Higher order Chebyshev basis functions for two-point boundary value problems

Cubic basis functions in one dimension for the solution of two-point boundary value problems are constructed based on the zeros of Chebyshev polynomials of the first kind. A general formula is derived for the construction of polynomial basis functions of degree r, where 1 ≤r < ∞. A Galerkin finite element method using the constructed basis functions for the cases r = 1, 2 and 3 is successfully applied to three different types of problem including a singular perturbation problem.     

On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs

For the interpolation of continuous functions and the solution of partial differential equation (PDE) by radial basis function (RBF) collocation, it has been observed that solution becomes increasingly more accurate as the shape of the RBF is flattened by the adjustment of a shape parameter. In the case of interpolation of continuous functions, it has been proven that in the limit of increasingly flat RBF, the interpolant reduces to Lagrangian polynomials. Does this limiting behavior implies that RBFs can perform no better than Lagrangian polynomials in the interpolation of a function, as well as in the solution of PDE? In this paper, arbitrary precision computation is used to test these and other conjectures. It is found that RBF in fact performs better than polynomials, as the optimal shape parameter exists not in the limit, but at a finite value.     

On the selection of coordinate functions for the solution of boundary problems by Galerkin's method

We propose a method for determining the coordinate functions to be employed in Galerkin's method for the solution of boundary problems, which increases significantly the precision of the calculations in a first approximation.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 18, No. 2, pp. 309–315, February, 1970.     

A method for the numerical solution of a class of boundary value problems governed by second order elliptic systems

A method for the numerical solution of a class of problems governed by a system of second order elliptic partial differential equations is derived. The solution to the boundary value problem is obtained in terms of an integral taken round part of the boundary of the region under consideration. Some numerical examples are considered and the results obtained are shown to be in excellent agreement with those obtained either analytically or by employing other numerical methods.     

Some comments on the use of radial basis functions in the dual reciprocity method

We comment on the use of radial basis functions in the dual reciprocity method (DRM), particularly thin plate splines as used in Agnantiaris, Polyzos and Beskos (1996). We note that the omission of the linear terms could have biased the numerical results as has occurred in several previous studies. Furthermore, we show that a full understanding of the convergence behavior of the DRM requires one to consider both interpolation and BEM errors, since the latter can offset the effect of improved data approximation. For a model Poisson problem this is demonstrated theoretically and the results confirmed by a numerical experiment.     

Numerical solution of fuzzy boundary value problems using Galerkin method

This paper proposes a new technique based on Galerkin method for solving nth order fuzzy boundary value problem. The proposed method has been illustrated by considering three different cases depending upon the sign of coefficients with benchmark example problems. To show the applicability of the proposed method, an application problem related to heat conduction has also been studied. The results obtained by the proposed methods are compared with the exact solution and other existing methods for demonstrating the validity and efficiency of the present method.     

Use of singularity capturing functions in the solution of problems with discontinuous boundary conditions

A method is proposed to improve the accuracy of the numerical solution of elliptic problems with discontinuous boundary conditions using both global and local meshless collocation methods with multiquadrics as basis functions. It is based on the use of special functions which capture the singular behavior near discontinuities in boundary conditions. In the case of global collocation, the method consists in enlarging the functional space spanned by the RBF basis functions, while in the case of local collocation, the method consists in modifying appropriately the problem in order to eliminate the singularities from the formulation. Numerical results for benchmark problems such as a stationary heat equation in a box (harmonic) and Stokes flow in a lid-driven square cavity, show significant improvements in accuracy and in compliance with the continuity equation.     

Transformation of elliptic partial differential equations for solving two-dimensional boundary value problems in fluid flow

It is possible to transform elliptic partial differential equations to exchange the dependent with one of the independent variables. The Laplace equation for a stream function ‘Ψ’ over the X and Y co-ordinate system, for example, can be transformed into a relationship expressing the Y position of streamlines in terms of strem function Ψ and X. Although the resulting new partial differential equation is much more complex it is much more convenient to use in a computer. An irregular Y boundary becomes, with the new relationship, merely the boundary values assigned to the outer streamlines and the computer always need only deal with a rectangular array. The resulting answer is in the form of the position of streamlines which is the information directly required for plotting flow maps.     

Application of direct variational method to the solution of mixed boundary value problems

A direct variational method has been considered for solving mixed boundary value problems for linear elliptical equations. The main idea of the method consists in searching the unknown solution in a class of functions satisfying the differential equation and some of the boundary conditions. The functional to be minimized may be expressed in terms of the boundary values of the unknown solution, and thus the dimension of the variational problem is reduced. Reduction of the functional defined on the domain to a functional on its boundary corresponds to the common in the elsaticity theory of application of Clapeyron's theorem for the transformation of functionals of minimum elastic potential energy and complementary work. The reduced problem may be solved by the Ritz direct method. Some examples from the elasticity and seepage theories have been given to demonstrate the effectiveness of the method. Among the problems that have been solved by the method we may mention the plane elastic problem for a crack in a strip, three-dimensional elastic problems for a plane rectangular crack in an infinite elastic medium and the problem of a rectangular die in sliding contact with an elastic half-space (numerical solutions exhibit correct asymptotic behaviour in the neighbourhood of corners of cracks or die), and the plane and axisymmetric seepage problems.     

A numerical solution of nonlinear parabolic-type Volterra partial integro-differential equations using radial basis functions

In this paper, an effective numerical method for solving nonlinear Volterra partial integro-differential equations is proposed. These equations include the partial differentiations of an unknown function and the integral term containing the unknown function which is the “memory” of problem. This method is based on radial basis functions (RBFs) and finite difference method (FDM) which provide the approximate solution. These techniques play the important role to reduce a nonlinear partial integro-differential equation to a linear system of equations. Some illustrative examples are shown to describe the method. Numerical examples confirm the validity and efficiency of the presented method.     

A meshless method of lines for the numerical solution of KdV equation using radial basis functions

In this paper, a meshless method of lines (MOL) is presented for the numerical solution of the Korteweg–de Vries (KdV) equation. This novel method has an advantage over the traditional method of lines which approximates the spatial derivatives using finite difference method (FDM) or finite element method (FEM), because it does not need the mesh in the domain, and it approximates the solution using the radial basis functions (RBFs) on a set of node scattered in problem domain. A comparison among some RBFs is made in numerical examples. Numerical examples demonstrate the accuracy and easy implementation of this novel method and it is an efficient method for the nonlinear time-dependent partial differential equations (PDEs).     

On the finite element solution of an exterior boundary value problem

This paper compares three methods for dealing with an exterior boundary value problem by the Finite Element Method, one of which involves using an infinite element. The methods are illustrated by application to the problem of ground water flow round a tunnel with permeable invert. The use of a special trial function with a variable parameter in the infinite element gives a particularly efficient method of solution.     

Macro-elements in the mixed boundary value problems

 The symmetric Galerkin boundary element method (SGBEM), applied to elastostatic problems, is employed in defining a model with BE macro-elements. The model is governed by symmetric operators and it is characterized by a small number of independent variables upon the interface between the macro-elements. The kinematical and mechanical transmission conditions across the interface are imposed in global form, whereas the response of the boundary discretized elastic problem is provided in terms of forces on the interface boundary sides and of displacements at the interface nodes. A variational formulation is presented in which the boundary transmission conditions are derived by Polizzotto's boundary min-max variational principle. Simple numerical applications are shown. Received 8 December 99     

Numerical solution of Bernoulli-type free boundary value problems by variable domain method

The variable domain method for a free surface problem is considered as a weighted residual method. This considerably simplifies discretization of the problem and at the same time allows implementation of any standard discretization technique. Discretization of the problem results in a non-linear system of equations for discrete values of a field variable and co-ordinates of the nodes on the free boundary. Variables corresponding to a field variable are eliminated and the resulting non-linear system is of considerable smaller size. Some new update strategies of the Jacobian together with discussion of the cost-effectiveness and covergence rate are introduced and compared to Broyden's method. Numerical results which confirm validity of the presented discussion are given for three potential free surface problems, (i) two-dimensional free jet impingement against an infinite wall, (ii) axisymmetric free jet impingement subjected to gravity and surface tension forces, and (iii) computation of critical flow over a weir. In case (i) comparison with known analytical results is given, while in case (iii) a good agreement with previously published results is established.     

Analytical solution of boundary value problems of heat conduction in composite regions with arbitrary convection boundary conditions

Summary A mathematical model is developed describing the mechanical behavior of an elastic-plastic composite on the basis of phase properties and the micro-structural geometry. The overall yield condition and hardening rule are obtained in general form for two-phase composites with different phase properties. The constituents may be elastic-perfectly plastic or strain hardening ones. The case of a plastic state of both components is investigated in detail. The loading surface in the space of macro stresses has singular points. Plastic micro strains cannot be eliminated from the system of equations, and must be considered as additional hardening parameters. The hardening of composites with ideally plastic constituents is limited. The limiting surface exists in the space of macro stresses which defines the condition of perfect plasticity. However, it is not associated with the plastic flow law.     

Analysis of thick plates by radial basis functions

In this paper, we use a higher-order shear deformation theory and a radial basis function collocation technique for predicting the static deformations and free vibration behavior of thick plates. Through numerical experiments, the capability and efficiency of this collocation technique for static and vibration problems are demonstrated, and the numerical accuracy and convergence are thoughtfully examined.