Laplace transform inversion using Bernstein operational matrix of integration and its application to differential and in
- PDF / 646,593 Bytes
- 29 Pages / 510.236 x 680.315 pts Page_size
- 19 Downloads / 185 Views
Laplace transform inversion using Bernstein operational matrix of integration and its application to differential and integral equations VINOD MISHRA and DIMPLE RANI∗ Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal 148 106, India *Corresponding author. E-mail: [email protected]; [email protected]; [email protected]
MS received 3 October 2018; revised 10 January 2020; accepted 29 January 2020 Abstract. In Rani et al. (Numerical inversion of Laplace transform based on Bernstein operational matrix, Mathematical Methods in the Applied Sciences (2018) pp. 1–13), a numerical method is developed to find the inverse Laplace transform of certain functions using Bernstein operational matrix. Here, we describe Bernstein operational matrix of integration and propose an algorithm to solve linear time-varying systems governing differential equations. Apart from discussing error estimate, the method is implemented to linear differential equations on Bessel equation of order zero, damped harmonic oscillator, some higher order differential equations, singular integral equation, Volterra integral and integro-differential equations and nonlinear Volterra integral equations of the first kind. A comparison with some existing methods like Haar operational matrix, block pulse operational matrix and others are discussed. The method is simple and easy to implement on a variety of problems. Relative errors estimate just for 5th or 6th approximation show high applicability of the method. Keywords. Numerical inverse Laplace transform; orthonormalized Bernstein polynomials; operational matrix of integration. Mathematics Subject Classification.
44A10, 65R10, 40A25.
1. Introduction Laplace transform is one of the most popular integral transforms that enables to solve different types of problems like differential and integral equations. Having considerable applications in various fields of science and engineering like design of engineering systems such as electrical circuits and mechanical vibrations, the Laplace transform is the subject of attention for many researchers. Inversion is an ill-posed problem. Hence, to calculate the inverse Laplace transform, numerical techniques are adopted. Several numerical techniques are available in literature to invert the Laplace transform. An extensive survey and comparison of methods were discussed in [11,14,21,41]. Murli and Rizzardi [44] employed Talbot’s algorithm, Murli et al. [43] proposed numerical approximation of the © Indian Academy of Sciences 0123456789().: V,-vol
60
Page 2 of 29
Proc. Indian Acad. Sci. (Math. Sci.)
(2020) 130:60
inverse Laplace function based on the Laplace transform eigenfunction expansion of the inverse function, in a real case. Dubner and Abate [23] determined the inverse Laplace transform numerically on the basis of evaluating the inverse Laplace transform integral. There exist a freedom in choosing the contour of integration. They expressed the inverse function as a Fourier cosine series. In [24
Data Loading...
