Investigating the Laplace Transform Method’s Efficiency to a Simple Engineering Problem

Year 2023, Volume: 15 Issue: 2, 326 - 333, 31.12.2023

Hüseyin Demir , İnci Çilingir Süngü , İbrahim Keles

https://doi.org/10.47000/tjmcs.1378857

Abstract

In this study, it is aimed to solve the differential equation that forms a simple engineering system and transform it into the Laplace domain, and then to investigate the effectiveness of the method used to compare the solutions with the exact solutions. For this purpose, first, the solutions of a given test function with analytical and numerical Laplace inverse transform methods (Durbin, Stehfest and Talbot) are given comparatively. Although the values obtained from these three methods overlap with each other but it is observed that the Talbot inverse transform method is more suitable than the other two methods due to its lower calculation time requirement. In addition, Talbot’s method and analytical solutions to engineering problems related to the vibratory mechanical system, heat conduction problem and a single matrix block in a fractured reservoir non-isothermal gravity drainage are numerically compared. It is understood that the Talbot inverse transform method is quite effective, and this is evident from the consistency of the numerical results and analytical results of the study. The findings show that the proposed method is very suitable and the method is easy to implement without much difficulty for solving a simple engineering problem.

Keywords

Laplace transform, numerical inversion, test function, Talbot method

References

• Abate, J., Valk´o, P.P., Multi-precision Laplace transform inversion, Int J Numer Methods Eng., 60(5)(2004), 979–993.
• Abdulazeez, S.T., Modanli, M., Analytic solution of fractional order Pseudo-Hyperbolic Telegraph equation using modified double Laplace transform method, Int J Math Comput Eng., 1(1)(2023), 105–114.
• Cohen, A.M., Numerical Methods for Laplace Transform Inversion, Springer Science and Business Media, New York, 2007.
• Dubner, H., Abate, J., Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform, J. ACM (JACM), 15(1)(1968), 115–123.
• Durbin, F., Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s method, Comput J, 17(4)(1974), 371–376.
• Egonmwan, A.O., The Numerical Inversion of the Laplace Transform, Eindhoven University of Technology, Master Thesis, 2012.
• Gaver Jr, D.P.,Observing stochastic processes, and approximate transform inversion, Oper. Res., 14(3)(1966), 444–459.
• Kitahara, N., Nagahara, D., Yano, H., A numerical inversion of Laplace transform and its application, J. Frank. Inst, 325(2)(1988), 221–233.
• Liu, Z., Bouklas, N., Hui, C.-Y., Coupled flow and deformation fields due to a line load on a poroelastic half space: effect of surface stress and surface bending, Proc. R. Soc. A, 476(2233)(2020), 20190761.
• Mashayekhizadeh, V., Dejam, M., Ghazanfari, M.H., The application of numerical Laplace inversion methods for type curve development in well testing: a comparative study, Pet Sci Technol, 29(7)(2011), 695–707.
• Papoulis, A., A new method of inversion of the Laplace transform, Q Appl Math, 14(4)(1957), 405–414.
• Pooladi-Darvish, M., Tortike,W.S., Ali, S.M.F., Steam heating of fractured formations containing heavy oil: basic premises and a single-block analytical model, Proc. - SPE Annu. Tech. Conf. Exhib., SPE-28642-MS(1994).
• Sepehri-Amin, S., Faal, R.T., Das, R., Analytical and numerical solutions for vibration of a functionally graded beam with multiple fractionally damped absorbers, Thin-Walled Struct., 157(2020), 106711.
• Stehfest, H., Algorithm 368: numerical inversion of Laplace transforms, Commun. ACM, 13(1)(1970), 47—49.
• Talbot, A.,The accurate numerical inversion of Laplace transforms, IMA J Appl Math, 23(1)(1979), 97–120.
• Valk, P.P., Vajda, S., Inversion of noise-flee Laplace transforms: Towards a standardized set of test problems, Inverse Probl Sci Eng ., 10(2002), 467–483.
• Valko, P.P., Abate, J., Comparison of sequence accelerators forthe Gaver method of numerical Laplace transform inversion, Comput. Math. with Appl., 48(3–4)(2004), 629–636.
• Weeks, W.T., Numerical inversion of Laplace transforms using Laguerre functions, J. ACM, 13(3)(1966), 419–429.