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.     

Review of inverse Laplace transform algorithms for Laplace-space numerical approaches

A boundary element method (BEM) simulation is used to compare the efficiency of numerical inverse Laplace transform strategies, considering general requirements of Laplace-space numerical approaches. The two-dimensional BEM solution is used to solve the Laplace-transformed diffusion equation, producing a time-domain solution after a numerical Laplace transform inversion. Motivated by the needs of numerical methods posed in Laplace-transformed space, we compare five inverse Laplace transform algorithms and discuss implementation techniques to minimize the number of Laplace-space function evaluations. We investigate the ability to calculate a sequence of time domain values using the fewest Laplace-space model evaluations. We find Fourier-series based inversion algorithms work for common time behaviors, are the most robust with respect to free parameters, and allow for straightforward image function evaluation re-use across at least a log cycle of time.     

A reliable treatment of Abel’s second kind singular integral equations

The central idea of this paper is to construct a new mechanism for the solution of Abel’s type singular integral equations that is to say the two-step Laplace decomposition algorithm. The two-step Laplace decomposition algorithm (TSLDA) is an innovative adjustment in the Laplace decomposition algorithm (LDA) and makes the calculation much simpler. In this piece of writing, we merge the Laplace transform and decomposition method and present a novel move toward solving Abel’s singular integral equations.     

The modified algorithm for the differential transform method to solution of Genesio systems

In this article, approximate analytical solution of chaotic Genesio system is acquired by the modified differential transform method (MDTM). The differential transform method (DTM) is mentioned in summary. MDTM can be obtained from DTM applied to Laplace, inverse Laplace transform and Padé approximant. The MDTM is used to increase the accuracy and accelerate the convergence rate of truncated series solution getting by the DTM. Results are given with tables and figures.     

Laplace’s transform of fractional order via the Mittag–Leffler function and modified Riemann–Liouville derivative

We propose a (new) definition of a fractional Laplace’s transform, or Laplace’s transform of fractional order, which applies to functions which are fractional differentiable but are not differentiable, in such a manner that they cannot be analyzed by using the Djrbashian fractional derivative. After a short survey on fractional analysis based on the modified Riemann–Liouville derivative, we define the fractional Laplace’s transform. Evidence for the main properties of this fractal transformation is given, and we obtain a fractional Laplace inversion theorem.     

Fluid model driven by an M/M/1 queue with multiple vacations and N-policy

This paper studies a fluid model driven by an M/M/1 queue with multiple exponential vacations and N-policy. The expression for the Laplace transform of the joint steady-state distribution of the fluid model is of a simple matrix power function form or matrix factorial form. Based on this fact, we introduce a new method of fluid model??modified matrix geometric solution method. The Laplace transform and Laplace-Stieltjes transform of the steady-state distribution of the buffer content are concisely expressed through the minimal positive solution to a crucial quadratic equation. Finally, we give concise expression for the performance measure??mean buffer content, which is useful in parameter design of fluid model and various practical applications.     

An electromagnetic free convection flow of a micropolar fluid with relaxation time

In the present investigation, we study the influence of a transverse magnetic field on the one-dimensional motion of an electrically conducting micropolar fluid through a porous medium. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the microrotation and the velocity distributions as well as for the induced magnetic and electric fields are carried out and represented graphically.     

Solving coupled pseudo-parabolic equation using a modified double Laplace decomposition method

In this paper, the modification of double Laplace decomposition method is proposed for the analytical approximation solution of a coupled system of pseudo-parabolic equation with initial conditions. Some examples are given to support our presented method. In addition, we prove the convergence of double Laplace transform decomposition method applied to our problems.     

The compound Poisson process perturbed by a diffusion with a threshold dividend strategy

In this paper, we consider the compound Poisson process perturbed by a diffusion in the presence of the so‐called threshold dividend strategy. Within this framework, we prove the twice continuous differentiability of the expected discounted value of all dividends until ruin. We also derive integro‐differential equations for the expected discounted value of all dividends until ruin and obtain explicit expressions for the solution to the equations. Along the same line, we establish explicit expressions for the Laplace transform of the time of ruin and the Laplace transform of the aggregate dividends until ruin. In the case of exponential claims, some examples are provided. Copyright © 2008 John Wiley & Sons, Ltd.     

Probabilistic Remarks in the Theory of Scattering by Soft Obstacles

Considering the Laplace transform in time of the solution to the acoustic wave equation, we make the problem of scattering by a soft obstacle amenable to probabilistic analysis. A uniqueness result is obtained, and a possible probabilistic interpretation of scattering amplitudes is presented.     

A Laplace transform technique for the analytical solution of a diffusion- convection equation over a finite domain

A Laplace transform technique has been utilized to obtain two different analytic solutions to a single diffusion-convection equation over a finite domain. One analytic solution is continuous at both ends of the domain of interest, while the other solution is discontinuous at the origin. This difference in the two solutions is explained. An application of the Laplace transform technique to a more complex system of equations, on a finite domain, is noted and an error apparent in a previous paper is corrected.     

A perturbed risk model with dependence between premium rates and claim sizes

This paper considers a dependent risk model with diffusion for the surplus of an insurer, in which a current premium rate will be adjusted after a claim occurs and the adjusted rate is determined by the amount of the claim. At the same time, the diffusion is changed correspondingly. Using Rouché’s theorem, we first derive the closed-form solution for the Laplace transform of the survival probability in the dependent risk model. Then, using the Laplace transform, we derive a defective renewal equation satisfied by the survival probability. For the exponential claim sizes, we present the explicit recursion expression for the survival probability, by which we can exactly solve the survival probability step-by-step. We also illustrate the influence of the model parameters in the dependent risk model on the survival probability by numerical examples.     

A new construction of the Clifford-Fourier kernel

In this paper, we develop a new method based on the Laplace transform to study the Clifford-Fourier transform. First, the kernel of the Clifford-Fourier transform in the Laplace domain is obtained. When the dimension is even, the inverse Laplace transform may be computed and we obtain the explicit expression for the kernel as a finite sum of Bessel functions. We equally obtain the plane wave decomposition and find new integral representations for the kernel in all dimensions. Finally we define and compute the formal generating function for the even dimensional kernels.     

The numerical solution of linear time-dependent partial differential equations by the Laplace transform and finite difference approximations

A feasible method is presented for the numerical solution of a large class of linear partial differential equations which may have source terms and boundary conditions which are time-varying. The Laplace transform is used to eliminate the time-dependency and to produce a subsidiary equation which is then solved in complex arithmetic by finite difference methods. An effective numerical Laplace transform inversion algorithm gives the final solution at each spatial mesh point for any specified set of values of t. The single-step property of the method obviates the need to evaluate the solution at a large number of unwanted intermediate time points. The method has been successfully applied to a variety of test problems and, with two alternative numerical Laplace transform inversion algorithms, has been found to give results of good to excellent accuracy. It is as accurate as other established finite difference methods using the same spatial grid. The algorithm is easily programmed and the same program handles equations of parabolic and hyperbolic type.     

Efficient Computation of First Passage Times in Kou’s Jump-diffusion Model

S. G. Kou and H. Wang [First passage times of a jump diffusion process, Adv. Appl. Probab. 35 (2003) 504–531] give expressions of both the real Laplace transform of the distribution of first passage time and the real Laplace transform of the joint distribution of the first passage time and the running maxima of a jump-diffusion model called Kou model. These authors invert the former Laplace transform by using Gaver-Stehfest algorithm, and for the latter they need a large computing time with an algebra computer system. In the present paper, we give a much simpler expression of the Laplace transform of the joint distribution, and we also show, using Complex Analysis techniques, that both Laplace transforms can be extended to the complex plane. Hence, we can use inversion methods based on the complex inversion formula or Bromwich integral which are very efficent. The improvement in the computing times and accuracy is remarkable.     

Stability and stabilization of a system with two constant delays

We consider the stabilization problem for a linear system of differential equations with two constant commensurable delays and with an exponential factor on the right-hand side. Furthermore, by using the Laplace transform, we obtain sufficient conditions for the instability of a solution of the considered system.     

Magnetoelectroelastodynamic interaction of multiple arbitrarily oriented planar cracks

The problem of multiple arbitrarily oriented planar cracks in an infinite magnetoelectroelastic space under dynamic loadings is considered. An explicit solution to the problem is given in the Laplace transform domain in terms of suitable exponential Fourier integral representations. The unknown functions in the Fourier integrals are directly related to the Laplace transform of the jumps in the displacements, electric potential and magnetic potential across opposite crack faces and are to be determined by solving a system of hypersingular integral equations. Once the hypersingular integral equations are solved, the displacements, electric potential, magnetic potential and other quantities of interest such as the crack tip intensity factors may be easily computed in the Laplace transform domain and recovered in the physical space with the help of a suitable algorithm for inverting Laplace transforms.     

A two-dimensional ruin problem on the positive quadrant

In this paper we study the joint ruin problem for two insurance companies that divide between them both claims and premia in some specified proportions (modeling two branches of the same insurance company or an insurance and re-insurance company). Modeling the risk processes of the insurance companies by Cramér-Lundberg processes we obtain the Laplace transform in space of the probability that either of the insurance companies is ruined in finite time. Subsequently, for exponentially distributed claims, we derive an explicit analytical expression for this joint ruin probability by explicitly inverting this Laplace transform. We also provide a characterization of the Laplace transform of the joint ruin time.     

A perturbed risk model with dependence between premium rates and claim sizes

This paper considers a dependent risk model with diffusion for the surplus of an insurer, in which a current premium rate will be adjusted after a claim occurs and the adjusted rate is determined by the amount of the claim. At the same time, the diffusion is changed correspondingly. Using Rouché’s theorem, we first derive the closed-form solution for the Laplace transform of the survival probability in the dependent risk model. Then, using the Laplace transform, we derive a defective renewal equation satisfied by the survival probability. For the exponential claim sizes, we present the explicit recursion expression for the survival probability, by which we can exactly solve the survival probability step-by-step. We also illustrate the influence of the model parameters in the dependent risk model on the survival probability by numerical examples.     

A Laplace variational iteration strategy for the solution of differential equations

The aim of this article is to introduce a novel Laplace variational numerical scheme, based on the variational iteration method (VIM) and Laplace transform, for the solution of certain classes of linear and nonlinear differential equations. The strategy is outlined and then illustrated through a number of test examples. The results assert that this alternative approach yields accurate results, converges rapidly and handles impulse functions and the ones with discontinuities.