The annihilator method for computing particular solutions to partial differential equations

We generalize the well-known annihilator method, used to find particular solutions for ordinary differential equations, to partial differential equations. This method is then used to find particular solutions of Helmholtz-type equations when the right hand side is a linear combination of thin plate and higher order splines. These particular solutions are useful in numerical algorithms for solving boundary value problems for a variety of elliptic and parabolic partial differential equations.     

The method of moments and its optimization

The method of moments is a semidiscrete numerical method for solving partial differential equations. The method approximates the solution of a partial differential equation by a finite sum of products of two functions. One function in the product is an unknown function of a single variable and the other function (moment function) is a prescribed function in the remaining variables. Using variational technique we obtain a finite system of boundary value problems of ordinary differential equations for the unknown functions. The main goal of this paper is the study of the theoretical background and numerical effectiveness of the method of moments for solving linear partial differential equations on rectangular-like domains. The mathematical formulation of the method together with error estimates and the theory of optimal moment functions are given. If for the one-dimensional moment functions piecewise polynomials of degree K are used then finite element type error bounds are obtained for the approximate solution in two dimensions. We also consider the numerical implementation of the method through the factorization method and efficient initial value methods. Several numerical examples showing the efficiency of the method are presented.     

The localized differential quadrature method for two-dimensional stream function formulation of Navier–Stokes equations

Localized differential quadrature (LDQ) method is employed to solve two-dimensional stream function formulation of incompressible Navier–Stokes equations. Being developed by introducing the localization concept to the general differential quadrature (GDQ) method, the employment of LDQ method becomes efficient and flexible, especially for the simulations of large scale computations. By introducing the Lagrange stream function to vorticity transport equation, the governing equation—the fourth-order partial differential equation (PDE)—is derived. To stably obtain the solutions of the fourth-order PDE, a fictitious point method is included to treat the boundary conditions. To examine the present scheme, two different types of classic benchmark fluid flow problems are proposed, including driven cavity flow problems and backward-facing step flow problems. The good agreement of solutions demonstrate the robustness and feasibility of the proposed scheme. Conclusively, the LDQ method is sufficient and appropriate enough to simulate the solutions of stream function formulation of Navier–Stokes equations with various Reynolds numbers.     

LEAST-SQUARES SOLUTIONS OF A GENERAL NUMERICAL METHOD FOR ARBITRARY IRREGULAR GRIDS

This article describes a computational method to calculate solutions of elliptic boundary value problems using arbitrary irregular grids. The main feature of the numerical method is its ability to approximate solutions of differential equations without co-ordinate mapping or metric tensor information. For linear differential equations, the numerical method yields coupled linear systems on local computational cells. Optimal least-squares solutions of coupled linear systems are discussed and applied to the scheme. The numerical method is used to simulate potential flow for two model problems. The computational results are in very good agreement with analytical solutions. © 1997 by John Wiley & Sons, Ltd.     

Applicability of the method of fundamental solutions

The condition number of a matrix is commonly used for investigating the stability of solutions to linear algebraic systems. Recent meshless techniques for solving partial differential equations have been known to give rise to ill-conditioned matrices, yet are still able to produce results that are close to machine accuracy. In this work, we consider the method of fundamental solutions (MFS), which is known to solve, with extremely high accuracy, certain partial differential equations, namely those for which a fundamental solution is known. To investigate the applicability of the MFS, either when the boundary is not analytic or when the boundary data are not harmonic, we examine the relationship between its accuracy and the effective condition number.Three numerical examples are presented in which various boundary value problems for the Laplace equation are solved. We show that the effective condition number, which estimates system stability with the right-hand side vector taken into account, is roughly inversely proportional to the maximum error in the numerical approximation. Unlike the proven theories in literature, we focus on cases when the boundary and the data are not analytic. The effective condition number numerically provides an estimate of the quality of the MFS solution without any knowledge of the exact solution and allows the user to decide whether the MFS is, in fact, an appropriate method for a given problem, or what is the appropriate formulation of the given problem.     

A one-stage meshless method for nonhomogeneous Cauchy problems of elliptic partial differential equations with variable coefficients

A one-stage meshless method is devised for solving Cauchy boundary value problems of elliptic partial differential equations (PDEs) with variable coefficients. The main idea is to approximate an unknown solution using a linear combination of fundamental solutions and radial basis functions. Compared with the two-stage method of particular solution, the proposed method can deal with more general elliptic PDEs with variable coefficients. Several numerical results in both two- and three-dimensional space show that our proposed method is accurate and effective.     

Continuous elements in the finite element method

The discretization of the media at all spatial co-ordinates but one is presented here. This partial discretization leads to continuous finite elements as opposed to fully discrete ones and the problem resolves, for the cases presented here, into a set of linear differential equations rather than algebraic equations. The general problem of first derivative functionals in elastostatics is considered and it is shown, in general, how the continuous finite elements required for the solution may be obtained. Plane states, axisymmetric and three-dimensional continuous elements are obtained to illustrate application to particular cases. Different methods of solution for the set of differential equations are discussed and it is shown that several existing and widely used finite element related techniques are particular cases of this local partial discretization. Three numerical examples are solved to demonstrate the good comparison obtained between the numerical and the exact solutions. The semi-infinite examples included also illustrate the treatment of these types of problems without the use of fictitious boundaries.     

Improved finite difference method for equilibrium problems based on differentiation of the partial differential equations and the boundary conditions

A numerical algorithm for producing high-order solutions for equilibrium problems is presented. The approximated solutions are improved by differentiating both the governing partial differential equations and their boundary conditions. The advantages of the proposed method over standard finite difference methods are: the possibility of using arbitrary meshes; the possibility of using simultaneously approximations with different (distinct) orders of accuracy at different locations in the problem domain; an improvement in approximating the boundary conditions; the elimination of the need for ‘fictitious’ or ‘external’ nodal points in treating the boundary conditions. Furthermore, the proposed method is capable of reaching approximate solutions which are more accurate than other finite difference methods, when the same number of nodal points participate in the local scheme. A computer program was written for solving two-dimensional problems in elasticity. The solutions of a few examples clearly illustrate these advantages.     

Microscale heat conduction in homogeneous anisotropic media: a dual-reciprocity boundary element method and polynomial time interpolation approach

In this paper, a dual-reciprocity boundary element method based on some polynomial interpolations to the time-dependent variables is presented for the numerical solution of a two-dimensional heat conduction problem governed by a third order partial differential equation (PDE) over a homogeneous anisotropic medium. The PDE is derived using a non-Fourier heat flux model which may account for thermal waves and/or microscopic effects. In the analysis, discontinuous linear elements are used to model the boundary and the variables along the boundary. The systems of algebraic equations are set up to solve all the unknowns. For the purpose of evaluating the proposed method, some numerical examples with known exact solutions are solved. The numerical results obtained agree well with the exact solutions.     

Homotopy method of fundamental solutions for solving certain nonlinear partial differential equations

In this study, the homotopy analysis method (HAM) is combined with the method of fundamental solutions (MFS) and the augmented polyharmonic spline (APS) to solve certain nonlinear partial differential equations (PDE). The method of fundamental solutions with high-order augmented polyharmonic spline (MFS–APS) is a very accurate meshless numerical method which is capable of solving inhomogeneous PDEs if the fundamental solution and the analytical particular solutions of the APS associated with the considered operator are known. In the solution procedure, the HAM is applied to convert the considered nonlinear PDEs into a hierarchy of linear inhomogeneous PDEs, which can be sequentially solved by the MFS–APS. In order to solve strongly nonlinear problems, two auxiliary parameters are introduced to ensure the convergence of the HAM. Therefore, the homotopy method of fundamental solutions can be applied to solve problems of strongly nonlinear PDEs, including even those whose governing equation and boundary conditions do not contain any linear terms. Therefore, it can greatly enlarge the application areas of the MFS. Several numerical experiments were carried out to validate the proposed method.     

An order-N complexity meshless algorithm for transport-type PDEs,based on local Hermitian interpolation

This work describes a new numerical method utilising radial basis function interpolants. Based on local Hermitian interpolation of function values and boundary operators, and using an explicit time advancement formulation, the method is of order-N complexity. Computational cost to advance the solution in time is minimal, and is largely dependent on local system support size. The explicit time advancement formulation allows a novel solution technique for many nonlinear partial differential equations.The performance of the method is examined for a variety of linear convection–diffusion–reaction problems, featuring both steady and unsteady solutions. The method is also demonstrated with a nonlinear Richards’ equation model, solving an unsaturated flow in porous media problem. The technique is named the local Hermitian interpolation (LHI) method.     

Solving initial and boundary value problems using learning automata particle swarm optimization

In this article, the particle swarm optimization (PSO) algorithm is modified to use the learning automata (LA) technique for solving initial and boundary value problems. A constrained problem is converted into an unconstrained problem using a penalty method to define an appropriate fitness function, which is optimized using the LA-PSO method. This method analyses a large number of candidate solutions of the unconstrained problem with the LA-PSO algorithm to minimize an error measure, which quantifies how well a candidate solution satisfies the governing ordinary differential equations (ODEs) or partial differential equations (PDEs) and the boundary conditions. This approach is very capable of solving linear and nonlinear ODEs, systems of ordinary differential equations, and linear and nonlinear PDEs. The computational efficiency and accuracy of the PSO algorithm combined with the LA technique for solving initial and boundary value problems were improved. Numerical results demonstrate the high accuracy and efficiency of the proposed method.     

三维无旋矢量场的一种新的可视化方法

提出了三维无旋矢量场的一种新的可视化方法,即构造空间曲面,使得矢量场在曲面上任意一点处垂直于该曲面。首先找到曲面所满足的偏微分方程组,通过采用类似于经典四阶龙格―库塔方法的数值解法对其求解,得到曲面上的离散点,然后进行三角剖分,从而得到逼近于曲面的空间三角网格。论文的偏微分方程组的求解借鉴了常微分方程求解算法的设计思想,构造出的曲面与传统的点图标和线图标相比,在更大程度上揭示了矢量场本身的连续性。     

Dispersive properties of the natural element method

 The Natural Element Method (NEM) is a mesh-free numerical method for the solution of partial differential equations. In the natural element method, natural neighbor coordinates, which are based on the Voronoi tesselation of a set of nodes, are used to construct the interpolant. The performance of NEM in two-dimensional linear elastodynamics is investigated. A standard Galerkin formulation is used to obtain the weak form and a central-difference time integration scheme is chosen for time history analyses. Two different applications are considered: vibration of a cantilever beam and dispersion analysis of the wave equations. The NEM results are compared to finite element and analytical solutions. Excellent dispersive properties of NEM are observed and good agreement with analytical solutions is obtained.     

Least-squares differential quadrature time integration scheme in the dual reciprocity boundary element method solution of diffusive–convective problems

Least-squares differential quadrature method (DQM) is used for solving the ordinary differential equations in time, obtained from the application of dual reciprocity boundary element method (DRBEM) for the spatial partial derivatives in diffusive–convective type problems with variable coefficients. The DRBEM enables us to use the fundamental solution of Laplace equation, which is easy to implement computationally. The terms except the Laplacian are considered as the nonhomogeneity in the equation, which are approximated in terms of radial basis functions. The application of DQM for time derivative discretization when it is combined with the DRBEM gives an overdetermined system of linear equations since both boundary and initial conditions are imposed. The least squares approximation is used for solving the overdetermined system. Thus, the solution is obtained at any time level without using an iterative scheme. Numerical results are in good agreement with the theoretical solutions of the diffusive–convective problems considered.     

A numerical method for coupled differential equations

This paper describes a numerical method for solving first-order coupled matrix differential equations. Recursive equations are used to find ‘reflection’, ‘transmission’ and ‘source’ coefficients; these coefficients are then used to construct the vector solution to the differential equations. The method can be used to solve linear and non-linear differential equations with specified initial or two-point boundary values. Numerical results for several initial value problems are given in the paper.     

Particular solutions of splines and monomials for polyharmonic and products of Helmholtz operators

This paper presents the particular solutions for the polyharmonic and the products of Helmholtz partial differential operators with polyharmonic splines and monomials right-hand side. By the application of the Hörmander linear partial differential operator theory, many of the systems can be reduced to a single equation involving the polyharmonic or the product of Helmholtz differential operators. If the inhomogeneous right-hand side of these operators can be removed by the method of particular solutions, then boundary-type numerical methods, such as the boundary element method, the method of fundamental solutions, and the Trefftz method, can be applied to solve these differential equations.     

Application of backward differentiation methods to the finite element solution of time-dependent problems

The application of finite element methods to parabolic partial differential equations leads to large linear systems of first-order ordinary differential equations. Very often these systems are stiff and difficulties arise in their numerical solution. We attempt to analyse the problem of how to select numerical methods for the solution of such linear systems.     

A meshless method using the Takagi–Sugeno fuzzy model

This article proposes a new strong-form meshless method using the Takagi–Sugeno fuzzy model (MTSF) for solving differential equations (DEs). Considering the conventional fuzzy model, the fuzzy inference system (FIS) can be categorized into two architectures, a simple rule base using the Euclidean distance in a multidimensional space (Simple-FIS), and an adaptive neuro-fuzzy inference system (ANFIS). Accordingly, MTSF also can be implemented using Simple-FIS and ANFIS. Based on the two architectures, an approximation scheme for continuous functions is drawn out first. In turn, the derivation is further proposed in which the differential functions are approximated using two independent sets of points, one for the collocation point and the other for the rule point. Solving higher-order DEs becomes possible by following the derivations, and eventually numerical solutions can be obtained. Several examples of one-dimensional ordinary and two-dimensional partial DEs (ODEs and PDEs) are presented to demonstrate the performance of the MTSF method. By MTSF, solutions solved using Simple-FIS and using ANFIS are compared. Variations in boundary conditions and membership function parameters are also studied to examine the agreement among numerical solutions.     

Group method solutions of the generalized forms of Burgers, Burgers�CKdV and KdV equations with time-dependent variable coefficients

Solutions for the generalized forms of Burgers, Burgers?CKdV, and KdV equations with time-dependent variable coefficients and with initial and boundary conditions are constructed. The analysis rests mainly on the standard group method. Similarity solutions are found which reduce the nonlinear system of partial differential equations to systems of ordinary differential equations to obtain some exact solutions and others as numerical solutions.