A new high order compact off-step discretization for the system of 3D quasi-linear elliptic partial differential equations

We present a new fourth order compact finite difference scheme based on off-step discretization for the solution of the system of 3D quasi-linear elliptic partial differential equations subject to appropriate Dirichlet boundary conditions. We also develop new fourth order methods to obtain the numerical solution of first order normal derivatives of the solution. In all the cases, we use only 19-grid points of a single computational cell to compute the problem. The proposed methods are directly applicable to singular problems and the problems in polar coordinates, without any modification required unlike the previously developed high order schemes of [14] and [30]. We discuss the convergence analysis of the proposed method in details. Many physical problems are solved and comparative results are given to illustrate the usefulness of the proposed methods.     

Discrete weighted least-squares method for the Poisson and biharmonic problems on domains with smooth boundary

In this article a discrete weighted least-squares method for the numerical solution of elliptic partial differential equations exhibiting smooth solution is presented. It is shown how to create well-conditioned matrices of the resulting system of linear equations using algebraic polynomials, carefully selected matching points and weight factors. Two simple algorithms generating suitable matching points, the Chebyshev matching points for standard two-dimensional domains and the approximate Fekete points of Sommariva and Vianello for general domains, are described. The efficiency of the presented method is demonstrated by solving the Poisson and biharmonic problems with the homogeneous Dirichlet boundary conditions defined on circular and annular domains using basis functions in the form satisfying and in the form not satisfying the prescribed boundary conditions.     

A new high accuracy finite difference discretization for the solution of 2D nonlinear biharmonic equations using coupled approach

In this article, using coupled approach, we discuss fourth order finite difference approximation for the solution of two dimensional nonlinear biharmonic partial differential equations on a 9‐point compact stencil. The solutions of unknown variable and its Laplacian are obtained at each internal grid points. This discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions. We require only system of two equations to obtain the solution and its Laplacian. The proposed fourth order method is used to solve a set of test problems and produce high accuracy numerical solutions. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

On the combination of some semi‐discretization methods and boundary integral equations for the numerical solution of initial boundary value problems

We consider initial boundary value problems for the homogeneous differential equation of hyperbolic or parabolic type in the unbounded two‐ or three‐dimensional spatial domain with the homogeneous initial conditions and with Dirichlet or Neumann boundary condition. The numerical solution is realized in two steps. At first using the Laguerre transformation or Rothe's method with respect to the time variable the non‐stationary problem is reduced to the sequence of boundary value problems for the non‐homogeneous Helmholtz equation. Further we construct the special integral representation for solutions and obtain the sequence of boundary integral equations (without volume integrals). For the full‐discretization of integral equations we propose some projection methods.     

Approximate analytical solutions of a class of boundary layer equations over nonlinear stretching surface

Third order nonlinear ordinary differential equations, subject to appropriate boundary conditions arising in fluid dynamics, are solved using three different methods viz., the Dirichlet series, method of stretching of variables, and asymptotic function method. Similarity transformations are used to convert the governing partial differential equations into nonlinear ordinary differential equations. The numerical results obtained from the above methods for various problems are given in terms of skin friction. Our study revealed that the results obtained from these methods agree well with those of direct numerical simulation of ordinary differential equations. Also, these methods have advantages over pure numerical methods in obtaining derived quantities such as velocity profile accurately for various values of the parameters at a stretch.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

A posteriori error estimation and adaptive mesh-refinement techniques

We analyse three different a posteriori error estimators for elliptic partial differential equations. They are based on the evaluation of local residuals with respect to the strong form of the differential equation, on the solution of local problems with Neumann boundary conditions, and on the solution of local problems with Dirichlet boundary conditions. We prove that all three are equivalent and yield global upper and local lower bounds for the true error. Thus adaptive mesh-refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities. Some numerical examples prove the efficiency of the error estimators and the mesh-refinement techniques.     

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.     

The numerical solution of linear time-dependent partial differential equations by the Laplace transform and finite difference approximations

A feasible method is presented for the numerical solution of a large class of linear partial differential equations which may have source terms and boundary conditions which are time-varying. The Laplace transform is used to eliminate the time-dependency and to produce a subsidiary equation which is then solved in complex arithmetic by finite difference methods. An effective numerical Laplace transform inversion algorithm gives the final solution at each spatial mesh point for any specified set of values of t. The single-step property of the method obviates the need to evaluate the solution at a large number of unwanted intermediate time points. The method has been successfully applied to a variety of test problems and, with two alternative numerical Laplace transform inversion algorithms, has been found to give results of good to excellent accuracy. It is as accurate as other established finite difference methods using the same spatial grid. The algorithm is easily programmed and the same program handles equations of parabolic and hyperbolic type.     

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.     

Influence of several factors in the generalized finite difference method

In this paper it is possible to appreciate the great efficiency of the generalized finite difference method (GFD), that is to say with an irregular arrangements of nodes, to solve second-order partial differential equations which represent the behaviour of many physical processes. The method solves any type of second-order differential equation, in any type of domain and boundary condition (Dirichlet, Neumann and mixed), and immediately obtains the values of derivatives of the nodes through the application of the formulae in differences obtained. This paper analyses the influences of key parameters of the method, such as the number of nodes of the star, the arrangement of the same, the weight function and the stability parameter in time-dependent problems. This analysis includes solutions obtained for different types of problems, represented by different differential equations, including time-dependent equations and under different boundary conditions.     

An Element-Free Galerkin Method for Time-Fractional Diffusion-Wave Equations北大核心CSCD

利用无单元Galerkin法,对Caputo意义下的时间分数阶扩散波方程进行了数值求解和相应误差理论分析。首先用L1逼近公式离散该方程中的时间变量,将时间分数阶扩散波方程转化成与时间无关的整数阶微分方程;然后采用罚函数方法处理Dirichlet边界条件,并利用无单元Galerkin法离散整数阶微分方程;最后推导该方程无单元Galerkin法的误差估计公式。数值算例证明了该方法的精度和效果。     

一类高阶椭圆型方程Dirichlet问题

本文讨论了一类奇摄动高阶椭圆型方程Dirichlet问题，利用伸长变量和变界层校正法，得到了问题解的形式渐近展开式．再用微分不等式理论，证明了解的一致有效性．     

A meshless Galerkin method for Dirichlet problems using radial basis functions

In this paper, a numerical method is given for partial differential equations, which combines the use of Lagrange multipliers with radial basis functions. It is a new method to deal with difficulties that arise in the Galerkin radial basis function approximation applied to Dirichlet (also mixed) boundary value problems. Convergence analysis results are given. Several examples show the efficiency of the method using TPS or Sobolev splines.     

Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations

This paper examines the numerical solution of the transient nonlinear coupled Burgers’ equations by a Local Radial Basis Functions Collocation Method (LRBFCM) for large values of Reynolds number (Re) up to 10³. The LRBFCM belongs to a class of truly meshless methods which do not need any underlying mesh but works on a set of uniform or random nodes without any a priori node to node connectivity. The time discretization is performed in an explicit way and collocation with the multiquadric radial basis functions (RBFs) are used to interpolate diffusion-convection variable and its spatial derivatives on decomposed domains. Five nodded domains of influence are used in the local support. Adaptive upwind technique [1] and [54] is used for stabilization of the method for large Re in the case of mixed boundary conditions. Accuracy of the method is assessed as a function of time and space discretizations. The method is tested on two benchmark problems having Dirichlet and mixed boundary conditions. The numerical solution obtained from the LRBFCM for different value of Re is compared with analytical solution as well as other numerical methods [15], [47] and [49]. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high Reynolds numbers.     

A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM.     

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.     

An asymptotic initial value method for second order singular perturbation problems of convection-diffusion type with a discontinuous source term

In this paper a numerical method is presented to solve singularly perturbed two points boundary value problems for second order ordinary differential equations consisting a discontinuous source term. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of a well known perturbation method WKB. Then some initial value problems and terminal value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial value problems are happened to be singularly perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples provided to illustrate the method.     

The boundary element method for stochastic potential problems

Stochastic Dirichlet and Neumann boundary value problems and stochastic mixed problems have been formulated. As a result the stochastic singular integral equations have been obtained. A way of solving these equations by means of discretization of a boundary using stochastic boundary elements has been presented, resulting in a set of random algebraic equations. It has been proved that for Dirichlet and Neumann problems probabilistic characteristics (i.e. moments: expected value and correlation function) fulfilled deterministic singular integral equations. A numerical method of evaluation of moments on a boundary and inside a domain has been presented.     

High Accuracy Solution of Three-Dimensional Biharmonic Equations

In this paper, we consider several finite-difference approximations for the three-dimensional biharmonic equation. A symbolic algebra package is utilized to derive a family of finite-difference approximations for the biharmonic equation on a 27 point compact stencil. The unknown solution and its first derivatives are carried as unknowns at selected grid points. This formulation allows us to incorporate the Dirichlet boundary conditions automatically and there is no need to define special formulas near the boundaries, as is the case with the standard discretizations of biharmonic equations. We exhibit the standard second-order, finite-difference approximation that requires 25 grid points. We also exhibit two compact formulations of the 3D biharmonic equations; these compact formulas are defined on a 27 point cubic grid. The fourth-order approximations are used to solve a set of test problems and produce high accuracy numerical solutions. The system of linear equations is solved using a variety of iterative methods. We employ multigrid and preconditioned Krylov iterative methods to solve the system of equations. Test results from two test problems are reported. In these experiments, the multigrid method gives excellent results. The multigrid preconditioning also gives good results using Krylov methods.