Saulyev'S Techniques For Solving A Parabolic Equation With A Non Linear Boundary Specification

In this paper a two-dimensional heat equation is considered. The problem has both Neumann and Dirichlet boundary conditions and one non-local condition in which an integral of the unknown solution u occurs. The Dirichlet boundary condition contains an additional unknown function \mu (t) . In this paper the numerical solution of this equation is treated. Due to the structure of the boundary conditions a reduced one-dimensional heat equation for the new unknown v(\hskip1pty, t) = \vint u(x, y, t)\,\hbox{d}x can be formulated. The resulting problem has a non-local boundary condition. This one-dimensional heat equation is solved by Saulyev's formula. From the solution of this one-dimensional problem an approximation of the function \mu (t) is obtained. Once this approximation is known, the given two-dimensional problem reduces to a standard heat equation with the usual Neumann's boundary conditions. This equation is solved by an extension of the Saulyev's techniques. Results of numerical experiments are presented.     

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 Dirichlet problem for a 3D elliptic equation with two singular coefficients

In the present work, we investigate the Dirichlet problem for a three-dimensional (3D) elliptic equation with two singular coefficients. We find four fundamental solutions of the equation, containing hypergeometric functions of Appell. Then using an “a-b-c” method, the uniqueness for the solution of the Dirichlet problem is proved. Applying a method of Green’s function, we are able to find the solution of the problem in an explicit form. Moreover, decomposition formulas, formulas of differentiation and some adjacent relations for Appell’s hypergeometric functions were used in order to find the explicit solution for the formulated problem.     

A method of harmonic extension for computing the generalized stress intensity factors for Laplace’s equation with singularities

The solution of the Dirichlet problem for Laplace’s equation on a special polygon is harmonically extended to a sector with the center at the singular vertex. This is followed by an integral representation of the extended function in this sector, which is approximated by the mid-point rule. By using the extension properties for the approximate values at the quadrature nodes, a well-conditioned and exponentially convergent, with respect to the number of nodes algebraic system of equations are obtained. These values determine the coefficients of the series representation of the solution around the singular vertex of the polygonal domain, which are called the generalized stress intensity factors (GSIFs). The comparison of the results with those existing in the literature, in the case of Motz’s problem, show that the obtained GSIFs are more accurate. Moreover, the extremely accurate series segment solution is obtained by taking an appropriate number of calculated GSIFs.     

Numerical solution of an integral equation for perpetual Bermudan options

We consider perpetual Bermudan options, which have no expiration and can be exercised every T time units. We use the Green's function approach to write down an integral equation for the value of a perpetual Bermudan call option on an expiration date; this integral equation leads to a Wiener–Hopf problem. We discretize the integral in the integral equation to convert the problem to a linear algebra problem, which is straightforward to solve, and this enables us to find the location of the free boundary and the value of the perpetual Bermudan call. We compare our results to earlier studies which used other numerical methods.     

B-polynomial multiwavelets approach for the solution of Abel's integral equation

A numerical method for solving Abel's integral equation as singular Volterra integral equations is presented. The method is based upon Bernstein polynomial (B-polynomial) multiwavelet basis approximations. The properties of B-polynomial multiwavelets are first presented. These properties are then utilized to reduce the singular Volterra integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

A Computational Method for Stochastic Optimal Control Problems in Financial Mathematics

Principle of optimality or dynamic programming leads to derivation of a partial differential equation (PDE) for solving optimal control problems, namely the Hamilton‐Jacobi‐Bellman (HJB) equation. In general, this equation cannot be solved analytically; thus many computing strategies have been developed for optimal control problems. Many problems in financial mathematics involve the solution of stochastic optimal control (SOC) problems. In this work, the variational iteration method (VIM) is applied for solving SOC problems. In fact, solutions for the value function and the corresponding optimal strategies are obtained numerically. We solve a stochastic linear regulator problem to investigate the applicability and simplicity of the presented method and prove its convergence. In particular, for Merton's portfolio selection model as a problem of portfolio optimization, the proposed numerical method is applied for the first time and its usefulness is demonstrated. For the nonlinear case, we investigate its convergence using Banach's fixed point theorem. The numerical results confirm the simplicity and efficiency of our method.     

Numerical computations for singular semilinear elliptic boundary value problems

The paper studies a class of Dirichlet problems with homogeneous boundary conditions for singular semilinear elliptic equations in a bounded smooth domain in

. A numerical method is devised to construct an approximate Green's function by using radial basis functions and the method of fundamental solutions. An estimate of the error involved is also given. A weak solution of the above given problem is a solution of its corresponding nonlinear integral equation. A computational method is given to find the minimal weak solution U, and the critical index λ* (such that a weak solution U exists for λ < λ*, and U does not exist for λ > λ*).     

Solution of Dirichlet problem for a rectangular region in terms of elliptic functions

In this study, the Green function of the (interior) Dirichlet problem for the Laplace (also Poisson) differential equation in a rectangular domain is expressed in terms of elliptic functions and the solution of the problem is based on the Green function and therefore on the elliptic functions. The method of solution for the Dirichlet problem by the Green function is presented; the Green function and transformation required for the solution of the Dirichlet problem in the rectangular region is found and the problem is solved in the rectangular region. An example for the problem in the rectangular region is given in order to present an application of the solution of Dirichlet problem. The equation is solved first by the known method of separation of variables and then in terms of elliptic functions; the results of both methods are compared. The results are found to be consistent but the advantage of this method is that the solution is obtained in terms of elementary functions.     

Direct and indirect boundary element methods for solving the heat conduction problem

The boundary element method is used to solve the stationary heat conduction problem as a Dirichlet, a Neumann or as a mixed boundary value problem. Using singularities which are interpreted physically, a number of Fredholm integral equations of the first or second kind is derived by the indirect method. With the aid of Green's third identity and Kupradze's functional equation further direct integral equations are obtained for the given problem. Finally a numerical method is described for solving the integral equations using Hermitian polynomials for the boundary elements and constant, linear, quadratic or cubic polynomials for the unknown functions.     

Double layer potential scheme for Dirichlet problems on smooth open arcs

In this paper, we present a new numerical scheme for the Dirichlet problem on the smooth open arcs in the plane. By using the double layer potential to construct the solution of the problem we induce a Fredholm integral equation of the second kind. Existence and uniqueness of the solution of this integral equation are proved in the case when the boundary arc is convex. A numerical example is given to show the validity of the present scheme.     

Computing Green's function of the initial-boundary value problem for the wave equation in a layered cylinder

A new analytical method for the approximate computation of the time-dependent Green's function for the initial-boundary value problem of the three-dimensional wave equation on multi-layered bounded cylinder is suggested in this paper. The method is based on the derivation of eigenvalues and eigenfunctions for an ordinary differential equation with piecewise constant coefficients, and an approximate computation of Green's function in the form of the Fourier series with a finite number of terms relative to the orthogonal set of the derived eigenfunctions. The computational experiment confirms the robustness of the method.     

Synchronization for general complex dynamical networks with sampled-data

In this paper, the sampled-data synchronization control problem is investigated for a class of general complex networks with time-varying coupling delays. A rather general sector-like nonlinear function is used to describe the nonlinearities existing in the network. By using the method of converting the sampling period into a bounded time-varying delay, the addressed problem is first transformed to the problem of stability analysis for a differential equation with multiple time-varying delays. Then, by constructing a Lyapunov functional and using Jensen's inequality, a sufficient condition is derived to ensure the exponential stability of the resulting delayed differential equation. Based on that, the desired sampled-data feedback controllers are designed in terms of the solution to certain linear matrix inequalities (LMIs) that can be solved effectively by using available software. Finally, a numerical simulation example is exploited to demonstrate the effectiveness of the proposed sampled-data control scheme.     

A Nonoverlapping Domain Decomposition Method for Legendre Spectral Collocation Problems

We consider the Dirichlet boundary value problem for Poisson’s equation in an L-shaped region or a rectangle with a cross-point. In both cases, we approximate the Dirichlet problem using Legendre spectral collocation, that is, polynomial collocation at the Legendre–Gauss nodes. The L-shaped region is partitioned into three nonoverlapping rectangular subregions with two interfaces and the rectangle with the cross-point is partitioned into four rectangular subregions with four interfaces. In each rectangular subregion, the approximate solution is a polynomial tensor product that satisfies Poisson’s equation at the collocation points. The approximate solution is continuous on the entire domain and its normal derivatives are continuous at the collocation points on the interfaces, but continuity of the normal derivatives across the interfaces is not guaranteed. At the cross point, we require continuity of the normal derivative in the vertical direction. The solution of the collocation problem is first reduced to finding the approximate solution on the interfaces. The discrete Steklov–Poincaré operator corresponding to the interfaces is self-adjoint and positive definite with respect to the discrete inner product associated with the collocation points on the interfaces. The approximate solution on the interfaces is computed using the preconditioned conjugate gradient method. A preconditioner is obtained from the discrete Steklov–Poincaré operators corresponding to pairs of the adjacent rectangular subregions. Once the solution of the discrete Steklov–Poincaré equation is obtained, the collocation solution in each rectangular subregion is computed using a matrix decomposition method. The total cost of the algorithm is O(N ³), where the number of unknowns is proportional to N ².      

An efficient multilayered shielded microwave circuit analysis method based on neural networks

In previous works, a neural network based technique to analyze multilayered shielded microwave circuits was developed. The method is based on the approximation of the shielded media Green's functions by radial‐basis‐function neural networks (RBFNNs). The trained neural networks, substitute the original Green's functions during the application of the integral equation approach, allowing a faster analysis than the direct solution. In this article, new and important improvements are applied to the training of the RBFNNs, which permit a reduction in the approximation error introduced by the neural networks. Furthermore, outstanding time reductions in the analysis of printed circuits are achieved, clearly outperforming the former technique. The main improvement consists on a better processing of the Green's function singularity near the source. The singularity produces rapid variations near the source that makes difficult the neural network training. In this work, the singularity is extracted in a more suitable fashion than in previous works. The functions resulting from the singularity extraction present a smooth behavior, so they can be easily approximated by neural networks. In addition, a new subdivision strategy for the input space is proposed to efficiently train the neural networks. Two practical microwave filters are analyzed using the new techniques. Comparisons with measured results are also presented for validation. © 2010 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2010.     

Integral Equation Formulation of the Biharmonic Dirichlet Problem

We present a novel integral representation for the biharmonic Dirichlet problem. To obtain the representation, the Dirichlet problem is first converted into a related Stokes problem for which the Sherman–Lauricella integral representation can be used. Not all potentials for the Dirichlet problem correspond to a potential for Stokes flow, and vice-versa, but we show that the integral representation can be augmented and modified to handle either simply or multiply connected domains. The resulting integral representation has a kernel which behaves better on domains with high curvature than existing representations. Thus, this representation results in more robust computational methods for the solution of the Dirichlet problem of the biharmonic equation and we demonstrate this with several numerical examples.     

Modified method for determining an approximate solution of the Fredholm–Volterra integral equations by Taylor’s expansion

A modified method for determining an approximate solution of the Fredholm–Volterra integral equations of the second kind is developed. Via Taylor’s expansion of the unknown function, the integral equation to be solved is approximately transformed into a system of linear equations for the unknown and its derivatives, which can be dealt with in an easy way. The obtained nth-order approximate solution is of high accuracy, and is exact for polynomials of degree n. In particular, an approximate solution with satisfactory accuracy of the weakly singular Volterra integral equation is also given. The efficiency of the method is illustrated by some numerical examples.     

A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem

In this paper we propose a new a posteriori error estimator for a boundary element solution related to a Dirichlet problem with a second order elliptic partial differential operator. The method is based on an approximate solution of a boundary integral equation of the second kind by a Neumann series to estimate the error of a previously computed boundary element solution. For this one may use an arbitrary boundary element method, for example, a Galerkin, collocation or qualocation scheme, to solve an appropriate boundary integral equation. Due to the approximate solution of the error equation the proposed estimator provides high accuracy. A numerical example supports the theoretical results. Received: June 1999 / Accepted: September 1999     

Solution of Electromagnetic Scattering Problems Involving Three-Dimensional Homogeneous Dielectric Objects by the Single Integral Equation Method

The problem of electromagnetic scattering by a homogeneous dielectric object is usually formulated as a pair of coupled integral equations involving two unknown currents on the surface S of the object. In this paper, however, the problem is formulated as a single integral equation involving one unknown current on S. Unique solution at resonance is obtained by using a combined field integral equation. The single integral equation is solved by the method of moments using a Galerkin test procedure. Numerical results for a dielectric sphere are in good agreement with the exact results. Furthermore, the single integral equation method is shown to have superior convergence speed of iterative solution compared with the coupled integral equations method.