A parallel implementation of overlapping Schwarz method for the dual reciprocity method

We present a parallel algorithm for the overlapping domain decomposition boundary integral equation method for two dimensional partial differential equations. In addition to the improvement of the ill-conditioning and the computational efficiency achieved by domain partitioning, using a parallel computer with p processors can offer up to p times efficiency. Assuming direct solution is used throughout, partitioning the domain into p subregions and employing a processor for each subproblem, overall, result in p² times efficiency over using a single domain and a single processor, taking into account that a sequential algorithm of the underlying method can improve the computational efficiency at least p times over using a single domain. Some numerical results showing the efficiency of the parallel technique will be presented.     

Analysis and numerical solution of Benjamin-Bona-Mahony equation with moving boundary

In this work we present the existence, the uniqueness and numerical solutions for a mathematical model associated with equations of Benjamin-Bona-Mahony type in a domain with moving boundary. We apply the Galerkin method, multiplier techniques, energy estimates and compactness results to obtain the existence and uniqueness. For numerical solutions, we shall employ the finite element method together with the Crank-Nicolson method. Some numerical experiments are presented to show the moving boundary for the problem.     

Solutions of a class of partial differential equations with application to the Black-Scholes equation

In this paper we shall derive the solutions of a class of partial differential equations and its application to the Black-Scholes equation.     

An exact solution of a quasilinear Fisher equation in cylindrical coordinates

Lie symmetry method is applied to analyse Fisher equation in cylindrical coordinates. Symmetry algebra is found and symmetry invariance is used to reduce the equation to a first-order ODE. The first-order ODE is further analysed to obtain exact solution of Fisher equation in explicit form.     

A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type

In this paper, we develop an accurate and efficient Chebyshev wavelets method for solution of partial differential equations with boundary conditions of the telegraph type. In the proposed method we have employed mutually the operational matrices of integration and differentiation to get numerical solutions of such equations. The power of this manageable method is confirmed. Moreover the use of Chebyshev wavelet is found to be accurate, simple and fast.     

Some solutions of the linear and nonlinear Klein-Gordon equations using the optimal homotopy asymptotic method

We investigate the effectiveness of the Optimal Homotopy Asymptotic Method (OHAM) in solving time dependent partial differential equations. To this effect we consider the homogeneous, non-homogeneous, linear and nonlinear Klein-Gordon equations with boundary conditions. The results reveal that the method is explicit, effective, and easy to use.     

Moving planes, moving spheres, and a priori estimates

In this paper, we use the method of moving planes and the method of moving spheres to obtain a priori estimates for the solutions of semi-linear elliptic equations.Using the ‘method of moving planes’, we establish a sharper estimate on the solutions for prescribing scalar curvature equations with indefinite curvature functions and thus generalized a resent result of Lin.Applying ‘the method of moving spheres’, we give a different proof for a well-known sup+inf inequality established by Brezis, Li and Shafrir.     

Some remarks on solutions to an overdetermined elliptic problem in divergence form in a ball

We prove the existence of a continuum of non-radial pairs (k,u) solutions to the following overposed problem div in B _r , u = 0 and on ∂B _r , where B _r is the Euclidean ball centered at zero of radius r in . Dedicato a Marco e Andrea Provera.     

Computing the first passage time density of a time-dependent Ornstein–Uhlenbeck process to a moving boundary

In this paper we use the method of images to derive the closed-form formula for the first passage time density of a time-dependent Ornstein–Uhlenbeck process to a parametric class of moving boundaries. The results are then applied to develop a simple, efficient and systematic approximation scheme to compute tight upper and lower bounds of the first passage time density through a fixed boundary.     

An exponential time differencing method of lines for the Burgers and the modified Burgers equations

A second–order exponential time differencing scheme using the method of lines is developed in this article for the numerical solution of the Burgers and the modified Burgers equations. For each case, the resulting nonlinear system is solved explicitly using a modified predictor‐corrector method. The efficiency of the method introduced is tested by comparing experimental results with others selected from the available literature.     

Solution of free vibration equations of beam on elastic soil by using differential transform method

Free vibration differential equations of motion of one end fixed, the other simply supported and axial loaded beams on elastic soil is solved using differential transform method (DTM), analytical solution and frequency factors are obtained.     

New results on the analytic summation of Adomian series for some classes of differential and integral equations

In this paper, we demonstrate that an infinite number of successive integration by parts can be written in a closed form. This closed form can be used directly to prove that the analytic summation of Adomian series becomes identical to the closed form solution for some classes of differential and integral equations.     

A characteristic line based method to build finite-dimensional medoels of heat exchangers

surface heat exchngers are typical simulated with simplified models obtained through segmentation of the heat exchanging fluid path in a number of consecutive lumps In order to aviod major drawbacks of this approach, which may be very misleading for control design purpose, we propose a method, based on the intergration of the PDE system by the method of characteristic lines, for the construction of numerical heat exchangers models. It can be proved that the time response of such new models is indeed rid of parasitic oscillation and suitable for the understanding of complex dynamic phenomena occurring and suitable for the understanding of complex dynamic phenomena occurring in long residence time heat exchangers, both with one- and two- phase flow. In this paper, particular attention is paid to the problem of generating finite dimensional dynamic system by application of the characteristic lines method and computing the frequency responce of such models. Actually, since the characteristic lines method is not naturally is not straightforward to define Finally, the accuracy of CL models is compared with classical models of comparable complexity, with special reference to real application cases, taken from the power generation field.     

Method of lines solutions of the parabolic inverse problem with an overspecification at a point

The present work is motivated by the desire to obtain numerical solution to a quasilinear parabolic inverse problem. The solution is presented by means of the method of lines. Method of lines is an alternative computational approach which involves making an approximation to the space derivatives and reducing the problem to a system of ordinary differential equations in the variable time, then a proper initial value problem solver can be used to solve this ordinary differential equations system. Some numerical examples and also comparison with finite difference methods will be investigated to confirm the efficiency of this procedure.     

On the selection of a good value of shape parameter in solving time-dependent partial differential equations using RBF approximation method

Radial basis function method is an effective tool for solving differential equations in engineering and sciences. Many radial basis functions contain a shape parameter c which is directly connected to the accuracy of the method. Rippa [1] proposed an algorithm for selecting good value of shape parameter c in RBF-interpolation. Based on this idea, we extended the proposed algorithm for selecting a good value of shape parameter c in solving time-dependent partial differential equations.     

On application of the Lanczos method to solution of some partial differential equations

Let A be a square symmetric n × n matrix, φ be a vector from ⁿ, and f be a function defined on the spectral interval of A. The problem of computation of the vector u = f(A)φ arises very often in mathematical physics.

We propose the following method to compute u. First, perform m steps of the Lanczos method with A and φ. Define the spectral Lanczos decomposition method (SLDM) solution as u_m = φ Qf(H)e₁, where Q is the n × m matrix of the m Lanczos vectors and H is the m × m tridiagonal symmetric matrix of the Lanczos method. We obtain estimates for u − u_m that are stable in the presence of computer round-off errors when using the simple Lanczos method.

We concentrate on computation of exp(− tA)φ, when A is nonnegative definite. Error estimates for this special case show superconvergence of the SLDM solution. Sample computational results are given for the two-dimensional equation of heat conduction. These results show that computational costs are reduced by a factor between 3 and 90 compared to the most efficient explicit time-stepping schemes. Finally, we consider application of SLDM to hyperbolic and elliptic equations.     

On the structure of fractional degree vortices in a spinor Ginzburg-Landau model

We consider a Ginzburg-Landau functional for a complex vector order parameter Ψ=(ψ₊,ψ₋), whose minimizers exhibit vortices with half-integer degree. By studying the associated system of equations in R² which describes the local structure of these vortices, we show some new and unconventional properties of these vortices. In particular, one component of the solution vanishes, but the other does not. We also prove the existence and uniqueness of equivariant entire solutions, and provide a second proof of uniqueness, valid for a large class of systems with variational structure.     

Novel numerical techniques based on Fokas transforms,for the solution of initial boundary value problems

The unified transform method of A. S. Fokas has led to important new developments, regarding the analysis and solution of various types of linear and nonlinear PDE problems. In this work we use these developments and obtain the solution of time-dependent problems in a straightforward manner and with such high accuracy that cannot be reached within reasonable time by use of the existing numerical methods. More specifically, an integral representation of the solution is obtained by use of the A. S. Fokas approach, which provides the value of the solution at any point, without requiring the solution of linear systems or any other calculation at intermediate time levels and without raising any stability problems. For instance, the solution of the initial boundary value problem with the non-homogeneous heat equation is obtained with accuracy 10⁻¹⁵, while the well-established Crank–Nicholson scheme requires 2048 time steps in order to reach a 10⁻⁸ accuracy.     

A remark on an integral equation via the method of moving spheres

In this paper, we study the following integral equation: