In order to find stable, accurate, and computationally efficient methods for performing the inverse Laplace transform, a new double transformation approach is proposed. To validate and improve the inversion solution o...In order to find stable, accurate, and computationally efficient methods for performing the inverse Laplace transform, a new double transformation approach is proposed. To validate and improve the inversion solution obtained using the Gaver-Stehfest algorithm, direct Laplace transforms are taken of the numerically inverted transforms to compare with the original function. The numerical direct Laplace transform is implemented with a composite Simpson’s rule. Challenging numerical examples involving periodic and oscillatory functions, are investigated. The numerical examples illustrate the computational accuracy and efficiency of the direct Laplace transform and its inverse due to increasing the precision level and the number of terms included in the expansion. It is found that the number of expansion terms and the precision level selected must be in a harmonious balance in order for correct and stable results to be obtained.展开更多
A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Four...A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite sum展开更多
This paper develops a numerical method to invert multi-dimensional Laplace transforms. By a variable transform, Laplace transforms are converted to multi-dimensional Hansdorff moment problems so that the numerical sol...This paper develops a numerical method to invert multi-dimensional Laplace transforms. By a variable transform, Laplace transforms are converted to multi-dimensional Hansdorff moment problems so that the numerical solution can be achieved. Stability estimation is also obtained. Numerical simulations show the efficiency and practicality of the method.展开更多
Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find on approximation to f(t) by the use of the dassical Jacobi polynomials. The main contribution of our work is the development of a...Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find on approximation to f(t) by the use of the dassical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series ex-pansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.展开更多
This paper examines the performance of five algorithms for numerically inverting the Laplace transform, in standard, 16-digit and multi-precision environments. The algorithms are taken from three of the four main clas...This paper examines the performance of five algorithms for numerically inverting the Laplace transform, in standard, 16-digit and multi-precision environments. The algorithms are taken from three of the four main classes of numerical methods used to invert the Laplace transform. Because the numerical inversion of the Laplace transform is a perturbed problem, rounding errors which are generated in numerical approximations can adversely affect the accurate reconstruction of the inverse transform. This paper demonstrates that working in a multi-precision environment can substantially reduce these errors and the resulting perturbations exist in transforming the data from the s-space into the time domain and in so doing overcome the main drawback of numerically inverting the Laplace transform. Our main finding is that both the Talbot and the accelerated Gaver functionals perform considerably better in a multi-precision environment increasing the advantages of using Laplace transform methods over time-stepping procedures in solving diffusion and more generally parabolic partial differential equations.展开更多
In this article, we have effectively used the Numerical Inversion of Laplace transform to study the time-dependent thin film flow of a second grade fluid flowing down an inclined plane through a porous medium. The sol...In this article, we have effectively used the Numerical Inversion of Laplace transform to study the time-dependent thin film flow of a second grade fluid flowing down an inclined plane through a porous medium. The solution to the governing equation is obtained by using the standard Laplace transform. However, to transform the obtained solutions from Laplace space back to the original space, we have used the Numerical Inversion of Laplace transform. Graphical results have been presented to show the effects of different parameters involved and to show how the fluid flow evolves with time.展开更多
文摘In order to find stable, accurate, and computationally efficient methods for performing the inverse Laplace transform, a new double transformation approach is proposed. To validate and improve the inversion solution obtained using the Gaver-Stehfest algorithm, direct Laplace transforms are taken of the numerically inverted transforms to compare with the original function. The numerical direct Laplace transform is implemented with a composite Simpson’s rule. Challenging numerical examples involving periodic and oscillatory functions, are investigated. The numerical examples illustrate the computational accuracy and efficiency of the direct Laplace transform and its inverse due to increasing the precision level and the number of terms included in the expansion. It is found that the number of expansion terms and the precision level selected must be in a harmonious balance in order for correct and stable results to be obtained.
文摘A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite sum
基金the Jiangxi Provincial Natural Scientific Foundation(0211014)Scientific Research Program from Education Office of Jiangxi Province([2005]213)East China Institute of Technology.
文摘This paper develops a numerical method to invert multi-dimensional Laplace transforms. By a variable transform, Laplace transforms are converted to multi-dimensional Hansdorff moment problems so that the numerical solution can be achieved. Stability estimation is also obtained. Numerical simulations show the efficiency and practicality of the method.
文摘Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find on approximation to f(t) by the use of the dassical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series ex-pansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.
文摘This paper examines the performance of five algorithms for numerically inverting the Laplace transform, in standard, 16-digit and multi-precision environments. The algorithms are taken from three of the four main classes of numerical methods used to invert the Laplace transform. Because the numerical inversion of the Laplace transform is a perturbed problem, rounding errors which are generated in numerical approximations can adversely affect the accurate reconstruction of the inverse transform. This paper demonstrates that working in a multi-precision environment can substantially reduce these errors and the resulting perturbations exist in transforming the data from the s-space into the time domain and in so doing overcome the main drawback of numerically inverting the Laplace transform. Our main finding is that both the Talbot and the accelerated Gaver functionals perform considerably better in a multi-precision environment increasing the advantages of using Laplace transform methods over time-stepping procedures in solving diffusion and more generally parabolic partial differential equations.
文摘In this article, we have effectively used the Numerical Inversion of Laplace transform to study the time-dependent thin film flow of a second grade fluid flowing down an inclined plane through a porous medium. The solution to the governing equation is obtained by using the standard Laplace transform. However, to transform the obtained solutions from Laplace space back to the original space, we have used the Numerical Inversion of Laplace transform. Graphical results have been presented to show the effects of different parameters involved and to show how the fluid flow evolves with time.
