A method of convolution of Fourier expansions as applied to solving boundary value problems with intersecting interface lines

An efficient method of construction of solutions to a set of boundary value problems with additional interface conditions, more complicated boundary conditions, and so on on the basis of known solutions to classical boundary value problems is proposed. The method is based on the representation of solutions to classical and more complicated problems in the form of expansions into Fourier series with subsequent reduction of one series to the other. As a result, formulas directly expressing solutions to more complicated problems in terms of solutions to classical problems are obtained. On the basis of the well-known solution to the Dirichlet problem on a half plane, solutions to boundary value problems with interface conditions (including generalized conditions of the type of a crack and a screen) on intersecting straight lines for boundary conditions of the first and the third kind are obtained.     

Removable singularities of solutions of degenerate nonlinear elliptic equations on the boundary of a domain

The removability of singularities of solutions for the Dirichlet problem for degenerate nonlinear elliptic equations on the boundary of a domain is studied. A method based on a priori energetic estimates of solutions to elliptic boundary value problems is used. The growth in the vicinity of a boundary point (finite or at infinity) for generalized solutions is studied.     

Numerical solution of special 12th-order boundary value problems using differential transform method

In this paper, a differential transform method (DTM) is used to find the numerical solution of a special 12th-order boundary value problems with two point boundary conditions. The analysis is accompanied by testing differential transform method both on linear and nonlinear problems from the literature [Wazwaz AM. Approximate solutions to boundary value problems of higher-order by the modified decomposition method. Comput Math Appl 2000:40;679–91; Siddiqi SS, Ghazala Akram. Solutions of 12th order boundary value problems using non-polynomial spline technique. Appl Math Comput 2007. doi:10.1016/j.amc.2007.10.015; Siddiqi SS, Twizell EH. Spline solutions of linear 12th-order boundary value problems. J Comput Appl Math 1997;78:371–90]. Numerical experiments and comparison with existing methods are performed to demonstrate reliability and efficiency of the proposed method.     

Artificial boundary conditions for Petrovsky systems of second order in exterior domains and in other domains of conical type

The approximation of solutions to boundary value problems on unbounded domains by those on bounded domains is one of the main applications for artificial boundary conditions. Based on asymptotic analysis, here a new method is presented to construct local artificial boundary conditions for a very general class of elliptic problems where the main asymptotic term is not known explicitly. Existence and uniqueness of approximating solutions are proved together with asymptotically precise error estimates. One class of important examples includes boundary value problems for anisotropic elasticity and piezoelectricity. Copyright © 2004 John Wiley & Sons, Ltd.     

The closed-form particular solutions for Laplace and biharmonic operators using a Gaussian function

Particular solutions play a critical role in solving inhomogeneous problems using boundary methods such as boundary element methods or boundary meshless methods. In this short article, we derive the closed-form particular solutions for the Laplace and biharmonic operators using the Gaussian radial basis function. The derived particular solutions are implemented numerically to solve boundary value problems using the method of particular solutions and the localized method of approximate particular solutions. Two examples in 2D and 3D are given to show the effectiveness of the derived particular solutions.     

Existence and location results for sign-changing solutions for three-point boundary value problems using Leray-Schauder degree

In this paper, by using the modification function technique and the Leray-Schauder degree method some existence and multiplicity results for sign- changing solutions of certain three-point boundary value problems are obtained. The main result of this paper is an improvement on a former existence result for sign-changing solutions of certain three-point boundary value problems.     

Existence and location results for sign-changing solutions for three-point boundary value problems using Leray-Schauder degree

In this paper, by using the modification function technique and the Leray-Schauder degree method some existence and multiplicity results for sign- changing solutions of certain three-point boundary value problems are obtained. The main result of this paper is an improvement on a former existence result for sign-changing solutions of certain three-point boundary value problems.     

非线性Sobolev-Galpern方程解的blow up

本文研究非线性Ｓｏｂｏｌｅｖ－Ｃａｌｐｅｒｎ方程的初边值问题整体解的不存性即解的爆破问题,用能量估计方法并借助于Ｊｅｎｓｅｎ不等式证明了非线性Ｓｏｂｏｌｉｖ－Ｇａｌｐｅｒｎ方程各种初边值问题在某些假设下不存在整体解。     

A dual-level method of fundamental solutions for three-dimensional exterior high frequency acoustic problems

Physical essence of the fictitious boundary of the method of fundamental solutions has been a mystery for a long time. In this study, we attempt to explain the reason why fictitious boundary has such a dramatic effect on numerical results. The influence law of the fictitious boundary on numerical results is revealed. Based on this understanding, a dual-level method of fundamental solutions with self-adaptive adjustment coefficients is proposed. The competitive attributes of the method are that it inherits the high numerical efficiency of the method of fundamental solutions, and it improves numerical stability significantly. The effect of the fictitious boundary on numerical results is eliminated by introducing the concept of equivalent slope. It should be noted that the dual-level method of fundamental solutions can simulate exterior high frequency acoustic problems under the lowest sampling frequency allowed by the Shannon's sampling theorem. Numerical experiment with up to non-dimensional wavenumber of 600 has successfully been conducted on a single laptop when one uses 100,000 degrees of freedom.     

The comparison between the DRBEM and DQM solution of nonlinear reaction–diffusion equation

In this study, both the dual reciprocity boundary element method and the differential quadrature method are used to discretize spatially, initial and boundary value problems defined by single and system of nonlinear reaction–diffusion equations. The aim is to compare boundary only and a domain discretization method in terms of accuracy of solutions and computational cost. As the time integration scheme, the finite element method is used achieving solution in terms of time block with considerably large time steps. The comparison between the dual reciprocity boundary element method and the differential quadrature method solutions are made on some test problems. The results show that both methods achieve almost the same accuracy when they are combined with finite element method time discretization. However, as a method providing very good accuracy with considerably small number of grid points differential quadrature method is preferrable.     

A subregion DRBEM formulation for the dynamic analysis of two-dimensional cracks

The dual reciprocity boundary element method employing the step by step time integration technique is developed to analyse two-dimensional dynamic crack problems. In this method the equation of motion is expressed in boundary integral form using elastostatic fundamental solutions. In order to transform the domain integral into an equivalent boundary integral, a general radial basis function is used for the derivation of the particular solutions. The dual reciprocity boundary element method is combined with an efficient subregion boundary element method to overcome the difficulty of a singular system of algebraic equations in crack problems. Dynamic stress intensity factors are calculated using the discontinuous quarter-point elements. Several examples are presented to show the formulation details and to demonstrate the computational efficiency of the method.     

Multiple solutions of mixed convection in a porous medium on semi-infinite interval using pseudo-spectral collocation method

One knows that calculation of all branches of solutions of nonlinear boundary value problems can be difficult even by numerical methods, especially when the boundary conditions occur at infinity. Regarding this matter, this paper considers a model of mixed convection in a porous medium with boundary conditions on semi-infinite interval which admits multiple (dual) solutions. Furthermore, pseudo-spectral collocation method is applied in erudite way to calculate both dual solutions analytically. Comparison to exact solutions reveals reliability and high accuracy of the procedure and convince to be used to obtain multiple solutions of these kind of nonlinear problems.     

Boundary value problems for a nonstrictly hyperbolic equation of the third order

We study classical solutions of boundary value problems for a nonstrictly hyperbolic third-order equation. The equation is posed in a half-strip and a quadrant of the plane of two independent variables. The Cauchy conditions are posed on the lower boundary of the domain, and the Dirichlet conditions are posed on the lateral boundaries. By using the method of characteristics, we find the analytic form of the solution of considered problems. The uniqueness of the solutions is proved.     

Stationary internal layers in a reaction-advection-diffusion integro-differential system

A class of singularly perturbed nonlinear integro-differential problems with solutions involving internal transition layers (contrast structures) is considered. An asymptotic expansion of these solutions with respect to a small parameter is constructed, and the stability of stationary solutions to the associated integro-parabolic problems is investigated. The asymptotics are substantiated using the asymptotic method of differential inequalities, which is extended to the new class of problems. The method is based on well-known theorems about differential inequalities and on the idea of using formal asymptotics for constructing upper and lower solutions in singularly perturbed problems with internal and boundary layers.     

Generalized solutions of initial-boundary value problems for second-order hyperbolic systems

The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the space of generalized functions followed by passing to integral representations and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems is proved under certain smoothness conditions imposed on the boundary functions. The Green’s matrix of the system and new fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions and solving singular boundary integral equations.     

Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers

A numerical method is proposed for solving singularly perturbed turning point problems exhibiting twin boundary layers based on the reproducing kernel method (RKM). The original problem is reduced to two boundary layers problems and a regular domain problem. The regular domain problem is solved by using the RKM. Two boundary layers problems are treated by combining the method of stretching variable and the RKM. The boundary conditions at transition points are obtained by using the continuity of the approximate solution and its first derivatives at these points. Two numerical examples are provided to illustrate the effectiveness of the present method. The results compared with other methods show that the present method can provide very accurate approximate solutions.     

边界伸缩法

本文提出边界伸缩原理,并在此基础上提出边界伸缩法.不但较好地解决了边界元方法中求解边界附近区域包括边界上解的问题,而且可以方便地利用迭代过程改善求解精度.计算实例表明.本文提出的方法是十分有效的.     

An analytical approach for solving nonlinear boundary value problems in finite domains

Based on the homotopy analysis method (HAM), a general analytical approach for obtaining approximate series solutions to nonlinear two-point boundary value problems in finite domains is proposed. To demonstrate its effectiveness, this approach is applied to solve three nonlinear problems, and the analytical solutions obtained are more accurate than the numerical solutions obtained via the shooting method and the sinc-Galerkin method.     

Application of reproducing kernel method to third order three-point boundary value problems

This paper investigates the analytical approximate solutions of third order three-point boundary value problems using reproducing kernel method. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used directly to solve third order three-point boundary value problems, since there is no method of obtaining reproducing kernel satisfying three-point boundary conditions. This paper presents a method for solving reproducing kernel satisfying three-point boundary conditions so that reproducing kernel method can be used to solve third order three-point boundary value problems. Results of numerical examples demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.     

Critical points and positive solutions of singular elliptic boundary value problems

Usually we do not think there is variational structure for singular elliptic boundary value problems, so it cannot be considered by using critical points theory. In this paper, we use critical theory on certain convex closed sets to solve positive solutions for singular elliptic boundary value problems, especially use the ordinary differential equation theory of Banach spaces to obtain new results on the existence of multiple positive solutions. The method is useful for other singular problems.