Numerical Method for Inverse Laplace Transform with Haar Wavelet Operational Matrix


  • Suazlan Mt Aznam
  • Amran Hussin



operational matrix, numerical inversion, Inversion of Laplace transform, Haar wavelet,


Wavelets have been applied successfully in signal and image processing. Many attempts have been made in mathematics to use orthogonal wavelet function as numerical computational tool. In this work, an orthogonal wavelet function namely Haar wavelet function is considered. We present a numerical method for inversion of Laplace transform using the method of Haar wavelet operational matrix for integration. We proved the method for the cases of the irrational transfer function using the extension of Riemenn-Liouville fractional integral. The proposed method extends the work of J.L.Wu et al. (2001) to cover the whole of time domain. Moreover, this work gives an alternative way to find the solution for inversion of Laplace transform in a faster way. The use of numerical Haar operational matrix method is much simpler than the conventional contour integration method and it can be easily coded. Additionally, few benefits come from its great features such as faster computation and attractiveness. Numerical results demonstrate good performance of the method in term of accuracy and competitiveness compare to analytical solution. Examples on solving differential equation by Laplace transform method are also given.


J.L. Wu,C. H. Chen, and C. F. Chen, IEEE Trans. on Cir And Syst.-I: Fund. Theory AndAppl, 48 (2001), 120-122.

K. Adem, and Zeyad Abdel Aziz Al Zhour, Applied Mathematics and Computation, 187 (2007), 250–265

C.F. Chen, Y.T. Tsay, and T.T. Wu,Franklin Institute, 503 (1977), 267–284,

Zhi Shi, L. Y. Deng, and Q. J. Chen, Proceedings of the 2007 ICWAPR, ion, Beijing, China, 2-4 Nov. 2007

C.F. Chen, and C.H. Hsiao, IEE Proc.-Control Theory Appl, 144 (1997), 87-94.

B. Ross, Lecture Notes in Mathematics, 457 (1975), 207-215.

C. H. Wang, J. Franklin Inst.,3 (1983), 91-102.

C. F. Chen, and C. H. Hsiao, J. Franklin Inst.,300 (1975), 265–279. [9] C. H. Wang, J. Franklin Inst., 315 (1983), 91-102.

ÜloLepik, Applied Mathematics and Computation, 185 (2007), 695-704.