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In the present paper, the initial value problem for the third order partial differential equation with time delay is studied. The first and second order of accuracy difference schemes for the numerical solution of the third order partial differential equation with time delay are presented. The illustrative numerical results are provided. [ABSTRACT FROM AUTHOR]

This paper poses a magnetic field problem in cylindrical coordinate for two regions with different permeability for each one. The linear partial differential equation governing this problem is in the form of Laplace and Poisson equation. This problem is then solved using classical Gauss-Seidel Algorithm for the Finite difference (FD) solution of linear partial differential equation. In order to obtain adequate solution for a reasonable number of grid points for the regions under consideration a considerable amount of time will take for the program to converge. The paper presents a different technique known as Weighted Multi-Grid which will speed up the convergence process. In this method, the solution to the differential equation between two grid points for obtaining the initial condition is considered to be linear in nature with different weight for the value of each grid point. This problem is then solved for the minimum number of grid points let’s say one point in each region plus the boundary points. The initial guess for each new point for every level of computation is found as the weighted average of the two adjacent points. It then continues with finding the optimum weighting values for Laplace’s or Poisson’s equation. The main contribution is made by regarding the effect of the optimum initial weighted values of the variable vector in the convergence time for the Gauss-Seidel algorithm. [ABSTRACT FROM AUTHOR]

Consider the following problem related to an ordinary differential equation: 'For given constans ua,ub,M,N, function f(x) in [a,b], find ∂u/∂n at x = a,b which satisfies u"+Mu′+Nu = f in (a,b),u(a) = ua,u(b) = ub'. This kind of problem is called the "Dirichlet-Neumann map problem". This problem is usually solved in two steps. The solution is obtained in the first steps. In the second step, u′(a), u′(b) is computed by differentiating the solution. However, this two-step procedure is inefficient because the solution in (a,b) obtained in the first step is not essentially required. In this paper, the author presents a new strategy for obtaining Neumann data directly via a boundary integral equation formulation. Using this strategy, an explicit analytical expression of the Dirichlet-Neumann map of this problem can be directly obtained by solving 2 × 2 matrices. Furthermore, an extension of the strategy to partial differential equations in two-dimensional space and numerical algorithms is also presented. [ABSTRACT FROM AUTHOR]

A comparison study of algorithm on a rectangular collocated mesh is conducted for variable density Navier-Stokes equations with continuous surface forces. The algorithms include the original projection method (AI-TI) with the surface force calculated only in the predictor steps, the named balanced-force projection method (AII-TIV) with the surface force and the pressure gradient calculated together, and a consistent projection method (AIII-TVII) developed in this paper. Detailed comparisons are also conducted among the techniques for calculation of the pressure gradient and surface force at a cell center. A consistent projection method updates the velocity at a cell center in a very difference way with the balanced-force projection formula. A conservative interpolation is used to update the velocity a cell center, which is further used to obtain the sum of the pressure gradient and the surface force. [ABSTRACT FROM AUTHOR]

In this paper we will present an effective numerical method for the Black-Scholes equation with transaction costs for the limiting price u(s, t;a). The technique combines the Rothe method with a two-grid (coarse-fine) algorithm for computation of numerical solutions to initial boundary-values problems to this equation. Numerical experiments for comparison the accuracy ant the computational cost of the method with other known numerical schemes are discussed. [ABSTRACT FROM AUTHOR]

This paper deals with numerical methods for high frequency wave scattering. It introduces a new hybrid technique that couples a directional fast multipole method for a subsection of a scattering surface to an asymptotic formulation over the rest of the scattering domain. The directional fast multipole method is new and highly efficient for the solution of the boundary integral formulation of a general scattering problem but it requires at least a few unknowns per wavelength on the boundary. The asymptotic method that was introduced by Bruno and collaborators requires much fewer unknowns. On the other hand the scattered field must have a simple structure. Hybridization of these two methods retains their best properties for the solution of the full problem. Numerical examples are given for the solution of the Helmholtz equation in two space dimensions. [ABSTRACT FROM AUTHOR]