A regularization method for the approximate particular solution of nonhomogeneous Cauchy problems of elliptic partial differential equations with variable coefficients

Radial basis functions (RBFs) have proved to be very flexible in representing functions. Based on the idea of the analog equation method and radial basis functions, in this paper, ill-posed Cauchy problems of elliptic partial differential equations (PDEs) with variable coefficients are considered for the first time using the method of approximate particular solutions (MAPS). We show that, using the Tikhonov regularization, the MAPS results an effective and accurate numerical algorithm for elliptic PDEs and irregular solution domains. Comparing the proposed MAPS with Kansa's method, numerical results show that the proposed MAPS is effective, accurate and stable to solve the ill-posed Cauchy problems.     

A meshless method for solving nonhomogeneous Cauchy problems

In this paper the method of fundamental solutions (MFS) and the method of particular solution (MPS) are combined as a one-stage approach to solve the Cauchy problem for Poisson's equation. The main idea is to approximate the solution of Poisson's equation using a linear combination of fundamental solutions and radial basis functions. As a result, we provide a direct and effective meshless method for solving inverse problems with inhomogeneous terms. Numerical results in 2D and 3D show that our proposed method is effective for Cauchy problems.     

A boundary element method for a second order elliptic partial differential equation with variable coefficients

A boundary element method is derived for solving a class of boundary value problems governed by an elliptic second order linear partial differential equation with variable coefficients. Numerical results are given for a specific test problem.     

Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations

A meshfree method namely, discrete least squares meshless (DLSM) method, is presented in this paper for the solution of elliptic partial differential equations. In this method, computational domain is discretized by some nodes and the set of simultaneous algebraic equations are built by minimizing a least squares functional with respect to the nodal parameters. The least squares functional is defined as the sum of squared residuals of the differential equation and its boundary condition calculated at a set of points called sampling points, generally different from nodal points. A moving least squares (MLS) technique is used to construct the shape functions. The proposed method automatically leads to symmetric and positive-definite system of equations. The proposed method does not need any background mesh and, therefore, it is a truly meshless method. The solutions of several one- and two-dimensional examples of elliptic partial differential equations are presented to illustrate the performance of the proposed method. Sensitivity analysis on the parameters of the method is also carried out and the results are presented.     

Regularized meshless method for nonhomogeneous problems

The regularized meshless method is a novel boundary-type meshless method but by now has largely been confined to homogeneous problems. In this paper, we apply the regularized meshless method to the nonhomogeneous problems in conjunction with the dual reciprocity technique in the evaluation of the particular solution. Numerical experiments of three benchmark nonhomogeneous problems demonstrate the accuracy and efficiency of the present strategy.     

A domain decomposition method for solving thin film elliptic interface problems with variable coefficients

A domain decomposition method is developed for solving thin film elliptic interface problems with variable coefficients. In this study, the elliptic equation with variable coefficients is discretized using second‐order finite differences while a discrete interface equation is obtained using the immersed interface method in order to obtain a second‐order global accuracy. The obtained linear system is solved using a preconditioned Richardson iteration, which is shown to converge fast when the grid size in the thickness direction is much smaller than the grid sizes in both the length and width directions. To simplify the computation, a domain decomposition algorithm is obtained based on a parallel Gaussian elimination procedure. The method is illustrated by a numerical example. Copyright © 1999 John Wiley & Sons, Ltd.     

Local integro-differential equations with domain elements for the numerical solution of partial differential equations with variable coefficients

A new approach (Domain-Element Local Integro-Differential-Equation Method -- DELIDEM) is developed and implemented for the solution of 2-D potential problems in materials with arbitrary continuous variation of the material parameters. The domain is discretized into conforming elements for the polynomial approximation and the local integro-differential equations (LIDE) are considered on subdomains determined by domain elements and collocated at interior nodes. At the boundary nodes, either the prescribed boundary conditions or the LIDE are collocated. The applicability and reliability of the method is tested for several numerical examples.     

A local meshless method for Cauchy problem of elliptic PDEs in annulus domains

This paper is concerned with the development of a meshless local approach based on the finite collocation method for solving Cauchy problems of 2-D elliptic PDEs in annulus domains. In the proposed approach, besides the collocation of unknown solution, the governing equation is also enforced in the local domains. Moreover, to improve the accuracy, the method considers auxiliary points in local subdomains and imposes the governing PDE operator at these points, without changing the global system size. Localization property of the method reduces the ill-conditioning of the problem and makes it efficient for Cauchy problem. To show the efficiency of the method, four test problems containing Laplace, Poisson, Helmholtz and modified Helmholtz equations are given. A numerical comparison with traditional local RBF method is given in the first test problem.     

A meshless local boundary integral equation method for two-dimensional steady elliptic problems

A novel meshless local boundary integral equation (LBIE) method is proposed for the numerical solution of two-dimensional steady elliptic problems, such as heat conduction, electrostatics or linear elasticity. The domain is discretized by a distribution of boundary and internal nodes. From this nodal points’ cloud a “background” mesh is created by a triangulation algorithm. A local form of the singular boundary integral equation of the conventional boundary elements method is adopted. Its local form is derived by considering a local domain of each node, comprising by the union of neighboring “background” triangles. Therefore, the boundary shape of this local domain is a polygonal closed line. A combination of interpolation schemes is taken into account. Interpolation of boundary unknown field variables is accomplished through boundary elements’ shape functions. On the other hand, the Radial Basis Point Interpolation Functions method is employed for interpolating the unknown interior fields. Essential boundary conditions are imposed directly due to the Kronecker delta-function property of the boundary elements’ interpolation functions. After the numerical evaluation of all boundary integrals, a banded stiffness matrix is constructed, as in the finite elements method. Several potential and elastostatic benchmark problems in two dimensions are solved numerically. The proposed meshless LBIE method is also compared with other numerical methods, in order to demonstrate its efficiency, accuracy and convergence.     

Levi functions for linear elliptic systems with variable coefficients including shell equations

This paper gives an algorithm to construct Levi functions of arbitrary degree for elliptic systems of linear partial differential equations with variable (real-analytic) coefficients. Further, an indirect method is described to transform elliptic boundary value problems into a system of integral equations. This method is applied to the shell equations in the non-shallow case. (In the shallow case the shell equations have constant coefficients.) Some questions of discretization are discussed and numerical results are presented.     

A method of solving boundary value problems in linear partial differential equations

A solution in the form of a series in time functions and polynomials of the space coordinate is obtained using as an example the moisture conduction equation with allowance for the finite rate of capillary motion.     

Finite-difference solution of an elliptic partial differential equation with discontinuous coefficients

An algoritm is presented for point relaxation of an elliptic partial diferential equation at a grid station where its coefficient are discontinuous. Such equations can arise from the use of skewed co-ordinate system based on discotinuous reference functions (lengths) to map a complicated physical geometry onto a simple computational domain. Efficary of the scheme is illustrated by examples involving potential flow in a sharply bent channel and in a conical diffuser.     

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.     

The method of approximate fundamental solutions (MAFS) for elliptic equations of general type with variable coefficients

The paper presents a meshless method for solving elliptic equations of general type with variable coefficients. It is based on the use of the delta-shaped functions and the method of approximate fundamental solutions first suggested for solving equations with constant coefficients. The method assumes that the solution domain is embedded in a square and the initial equation is extended onto the square with the help of the CICE −(Chebyshev interpolation + C-expansion) approximation scheme. As a result the coefficients of the equation are approximated by the truncated Fourier series over some orthogonal system in the square. The approximate fundamental solutions (AFSs) satisfy L[u]=I(x), where I(x) is the delta shaped function in the form of the truncated Fourier series. Thus, the AFSs due to the special form of the operator can be obtained in the similar form of truncated series. The next part of the MAFS follows the general scheme of the MFS. The numerical examples are presented and the results are compared with the analytical solutions. The comparison shows that the method presented provides a very high precision in solution of two-dimensional elliptic equations of general type with different boundary conditions (Dirichlet, Neumann, mixed) in arbitrary domains.     

An integral equation method for the numerical solution of a Dirichlet problem for second-order elliptic equations with variable coefficients

We study how fiber-reinforced materials will naturally undergo swelling deformations in which a relatively greater stretch occurs transverse to the fibers than in the fiber direction. This means that a pattern of initially curved fibers prior to swelling will tend to straighten out as swelling proceeds. This can lead to swelling-induced deformations with a high degree of localized shearing and significant overall twisting. Such a process is examined for a plane strain swelling deformation that combines twist with radial expansion. Analytical results are obtained for both types: small and large swelling. Of particular interest is the relation of the extensible fiber theory to a theory for inextensible fibers. We examine the extent to which the former approaches the latter in the limit as the fibers are taken to be progressively stiffer.     

Fundamental solutions for second order linear elliptic partial differential equations

This paper is concerned with outlining some fundamental solutions and Green's functions for a system of second order linear elliptic partial differential equations in two independent variables. The fundamental solution and a number of Green's functions are given in relatively elementary closed form for some cases when the coefficients in the equations are constant. When the coefficients are variable the fundamental solution is obtained for some particular classes of equations.     

A fundamental solution for linear second-order elliptic systems with variable coefficients

A fundamental solution and a Green's function are obtained for a system of second-order elliptic partial differential equations with variable coefficients. Both the fundamental solution and the Green's function are suitable for facilitating the numerical solution of boundary-value problems in a number of practical areas. Some particular areas of application are outlined.     

A meshless method based on RBFs method for nonhomogeneous backward heat conduction problem

Based on the idea of radial basis functions approximation and the method of particular solutions, we develop in this paper a new meshless computational method to solve nonhomogeneous backward heat conduction problem. To illustrate the effectiveness and accuracy of the proposed method, we solve several benchmark problems in both two- and three-dimensions. Numerical results indicate that this novel approach can achieve an efficient and accurate solution even when the final temperature data is almost undetectable or disturbed with large noises. It has also been shown that the proposed method is stable to recover the unknown initial temperature from scattered final temperature data.     

An iterative asymptotic expansion method for elliptic eigenvalue problems with oscillating coefficients

In this work we propose a method for obtaining fine-scale eigensolution based on the coarse-scale eigensolution in elliptic eigenvalue problems with oscillating coefficient. This is achieved by introducing a 2-scale asymptotic expansion predictor in conjunction with an iterative corrector. The eigensolution predictor equation is formulated using the weak form of an auxiliary problem. It is shown that large errors exist in the higher eigenmodes when the 2-scale asymptotic expansion is used. The predictor solution is then corrected by the combined inverse iteration and Rayleigh quotient iteration. The numerical examples demonstrate the effectiveness of this approach.