A method for the numerical solution of some elliptic boundary value problems for a strip

A boundary integral equation for the numerical solution of a class of elliptic boundary value problems for a strip is derived. The equation should be particularly useful for the solution of an important class of problems governed by Laplace's equation and also for the solution of relevant problems in anisotropic thermostatics and elastostatics     

具Cauchy-Ventcel边界的阻尼波方程的对数衰减性

本文研究有界区域上具Cauchy-Ventcel边界条件的波动方程的解的衰减性质。在不要求耗散区域满足几何控制条件的情形下,我们得到了波方程的对数衰减结果。 主要结果的证明依赖于具Cauchy-Ventcel边界条件的椭圆方程的插值不等式以及关于该椭圆方程的预解式估计。为得到期望的插值不等式, 我们采用的工具是整体Carleman估计。     

Variable-coefficient auxiliary equation method for exact solutions of non-linear evolution equations

In this paper, a variable-coefficient auxiliary equation method is proposed to seek more general exact solutions of non-linear evolution equations. Being concise and straightforward, this method is applied to the Kawahara equation, Sawada–Kotera equation and (2+1)-dimensional Korteweg–de Vries equations. As a result, many new and more general exact solutions are obtained including Jacobi elliptic, hyperbolic and trigonometric function solutions. It is shown that the proposed method provides a straightforward and effective method for non-linear evolution equations in mathematical physics.     

Iterative solutions to non-linear fourth order differential equations through multi-integral methods

Implicit solution to a certain class of non-linear fourth order differential equation is presented. Bounds on sup norm for the derivative of a certain function f involved in the equation with different boundary values, are computed. These bounds provide the rate of convergence of the iterative sequence of approximate solutions obtained by Picard method, to the exact solution.     

Numerical solutions of 3D Cauchy problems of elliptic operators in cylindrical domain using local weak equations and radial basis functions

This paper is concerned with the numerical solutions of 3D Cauchy problems of elliptic differential operators in the cylindrical domain. We assume that the measurements are only available on the outer boundary while the interior boundary is inaccessible and the solution should be obtained from the measurements from the outer layer. The proposed discretization approach uses the local weak equations and radial basis functions. Since the Cauchy problem is known to be ill-posed, the Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system of equations. Numerical results of a different kind of test problems reveal that the method is very effective.     

A numerical study of variable coefficient elliptic Cauchy problem via projection method

In this paper, we investigate a numerical method for the solution of an inverse problem of recovering lacking data on some part of the boundary of a domain from the Cauchy data on other part for a variable coefficient elliptic Cauchy problem. In the process, the Cauchy problem is transformed into the problem of solving a compact linear operator equation. As a remedy to the ill-posedness of the problem, we use a projection method which allows regularization solely by discretization. The discretization level plays the role of regularization parameter in the case of projection method. The balancing principle is used for the choice of an appropriate discretization level. Several numerical examples show that the method produces a stable good approximate solution.     

Chebyshev pseudospectral method of viscous flows with corner singularities

Chebyshev pseudospectral solutions of the biharmonic equation governing two-dimensional Stokes flow within a driven cavity converge poorly in the presence of corner singularities. Subtracting the strongest corner singularity greatly improves the rate of convergence. Compared to the usual stream function/ vorticity formulation, the single fourth-order equation for stream function used here has half the number of coefficients for equivalent spatial resolution and uses a simpler treatment of the boundary conditions. We extend these techniques to small and moderate Reynolds numbers.     

A Fast Spectral Subtractional Solver for Elliptic Equations

The paper presents a fast subtractional spectral algorithm for the solution of the Poisson equation and the Helmholtz equation which does not require an extension of the original domain. It takes O(N ² log N) operations, where N is the number of collocation points in each direction. The method is based on the eigenfunction expansion of the right hand side with integration and the successive solution of the corresponding homogeneous equation using Modified Fourier Method. Both the right hand side and the boundary conditions are not assumed to have any periodicity properties. This algorithm is used as a preconditioner for the iterative solution of elliptic equations with non-constant coefficients. The procedure enjoys the following properties: fast convergence and high accuracy even when the computation employs a small number of collocation points. We also apply the basic solver to the solution of the Poisson equation in complex geometries.     

The Approximation and Computation of a Basis of the Trace Space <Emphasis Type="Italic">H</Emphasis><Superscript>1/2</Superscript>

We present a method for construction of an approximate basis of the trace space H ^1/2 based on a combination of the Steklov spectral method and a finite element approximation. Specifically, we approximate the Steklov eigenfunctions with respect to a particular finite element basis. Then solutions of elliptic boundary value problems with Dirichlet boundary conditions can be efficiently and accurately expanded in the discrete Steklov basis. We provide a reformulation of the discrete Steklov eigenproblem as a generalized eigenproblem that we solve by the implicitly restarted Arnoldi method of ARPACK. We include examples highlighting the computational properties of the proposed method for the solution of elliptic problems on bounded domains using both a conforming bilinear finite element and a non-conforming harmonic finite element. In addition, we document the efficiency of the proposed method by solving a Dirichlet problem for the Laplace equation on a densely perforated domain.     

求解任意波数的三维Helmholtz方程

提出通过Adomian分解法求解任意波数的三维Helmholtz方程。通过Adomian分解法可以把三维Helmholtz微分方程转换成递归代数公式,并进一步把其边界条件转换成适用符号计算的简单代数公式。利用边界条件可以很容易得到方程的解析解表达式。Adomian分解法的主要特点在于计算简单快速,并且不需要进行线性化或离散化。最后通过数值计算以验证Adomian分解法求解任意波数下三维Helmholtz方程的有效性。数值计算结果表明：Adomian分解法的计算结果非常接近精确解,并且该方法在大波数情况下还具有良好的收敛性。     

Modelling and effective modification of smooth boundary geometry in boundary problems using B-spline curves

To create curves in computer graphics, we use, among others, B-splines since they make it possible to effectively produce curves in a continuous way using a small number of de Boor’s control points. The properties of these curves have also been used to define and create boundary geometry in boundary problems solving using parametric integral equations system (PIES). PIES was applied for resolution 2D boundary-value problems described by Laplace’s equation. In this PIES, boundary geometry is theoretically defined in its mathematical formalism, hence the numerical solution of the PIES requires no boundary discretization (such as in BEM) and is simply reduced to the approximation of boundary functions. To solve this PIES a pseudospectral method has been proposed and the results obtained were compared with both exact and numerical solutions.     

Subanalytic solutions of linear difference equations and multidimensional hypergeometric sequences

We consider linear difference equations with polynomial coefficients over C and their solutions in the form of sequences indexed by the integers (sequential solutions). We investigate the C-linear space of subanalytic solutions, i.e., those sequential solutions that are the restrictions to Z of some analytic solutions of the original equation. It is shown that this space coincides with the space of the restrictions to Z of entire solutions and that the dimension of this space is equal to the order of the original equation.We also consider d-dimensional (d≥1) hypergeometric sequences, i.e., sequential and subanalytic solutions of consistent systems of first-order difference equations for a single unknown function. We show that the dimension of the space of subanalytic solutions is always at most 1, and that this dimension may be equal to 0 for some systems (although the dimension of the space of all sequential solutions is always positive).Subanalytic solutions have applications in computer algebra. We show that some implementations of certain well-known summation algorithms in existing computer algebra systems work correctly when the input sequence is a subanalytic solution of an equation or a system, but can give incorrect results for some sequential solutions.     

Non-equivalence of the conventional boundary integral formulation and its elimination for plane elasticity problems

Compared with a given boundary value problem of plane elasticity, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions which are dependent upon Poisson's ratio and geometry. Such a non-equivalence of solutions of boundary integral equations can be eliminated by using a necessary and sufficient boundary integral formulation proposed by He [Necessary and sufficient BIE-BEM: its theory and practice. Ph.D. Dissertation, Zhejiang University, Hangzhou, China (1993)]. Numerical analysis shows that the conventional boundary integral equation yields incorrect non-equivalent results when the scale in the fundamental solution is near its degenerate scale value. Also, this non-equivalence can be remedied by using the necessary and sufficient boundary integral equation.     

New exact solutions of the (n+1)-dimensional sine-Gordon equation using double elliptic equation method

In this paper, the (n+1)-dimensional sine-Gordon equation is studied using double elliptic equation method. With the aid of Maple, more exact solutions expressed by Jacobi elliptic functions are obtained. When the modulus m of Jacobi elliptic function is driven to the limit 1 and 0, some exact solutions expressed by hyperbolic function solutions and trigonometric functions can also be obtained, respectively.     

Non-equivalence of the conventional boundary integral formulation and its elimination for two-dimensional mixed potential problems

For a given mixed type potential problem, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions. Numerical results show that the conventional boundary integral formulation yields incorrect potential and flux results when the distance scale in the fundamental solution approaches its degenerate value. Such a kind of non-equivalence of the conventional boundary integral equation can be eliminated by the use of the necessary and sufficient boundary integral formulation which always ensures the equivalence of solutions.     

Asymptotic properties of solutions to discrete Coulomb equations

We construct two finite-difference models for the Coulomb differential equation which arises in the quantum mechanics analysis of the scattering of two charged point particles. These difference equations correspond to the standard and Mickens-Ramadhani schemes for the Coulomb equation. Our major goal is to determine the first two terms in the asymptotic solutions and compare them to the corresponding solutions of the Coulomb differential equation. In particular, the form of the anomalous phase term is examined.     

On fuzzy boundary value problems

The statement that a two-point boundary value problem of fuzzy differential equation is equivalent to a fuzzy integral equation was pointed out by Lakshmikantham et al. and O’Regan et al. Recently Bede gave a counterexample to show that this statement does not hold and he also argued that in many cases two-point boundary value problems have no solutions. Under a new structure and certain conditions we show that a two-point boundary value problem is equivalent to a fuzzy integral equation. We also prove the existence of solutions to the two-point boundary value problem. In some sense, this is an amendment to results of Lakshmikantham et al. and O’Regan et al., and it is an answer to one of Bede’s problems.     

Asymptotic behavior of a nonlinear Kirchhoff type equation with spring boundary conditions

In this paper, we consider the initial-boundary value problem for a nonlinear Kirchhoff type equation with the damping term and spring boundary conditions. We establish the global existence and uniqueness of solutions to this problem in time, and give an example and simulation to illustrate our results. For the proof, we use the Faedo–Galerkin approximate method. Finally, we study the asymptotic behavior of solutions and some of its simulation results. Results of this paper are able to apply industrial parts such as a typical model widely used to represent threads, wires, magnetic tapes, belts, band saws, and so on.     

Auxiliary equation method for time-fractional differential equations with conformable derivative

In this paper, the auxiliary equation method is applied to obtain analytical solutions of (2 + 1)-dimensional time-fractional Zoomeron equation and the time-fractional third order modified KdV equation in the sense of the conformable fractional derivative. Given equations are converted to the nonlinear ordinary differential equations of integer order; and then, the resulting equations are solved using a novel analytical method called the auxiliary equation method. As a result, some exact solutions for them are successfully established. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective.