High order convergent modified nodal bi-cubic spline collocation method for elliptic partial differential equation

A high order modified nodal bi-cubic spline collocation method is proposed for numerical solution of second-order elliptic partial differential equation subject to Dirichlet boundary conditions. The approximation is defined on a square mesh stencil using nine grid points. The solution of the method exists and is unique. Convergence analysis has been presented. Moreover, the superconvergent phenomena can be seen in proposed one step method. The numerical results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency.     

Modified nodal cubic spline collocation for three-dimensional variable coefficient second order partial differential equations

We formulate a fourth order modified nodal cubic spline collocation scheme for variable coefficient second order partial differential equations in the unit cube subject to nonzero Dirichlet boundary conditions. The approximate solution satisfies a perturbed partial differential equation at the interior nodes of a uniform $N\times N\times N$ partition of the cube and the partial differential equation at the boundary nodes. In the special case of Poisson’s equation, the resulting linear system is solved by a matrix decomposition algorithm with fast Fourier transforms at a cost $O(N^3\log N)$ . For the general variable coefficient diffusion-dominated case, the system is solved using the preconditioned biconjugate gradient stabilized method.     

Generalized quasilinearization method for a second order ordinary differential equation with Dirichlet boundary conditions

We study the existence and approximation of solutions for a nonlinear second order ordinary differential equation with Dirichlet boundary value conditions. We present a generalized quasilinearization technique to obtain a sequence of approximate solutions converging quadratically to a solution.

     

Approximate grid solution of a nonlocal boundary value problem for Laplace’s equation on a rectangle

A nonlocal boundary value problem for Laplace’s equation on a rectangle is considered. Dirichlet boundary conditions are set on three sides of the rectangle, while the boundary values on the fourth side are sought using the condition that they are equal to the trace of the solution on the parallel midline of the rectangle. A simple proof of the existence and uniqueness of a solution to this problem is given. Assuming that the boundary values given on three sides have a second derivative satisfying a Hölder condition, a finite difference method is proposed that produces a uniform approximation (on a square mesh) of the solution to the problem with second order accuracy in space. The method can be used to find an approximate solution of a similar nonlocal boundary value problem for Poisson’s equation.     

Powell–Sabin splines with boundary conditions for polygonal and non-polygonal domains

Powell–Sabin splines are piecewise quadratic polynomials with a global C¹$C^1$-continuity, defined on conforming triangulations. Imposing boundary conditions on such a spline leads to a set of constraints on the spline coefficients. First, we discuss boundary conditions defined on a polygonal domain, before we treat boundary conditions on a general curved domain boundary. We consider Dirichlet and Neumann conditions, and we show that a particular choice of the PS-triangles at the boundary can greatly simplify the corresponding constraints. Finally, we consider an application where the techniques developed in this paper are used: the numerical solution of a partial differential equation by the Galerkin and collocation method.     

On Schwarz's domain decomposition methods for elliptic boundary value problems

Summary. We study the additive and multiplicative Schwarz domain decomposition methods for elliptic boundary value problem of order 2 r based on an appropriate spline space of smoothness . The finite element method reduces an elliptic boundary value problem to a linear system of equations. It is well known that as the number of triangles in the underlying triangulation is increased, which is indispensable for increasing the accuracy of the approximate solution, the size and condition number of the linear system increases. The Schwarz domain decomposition methods will enable us to break the linear system into several linear subsystems of smaller size. We shall show in this paper that the approximate solutions from the multiplicative Schwarz domain decomposition method converge to the exact solution of the linear system geometrically. We also show that the additive Schwarz domain decomposition method yields a preconditioner for the preconditioned conjugate gradient method. We tested these methods for the biharmonic equation with Dirichlet boundary condition over an arbitrary polygonal domain using cubic spline functions over a quadrangulation of the given domain. The computer experiments agree with our theoretical results. Received December 28, 1995 / Revised version received November 17, 1998 / Published online September 24, 1999     

Finite Element Approximation of Elliptic Dirichlet Optimal Control Problems

We develop a priori error analysis for the finite element Galerkin discretization of elliptic Dirichlet optimal control problems. The state equation is given by an elliptic partial differential equation and the finite dimensional control variable enters the Dirichlet boundary conditions. We prove the optimal order of convergence and present a numerical example confirming our results.     

Bernstein Ritz‐Galerkin method for solving an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

In this article, the Ritz‐Galerkin method in Bernstein polynomial basis is implemented to give an approximate solution of a hyperbolic partial differential equation with an integral condition. We will deal here with a type of nonlocal boundary value problem, that is, the solution of a hyperbolic partial differential equation with a nonlocal boundary specification. The nonlocal conditions arise mainly when the data on the boundary cannot be measured directly. The properties of Bernstein polynomial and Ritz‐Galerkin method are first presented, then Ritz‐Galerkin method is used to reduce the given hyperbolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique presented in this article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Singular boundary characteristics of the Hamilton–Jacobi equation

Situations exist in boundary value problems for first order partial differential equations arising in physics (the Hamilton–Jacobi equation), optimal control theory (the Bellman equation) and the theory of differential games (the Isaacs equation) when the value of the required function is not given on a part of the boundary or not at all, or it is not the limit of the (generalized) solution of the problem. Nevertheless, such conditions are required for constructing the solution (by the method of characteristics, for example). It is shown that the required boundary values can be exposed as a specific continuation of the conditions that are known in the boundary submanifolds of the given part of the boundary. This extension of the conditions is accomplished using the characteristic curves starting in a known submanifold of the boundary and running along the boundary. The characteristics are a generalization of the classical characteristics associated with a partial differential equation. They are called singular characteristics, and the theory of these has been developed in a number of the author's papers. After obtaining these “natural” boundary conditions, the solution is constructed using the conventional method of integrating the equations of the classical characteristics. Conditions of the Dirichlet and Neumann type are considered. The technique is illustrated using a numerical example from the theory of differential games containing a number of parameters.     

Stochastic methods for Dirichlet problems

Using an equivalent expression for solutions of second order Dirichlet problems in terms of Ito type stochastic differential equations, we develop a numerical solution method for Dirichlet boundary value problems. It is possible with this idea to solve for solution values of a partial differential equation at isolated points without having to construct any kind of mesh and without knowing approximations for the solution at any other points. Our method is similar to a recently published approach, but differs primarily in the handling of the boundary. Some numerical examples are presented, applying these techniques to model Laplace and Poisson equations on the unit disk. Visiting Professor, Universidad de Salamanca.     

Higher order B-spline differential quadrature rule to approximate generalized Rosenau-RLW equation

In this article, B-spline-based collocation method is employed to approximate the usual and modified Rosenau-RLW nonlinear equations. The weighted extended B-spline (WEB-spline) is used as the modified form of B-spline as the usual B-splines fail to obey the Dirichlet boundary conditions. The WEB method is more general method that allows to discretize the domain into finite number of elements not necessarily start from the boundary points of the domain. Our method omits the linearization process of the nonlinear partial differential equation (PDE). Different cases are discussed by setting the parameter p=2,3,4, and 6 that appears in Rosenau-RLW equations. The error estimation is calculated, which gives good agreement of the exact solution.     

Initial‐Boundary Value Problems for the Defocusing Nonlinear Schrödinger Equation in the Semiclassical Limit

Initial‐boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so‐called unified transform techniques. The main obstruction to applying these methods in practice is that calculation of the spectral transforms of the initial and boundary data requires knowledge of too many boundary conditions, more than are required to make the problem well‐posed. The elimination of the unknown boundary values is frequently addressed in the spectral domain via the so‐called global relation, and types of boundary conditions for which the global relation can be solved are called linearizable. For the defocusing nonlinear Schrödinger equation, the global relation is only known to be explicitly solvable in rather restrictive situations, namely homogeneous boundary conditions of Dirichlet, Neumann, and Robin (mixed) type. General nonhomogeneous boundary conditions are not known to be linearizable. In this paper, we propose an explicit approximation for the nonlinear Dirichlet‐to‐Neumann map supplied by the defocusing nonlinear Schrödinger equation and use it to provide approximate solutions of general nonhomogeneous boundary value problems for this equation posed as an initial‐boundary value problem on the half‐line. Our method sidesteps entirely the solution of the global relation. The accuracy of our method is proven in the semiclassical limit, and we provide explicit asymptotics for the solution in the interior of the quarter‐plane space‐time domain.     

The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass

Some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications. In this paper, a numerical method which employs the Bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions. The properties of Bernstein polynomials, and the operational matrices for integration, differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

The Abreu equation with degenerated boundary conditions

The Abreu equation is a fully nonlinear 4th order partial differential equation that arises from the study of the extremal metrics on toric manifolds. We study the Dirichlet problem of the Abreu equation with degenerated boundary conditions. The solutions provide the Kähler metrics of constant scalar curvature on the complex torus.     

Quadratic spline solution for boundary value problem of fractional order

Fractional differential equations are widely applied in physics, chemistry as well as engineering fields. Therefore, approximating the solution of differential equations of fractional order is necessary. We consider the quadratic polynomial spline function based method to find approximate solution for a class of boundary value problems of fractional order. We derive a consistency relation which can be used for computing approximation to the solution for this class of boundary value problems. Convergence analysis of the method is discussed. Four numerical examples are included to illustrate the practical usefulness of the proposed method.     

Solution of high‐order linear Fredholm integro‐differential equations with piecewise intervals

In this study, a practical matrix method is presented to find an approximate solution for high‐order linear Fredholm integro‐differential equations with piecewise intervals under the initial boundary conditions in terms of Taylor polynomials. The method converts the integro differential equation to a matrix equation, which corresponds to a system of linear algebraic equations. Error analysis and illustrative examples are included to demonstrate the validity and applicability of the technique. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 27: 1327–1339, 2011     

Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions

We present a model for nonlocal diffusion with Dirichlet boundary conditions in a bounded smooth domain. We prove that solutions of properly rescaled nonlocal problems approximate uniformly the solution of the corresponding Dirichlet problem for the classical heat equation.     

Problems for elliptic singular equations with a gradient term

We investigate the homogeneous Dirichlet problem for a class of second-order nonlinear elliptic partial differential equations with singular data. In particular, we study the asymptotic behaviour of the solution near the boundary up to the second order under various assumptions on the growth of the coefficients of the equation.     

On Integral Representation Formulae for Biharmonic Functions on the Ball

Biharmonic functions are solutions of the fourth order partial differential equation ΔΔu = 0. A simple method is proposed for deriving integral representation formulae for these functions u on the n‐dimensional ball. Poisson‐type representations in the setting of Hardy Spaces are obtained for biharmonic functions subject to Dirichlet, Riquier and other boundary conditions. The approach exploits algebraic properties of a first order partial differential operator and its resolvent.     

On an approximate solution of a class of boundary integral equations of the first kind

We construct a method for computing an approximate solution of the boundary integral equation of the first kind corresponding to the Dirichlet boundary value problems for the Helmholtz equation.