Shivamoggi, Integral Transforms for Engineers, PHI, 2003.
[14] J. L. Wu, C. H. Chen and C. F. Chen, “Numerical Inversion of Laplace Transform using Haar Wavelet Operational Matrices,” IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, vol. 48, pp. 120-122, 2001.