A compression technique for the boundary integral equation reduced from Helmholtz equation

Applying the trigonometric wavelets and the multiscale Galerkin method, we investigate the numerical solution of the boundary integral equation reduced from the exterior Dirichlet problem of Helmholtz equation by the potential theory. Consequently, we obtain a matrix compression strategy, which leads us to a fast algorithm. Our truncated treatment is simple, the computational complexity and the condition number of the truncated coefficient matrix are bounded by a constant. Furthermore, the entries of the stiffness matrix can be evaluated from the Fourier coefficients of the kernel of the boundary integral equation. Examples given for demonstrating our numerical method shorten the runtime obviously.     

The exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions

In this article, a new integral equation is derived to solve the exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions, and existence and uniqueness is proven for all wave numbers. We apply the boundary element collocation method to solve the system of Fredholm integral equations of the second kind, where we use constant interpolation. We observe superconvergence at the collocation nodes and illustrate it with numerical results for several smooth surfaces.     

Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with a variable coefficient

In this paper, a numerical implementation of a direct united boundary-domain integral equation (BDIE) related to the Neumann boundary value problem for a scalar elliptic partial differential equation with a variable coefficient is discussed. The BDIE is reduced to a uniquely solvable one by adding an appropriate perturbation operator. The mesh-based discretization of the BDIEs with quadrilateral domain elements leads to a system of linear algebraic equations (discretized BDIE). Then, the system is solved by LU decomposition and Neumann iterations. Convergence of the iterative method is discussed in relation to the distribution of eigenvalues of the corresponding discrete operators calculated numerically.     

A matrix formulation to the wave equation with non-local boundary condition

Recently, various sequential numerical schemes have been proposed for the solution of non-classical hyperbolic initial value problems which involve non-local integral terms over the spatial domain. In this paper, we focus on the wave equation with the non-local boundary condition. Two matrix formulation techniques based on the shifted standard and shifted Chebyshev bases are proposed for the numerical solution. Several numerical examples and also some comparisons with another methods will be investigated to confirm the efficiency of this procedure.     

Global and local Trefftz boundary integral formulations for sound vibration

The sound-pressure field harmonically varying in time is governed by the Helmholtz equation. The Trefftz boundary integral equation method is presented to solve two-dimensional boundary value problems. Both direct and indirect BIE formulations are given. Non-singular Trefftz formulations lead to regular integrals counterpart to the conventional BIE with the singular fundamental solution. The paper presents also the local boundary integral equations with Trefftz functions as a test function. Physical fields are approximated by the moving least-square in the meshless implementation. Numerical results are given for a square patch test and a circular disc.     

Richardson extrapolation applied to boundary element method results in a Dirichlet problem for the Laplace equation

Richardson extrapolation is used to improve the accuracy of the numerical solutions for the normal boundary flux and for the interior potential resulting from the boundary element method. The boundary integral equations arise from a direct boundary integral formulation for solving a Dirichlet problem for the Laplace equation. The Richardson extrapolation is used in two different applications: (i) to improve the accuracy of the collocation solution for the normal boundary flux and, separately, (ii) to improve the solution for the potential in the domain interior. The main innovative aspects of this work are that the orders of dominant error terms are estimated numerically, and that these estimates are then used to develop an a posteriori technique that predicts if the Richardson extrapolation results for applications (i) and (ii) are reliable. Numerical results from test problems are presented to demonstrate the technique.     

Nonreflecting boundary condition for the Helmholtz equation

To solve the Helmholtz equation in an infinite three-dimensional domain a spherical artificial boundary is introduced to restrict the computational domain Ω. To determine the nonreflecting boundary condition on ∂Ω, we start with a finite number of spherical harmonics for the Helmholtz equation. With a precise choice of (primary) nodes on the sphere, the theorem on Gauss-Jordan quadrature establishes the discrete orthogonality of the spherical harmonics when summed over these nodes. An approximate nonreflecting boundary condition for the Helmholtz equation follows readily upon solving the exterior Dirichlet problem. The accuracy of the boundary condition is determined using a point source, and the computational results are presented for the scattering of a wave from a sphere.     

Efficient solution of a class of partial integro-differential equation in reproducing kernel space

In [Y.l. Wang, T. Chaolu, Z. Chen, Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems, Int. J. Comput. Math. 87(2) (2010), pp. 367–380], we present three algorithms to solve a class of ordinary differential equations boundary value problems in reproducing kernel space. It is worth noting that our methods can get the solution of partial integro-differential equation. In this note, we use method 2 [M. Dehghan, Solution of a partial integro-differential equation arising from viscoelasticity, Int. J. Comput. Math. 83(1) (2006), pp. 123–129] to solve a class of partial integro-differential equation in reproducing kernel space. Numerical example shows our method is effective and has high accuracy.     

The boundary element-free method for 2D interior and exterior Helmholtz problems

The boundary element-free method (BEFM) is developed in this paper for numerical solutions of 2D interior and exterior Helmholtz problems with mixed boundary conditions of Dirichlet and Neumann types. A unified boundary integral equation is established for both interior and exterior problems. By using the improved interpolating moving least squares method to form meshless shape functions, mixed boundary conditions in the BEFM can be satisfied directly and easily. Detailed computational formulas are derived to compute weakly and strongly singular integrals over linear and higher order integration cells. Three numerical integration procedures are developed for the computation of strongly singular integrals. Numerical examples involving acoustic scattering and radiation problems are presented to show the accuracy and efficiency of the meshless method.     

Three-dimensional eigenvalue analysis of the Helmholtz equation by multiple reciprocity boundary element method

The eigenvalue of the three-dimensional Helmholtz equation is determined efficiently by extending the previously developed method for the two-dimensional problem. Boundary integral equation is formulated in the realm of the multiple reciprocity method, using higher order fundamental solutions for the Laplace equation; yielding polynomial coefficient matrices in terms of unknown wavenumber (eigenvalue). The Newton iteration method with the help of LU decomposition is employed to search eigenvalue, which can reduce the computational task significantly.     

A direct boundary integral equation method for the numerical construction of harmonic functions in three-dimensional layered domains containing a cavity

We consider boundary value problems for the Laplace equation in three-dimensional multilayer domains composed of an infinite strip layer of finite height and a half-space containing a bounded cavity. The unknown (harmonic) function satisfies the Neumann boundary condition on the exterior boundary of the strip layer (i.e. at the bottom of the first layer), the Dirichlet, Neumann or Robin boundary condition on the boundary surface of the cavity and the corresponding transmission (matching) conditions on the interface layer boundary. We reduce this boundary value problem to a boundary integral equation over the boundary surface of the cavity by constructing Green's matrix for the corresponding transmission problem in the domain consisting of the infinite layer and the half-space (not with the cavity). This direct integral equation approach leads, for any of the above boundary conditions, to boundary integral equations with a weak singularity on the cavity. The numerical solution of this equation is realized by Wienert's [Die Numerische approximation von Randintegraloperatoren für die Helmholtzgleichung im R ³, Ph.D. thesis, University of Göttingen, Germany, 1990] method. The reduction of the problem, originally set in an unbounded three-dimensional region, to a boundary integral equation over the boundary of a bounded domain, is computationally advantageous. Numerical results are included for various boundary conditions on the boundary of the cavity, and compared against a recent indirect approach [R. Chapko, B.T. Johansson, and O. Protsyuk, On an indirect integral equation approach for stationary heat transfer in semi-infinite layered domains in R ³ with cavities, J. Numer. Appl. Math. (Kyiv) 105 (2011), pp. 4–18], and the results obtained show the efficiency and accuracy of the proposed method. In particular, exponential convergence is obtained for smooth cavities.     

The dual reciprocity boundary element formulation for nonlinear diffusion problems

This paper presents an extension of the dual reciprocity boundary element method (DRBEM) to deal with nonlinear diffusion problems in which thermal conductivity, specific heat, and density coefficients are all functions of temperature. The DRBEM, recently applied to the solution of problems governed by parabolic and hyperbolic equations, consists in the transformation of the differential equation into an integral equation involving boundary integrals only, the solution of which is achieved by employing a standard boundary element discretization coupled with a two-level finite difference time integration scheme. Contrary to previous formulations for the diffusion equation, the dual reciprocity BEM utilizes the well-known fundamental solution to Laplace's equation, which is space-dependent only. This avoids complex time integrations that normally appear in formulations employing time-dependent fundamental solutions, and permits accurate numerical solutions to be obtained in an efficient way. For nonlinear problems, the integral of conductivity is introduced as a new variable to obtain a linear diffusion equation in the Kirchhoff transform space. This equation involves a modified time variable which is itself a function of position. The problem is solved in an iterative way by using an efficient Newton-Raphson technique which is shown to be rapidly convergent.     

Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation

In this paper, the Cauchy problem for the modified Helmholtz equation in a rectangular domain is investigated. We use a quasi-reversibility method and a truncation method to solve it and present convergence estimates under two different a priori boundedness assumptions for the exact solution. The numerical results show that our proposed numerical methods work effectively.     

Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes

The meshless local boundary integral equation (LBIE) method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular and circular sections with non-conducting walls. Computations have been carried out for different Hartmann numbers and at various time levels. The method is based on the local boundary integral equation with moving least squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain, are utilized to approximate the interior and boundary variables. A time stepping method is employed to deal with the time derivative. Finally, numerical results are presented to show the behaviour of velocity and induced magnetic field.     

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.     

Numerical scheme based on similarity reductions for the regularized long wave equation

Numerical solution based on similarity reductions for partial differential equations used to get the numerical scheme for the regularized long wave (RLW) equation. The similarity reductions for RLW equation are obtained locally on subdomains defined by the classical three-point stencil. The ordinary differential equation, which deduced from the similarity reduction can be linearized, integrated analytically and then obtain the solution. This approch eliminates the difficulties associated with boundary conditions for the similarity reduction over the whole solution domain. Numerical results are obtained for test problem. The computed results using our scheme confirm the accuracy of our scheme.     

Exact and fast numerical algorithms for the stochastic wave equation

On the basis of integral representations we propose fast numerical methods to solve the Cauchy problem for the stochastic wave equation without boundaries and with Dirichlet boundary conditions. The algorithms are exact in a probabilistic sense.     

Solution of discontinuous interior Helmholtz problems by the boundary and shell element method

The Helmholtz equation governing an interior domain with shell discontinuities is not efficiently solvable by the traditional boundary element method. In this paper it is shown how the Helmholtz equation can be recast as an integral equation known as the boundary and shell integral equation. The application of collocation to the integral equation gives rise to a method termed the boundary and shell element method.

The associated problem of finding the eigenvalues and eigenfunctions of the Helmholtz equation in a discontinuous domain via the same method is also considered. This leads to a non-linear eigenvalue problem. Such a problem may be solved through polynomial interpolation of the matrix components. In this paper methods for solving the Helmholtz equation and the associated eigenvalue problem are implemented and applied to a test problem.     

A combined boundary integral and vortex method for the numerical study of three-dimensional fluid flow systems

Vortex methods using vorticity–velocity formulations have become an increasingly powerful and popular means of studying complex fluid flow systems. The problem of combining an integral equation method and grid-free discrete vortex method (DVM) when studying three-dimensional wall-bounded flows is considered. While the normal boundary condition is satisfied by means of a boundary integral equation (BIE), we also consider the problem of recovering pressure from given vorticity and velocity fields when using Lagrangian DVMs in terms of a BIE. For validation purposes, vortical flow past a sphere and past a flat plate are considered, for which the commonly used method of images is available. Results of near-wall boundary-layer flow simulations are then presented as an illustration of the numerical scheme. The importance of hairpin vortices is highlighted. Finally, results on wall compliance fluid flow are displayed emphasizing the versatility of the numerical method.     

Numerical approach for solving fractional Fredholm integro-differential equation

In this article, we present a new method which is based on the Taylor Matrix Method to give approximate solution of the linear fractional Fredholm integro-differential equations. This method is based on first taking the truncated Taylor expansions of the functions in the linear fractional differential part and Fredholm integral part then, substituting their matrix forms into the equation. We solve this matrix equation with the assistance of Maple 13. In addition, illustrative examples are presented to demonstrate the effectiveness of the proposed method.