Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This is the author's accepted manuscript. The final published article is available from the link below. Copyright @ 2012 Taylor & Francis.This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This is the author's accepted manuscript. The final published article is available from the link below. Copyright @ 2012 Taylor & Francis.This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods

A reliable iterative method for Cauchy problems

In the present paper, the new iterative method proposed by Daftardar-Gejji and Jafari (NIM or DJM) [V. Daftardar-Gejji, H. Jafari, An iterative method for solving non linear functional equations, J. Math. Anal. Appl. 316 (2006) 753-763] is used to solve the Cauchy problems. In this iterative method the solution is obtained in the series form that converge to the exact solution with easily computed components. The results demonstrate that the method has many merits such as being derivative-free, overcome the difficulty arising in calculating calculating Adomian polynomials to handle the nonlinear terms in Adomian Decomposition Method (ADM). It does not require to calculate Lagrange multiplier in Variational Iteration Method (VIM) and no needs to construct a homotopy and solve the corresponding algebraic equations in Homotopy Perturbation Method (HPM) and can be easily comprehended with only a basic knowledge of Calculus. The results show that the present method is very effective, simple and provide the analytic solutions. The software used for the calculations in this study was MATHEMATICA r 8.0.Keywords: New iterative method; Cauchy problems; Inviscid Burger’s equation; transport equatio

Analytical approximate solutions for linear and nonlinear Volterra integral and integrodifferential equations and some applications for the Lane-Emden equations using a power series method

In the present paper, we have presented a recursive method namely the Power Series Method (PSM)to solve first the linear and nonlinear Volterra integral and integro-differential equations. The PSM isemployed then to solve resulting equations of the nonlinear Volterra integral and integro-differentialforms of the Lane-Emden equations. The Volterra integral and integro-differential equations formsof the Lane-Emden equation overcome the singular behavior at the origin x = 0. Some examplesare solved and different cases of the Lane-Emden equations of first kind are presented. The resultsdemonstrate that the method has many merits such as being derivative-free, and overcoming thedifficulty arising in calculating Adomian polynomials to handle the nonlinear terms in Adomian DecompositionMethod (ADM). It does not require to calculate Lagrange multiplier as in VariationalIteration Method (VIM) and no need to construct a homotopy as in Homotopy Perturbation Method(HPM). The results prove that the present method is very effective and simple and does not requireany restrictive assumptions for nonlinear terms. The software used for the calculations in this studywas MATHEMATICA

Analytic approximate solutions of Volterra’s population and some scientific models by power series method

In this paper, we have implement an analytic approximate method based on power series method (PSM) to obtain asolutions for Volterra’s population model of population growth of a species in a closed system. The numerical solution isobtained by combining the PSM and Pad´e technique. The Pad´e approximation that often show superior performance overseries approximation are effectively used in the analysis to capture essential behavior of the population u(t) of identicalindividuals. The results demonstrate that the method has many merits such as being derivative-free, overcome the difficultyarising in calculating Adomian polynomials to handle the nonlinear terms in Adomian Decomposition Method (ADM).It does not require to calculate Lagrange multiplier as in Variational Iteration Method (VIM) and no needs to construct ahomotopy and solve the corresponding algebraic equations as in Homotopy Perturbation Method (HPM). Moreover, weused this method to solve some scientific models, namely, the hybrid selection model, the Riccati model and the logisticmodel to provide the analytic solutions. The obtained analytic approximate solutions of applying the PSM is in fullagreement with the results obtained with those methods available in the literature. The software used for the calculationsin this study was MATHEMATICAr 8.0