Numerical inversion of Laplace transform based on Bernstein operational matrix

Abstract

This paper provides a technique to investigate the inverse Laplace transform by using orthonormal Bernstein operational matrix of integration. The proposed method is based on replacing the unknown function through a truncated series of Bernstein basis polynomials and the coefficients of the expansion are obtained using the operational matrix of integration. This is an alternative procedure to find the inversion of Laplace transform with few terms of Bernstein polynomials.Numerical tests on various functions have been performed to check the applicability and efficiency of the technique. The root mean square error between exact and numerical results is computed, which shows that the method produces the satisfactory results. A rough upper bound for errors is also estimated. KEYWORDSLaplace transform, numerical inverse Laplace transform, operational matrices, orthonormal Bernstein polynomialswhere c is a positive real number and all the poles of F(s) lie at the left side of the line s = c. Sometimes, inverse Laplace transform becomes complicated when the analytic inverse cannot be found. Hence, to calculate the inverse Laplace transform, numerical techniques are adopted.Recently, numerous different methods have been adopted for analytically and numerically inverting the Laplace transform, which is well suited to different types of functions. Lee and Sheen 5 developed analytic continuation of Bromwich integral and determined the integral by using quadrature rule, Dubner and Abate 6 expressed the function in Fourier cosine series, further it was improved by Durbin 7 that the trapezoidal rule can be applied and the function can be expressed in trigonometric series, Iqbal 8 studied the regularization method, Cuomo et al 9 have described a numerical technique based Math Meth Appl Sci. 2018;41:9231-9243.wileyonlinelibrary.com/journal/mma

Numerical Inverse Laplace Transform for Solving a Class of Fractional Differential Equations

“…Enormous efforts and advances have been conducted to obtain the numerical solutions of fractional differential equations. An operational matrix of orthonormal Bernstein polynomials is derived to find the inverse Laplace transform in [50]. Here, the practical use of our proposed method is discussed for finding the solutions of some fractional order ordinary differential equations including the mathematical model of instrument (MEMS) and partial differential equations (particularly wave equation) that converts the problem to system of linear algebraic equations.…”

“…In this section, we describe the algorithm proposed in [50] by considering the linear time varying system…”

“…But these are not adopted with Laplace transform for inverting the Laplace transform. A Bernstein operational matrix of integration has been developed to find the numerical inverse Laplace transform of certain functions in our paper [50]. The proposed method expresses the solution of equations in terms of truncated Bernstein expansion and then using its operational matrix of integration, numerical inverse Laplace transform is obtained.…”

“…Here in our method, we achieve the accuracy using a matrix of order 6 or 7. In the present research, our aim is to find the numerical solutions of linear fractional ordinary and partial differential equations, nonlinear fractional differential equations and system of fractional differential equations using numerical inverse Laplace transform method based on Bernstein operational matrix of integration developed in [50]. At first, for linear problems, we transform the fractional differential equations to system of linear algebraic equations using Laplace transform then the numerical approach for calculating the inverse Laplace transform is used to retrieve the time-domain.…”

Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs

“…These approaches are used to solve a variety of PDEs and PIDEs, including a fourth-order PIDE with a weakly singular kernel [15], fractional PIDE of Volterra type [16], the PIDE that derives by financial stochastic processes [17], the PIDE obtained from the filtration model [18], and the time fractional PIDE [19]. Other approaches based on the matrix transform have been developed, such as Fourier-Bessel matrix transforms [20], Bernstein operational matrix [21], etc.…”