The text of Laplace, Sur l’application du calcul des probabilités à la philosophie naturelle, (Théorie Analytique des Probabilités. Troisième Édition. Premier Supplément), 1820, is quoted in the context of the Gram-Schmidt algorithm. We provide an English translation of Laplace’s manuscript (originally in French) and interpret the algorithms of Laplace in a contemporary context. The two algorit...

In order to overcome the difficulties of low computational efficiency and high memory requirement in the conventional boundary element method for solving large-scale potential problems, a fast multipole boundary element method for the problems of Laplace equation is presented. through the multipole expansion and local expansion for the basic solution of the kernel function of the Laplace equati...

Reported are Laplace barriers consisting of arrayed posts or ridges that impart ∼100 to 1000 s of N/m(2) Laplace pressure for fluid confinement, but the Laplace pressure is also small enough such that the barriers are porous to electrowetting control. As a result, the barriers are able to provide electrowetting flow thresholding and virtual fluid confinement in noncircular fluid geometries. A s...

In the present paper the authors introduce the exponential integral transform and the complementary error transform, then it is shown that the third iterate of the Laplace transform is the exponential integral transform and a modified third iterate of the Laplace transform is the complementary error transform. Some identities involving these new transforms, the Laplace transform and the Stieltj...

Abstract: In this paper, Laplace decomposition method is developed to solve linear and nonlinear fractional integrodifferential equations. The proposed method is based on the application of Laplace transform to nonlinear fractional integro-differential equation. The nonlinear term can easily be handled with the help of Adomian polynomials. The fractional derivative is described in the Caputo se...

In this paper, we analyze the convergence of the preconditioned GMRESmethod for the first order finite element discretizations of the Helmholtz equation in media with losses. We consider a Laplace preconditioner, an inexact Laplace preconditioner and a two-level preconditioner. Our analysis is based on bounding the field of values of the preconditioned system matrix in the complex plane. The an...

Laplace transform is a very powerful mathematical tool applied in various areas of engineering and science. With the increasing complexity of engineering problems, Laplace transforms help in solving complex problems with a very simple approach just like the applications of transfer functions to solve ordinary differential equations. This paper will discuss the applications of Laplace transforms...