On applying the Laplace transform method to an equation describing an axially moving string

In this paper an initial‐boundary value problem for a linear, nonhomogeneous axially moving string equation will be considered. The velocity of the string is assumed to be constant, and the nonhomogeneous terms in the string equation are due to external forces acting on the string. The Laplace transform method will be used to construct the solution of the problem. It will turn out that the method has considerable, computational advantages compared to the usually applied method of modal analysis based on eigenfunction expansions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions

We propose a method to construct numerical solutions of parabolic equations on the unit sphere. The time discretization uses Laplace transforms and quadrature. The spatial approximation of the solution employs radial basis functions restricted to the sphere. The method allows us to construct high accuracy numerical solutions in parallel. We establish L ₂ error estimates for smooth and nonsmooth initial data, and describe some numerical experiments.     

Solution of fractional order heat equation via triple Laplace transform in 2 dimensions

In this article, we introduce the triple Laplace transform for the solution of a class of fractional order partial differential equations. As a consequence, fractional order homogeneous heat equation in 2 dimensions is investigated in detail. The corresponding solution is obtained by using the aforementioned triple Laplace transform, which is the generalization of double Laplace transform. Numerical plots to the concerned solutions are provided to demonstrate our results.     

Riesz transform,Gaussian bounds and the method of wave equation

For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL^– on L^p for some > 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature.As an application of the obtained results we prove boundedness of the Riesz transform on L^p for all p (1,2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L^p of the Laplace-Beltrami operator on Riemannian manifolds for p > 2.Mathematics Subject Classification (1991): 42B20The author was partially supported by Summer Research Award from New Mexico State University.in final form: 8 June 2003     

A regularization method for approximating the inverse Laplace transform

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     

A new analytical modelling for fractional telegraph equation via Laplace transform

The main aim of the present work is to propose a new and simple algorithm for space-fractional telegraph equation, namely new fractional homotopy analysis transform method (HATM). The fractional homotopy analysis transform method is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method (HPTM). The beauty of the paper is error analysis which shows that our solution obtained by proposed method converges very rapidly to the known exact solution. The numerical solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. Finally, several numerical examples are given to illustrate the accuracy and stability of this method.     

On copson's application of riesz's method to the damped wave equation

Copson has shown that, in the two dimensional case, the Cauchy problem for the damped wave equation can be solved, using the original Riesz kernel, without introducing a new kernel. Copson stated that his approach could be used in any number of independent variables; this statement is verified in the present paper     

A comparative approach to Laplace transform inversion

A complex Laplace transform function was inverted by three numerical methods and compared to the small time and large time approximation curves. This technique enabled the best choice of an inversion method to be made, since one method gave excellent results, at both small and large times and moved smoothly from one curve to the other.     

A Tauberian theorem with remainder for the Laplace transform and its application to the riemann zeta-function

A new variant of a Tauberian theorem with remainder is proved for the Laplace transform in the plane, using the L₁ metric. A connection between the divisor problem and the growth of the zeta-function in the critical strip is established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 10, pp. 1368–1378, October, 1991.     

A well-conditioned method of fundamental solutions for Laplace equation

Numerical Algorithms - The method of fundamental solutions (MFS) is a numerical method for solving boundary value problems involving linear partial differential equations. It is well-known that it...     

A Laplace transform method for order statistics from nonidentical random variables and its application in Phase-type distribution

In this paper, we derive a method for obtaining the Laplace transform of order statistics (o.s.) arising from general independent nonidentically distributed random variables (r.v.’s). A survey of the most important properties, applications and the o.s. of a Phase-type (PH) distribution are also presented. Two illustrative examples are provided.     

A Trefftz/MFS mixed-type method to solve the Cauchy problem of the Laplace equation

By using the multiple-scale Trefftz method (MSTM) to solve the Cauchy problem of the Laplace equation in an arbitrary bounded domain, we may lose the accuracy several orders when the noise being imposed on the specified Cauchy data is quite large. In addition to the linear equations obtained from the MSTM, the fundamental solutions play as the test functions being inserted into a derived boundary integral equation. Therefore, after merely supplementing a few linear equations in the mixed-type method (MTM), which is a well organized combination of the Trefftz method and the method of fundamental solutions (MFS), we can improve the ill-conditioned behavior of the linear equations system and hence increase the accuracy of the solution for the Cauchy problem significantly, as explored by two numerical examples.     

A coupled method of Laplace transform and Legendre wavelets for nonlinear Klein–Gordon equations

Klein–Gordon equation models many phenomena in both physics and applied mathematics. In this paper, a coupled method of Laplace transform and Legendre wavelets, named (LLWM), is presented for the approximate solutions of nonlinear Klein–Gordon equations. By employing Laplace operator and Legendre wavelets operational matrices, the Klein–Gordon equation is converted into an algebraic system. Hence, the unknown Legendre wavelets coefficients are calculated in the form of series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient vectors of nonlinear terms. The convergence analysis of the LLWM is discussed. The results show that LLWM is very effective and easy to implement. Copyright © 2013 John Wiley & Sons, Ltd.     

A reduced basis method applied to the Restricted Hartree–Fock equations

In this Note, we describe a reduced basis approximation method for the computation of some electronic structure in quantum chemistry, based on the Restricted Hartree–Fock equations. Numerical results are presented to show that this approach allows for reducing the complexity and potentially the computational costs. To cite this article: Y. Maday, U. Razafison, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

The application of dimension split method in the three‐dimensional heat equation

A dimension splitting method (DSM) with Crank–Nicolson time discrete strategy for a three‐dimensional heat equation is proposed. The basic idea is to simulate the three‐Dimensional problem by numerically solving a series of two‐dimensional problems in parallel fashion. Convergence and error estimation for the DSM scheme are derived in the paper. Numerical experiments demonstrate the feasibility and efficiency of the DSM scheme. Copyright © 2016 John Wiley & Sons, Ltd.