A Taylor expansion approach for solving partial differential equations with random Neumann boundary conditions

Nonlinear partial differential equation with random Neumann boundary conditions are considered. A stochastic Taylor expansion method is derived to simulate these stochastic systems numerically. As examples, a nonlinear parabolic equation (the real Ginzburg-Landau equation) and a nonlinear hyperbolic equation (the sine-Gordon equation) with random Neumann boundary conditions are solved numerically using a stochastic Taylor expansion method. The impact of boundary noise on the system evolution is also discussed.     

Singular boundary characteristics of the Hamilton–Jacobi equation

Situations exist in boundary value problems for first order partial differential equations arising in physics (the Hamilton–Jacobi equation), optimal control theory (the Bellman equation) and the theory of differential games (the Isaacs equation) when the value of the required function is not given on a part of the boundary or not at all, or it is not the limit of the (generalized) solution of the problem. Nevertheless, such conditions are required for constructing the solution (by the method of characteristics, for example). It is shown that the required boundary values can be exposed as a specific continuation of the conditions that are known in the boundary submanifolds of the given part of the boundary. This extension of the conditions is accomplished using the characteristic curves starting in a known submanifold of the boundary and running along the boundary. The characteristics are a generalization of the classical characteristics associated with a partial differential equation. They are called singular characteristics, and the theory of these has been developed in a number of the author's papers. After obtaining these “natural” boundary conditions, the solution is constructed using the conventional method of integrating the equations of the classical characteristics. Conditions of the Dirichlet and Neumann type are considered. The technique is illustrated using a numerical example from the theory of differential games containing a number of parameters.     

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second‐order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one‐dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Universal boundary value problem for equations of mathematical physics

A new statement of a boundary value problem for partial differential equations is discussed. An arbitrary solution to a linear elliptic, hyperbolic, or parabolic second-order differential equation is considered in a given domain of Euclidean space without any constraints imposed on the boundary values of the solution or its derivatives. The following question is studied: What conditions should hold for the boundary values of a function and its normal derivative if this function is a solution to the linear differential equation under consideration? A linear integral equation is defined for the boundary values of a solution and its normal derivative; this equation is called a universal boundary value equation. A universal boundary value problem is a linear differential equation together with a universal boundary value equation. In this paper, the universal boundary value problem is studied for equations of mathematical physics such as the Laplace equation, wave equation, and heat equation. Applications of the analysis of the universal boundary value problem to problems of cosmology and quantum mechanics are pointed out.     

On the number of interior zeros of a one parameter family of solutions to a second order differential equation satisfying a boundary condition at one endpoint

A method for obtaining the existence of eigenvalues of an ordinary differential equation with separated boundary conditions is introduced. The method is based on counting the number of interior zeros of a one-parameter family of solutions which satisfy the boundary conditions at one of the end points. The coefficients of the differential equation depend continuously on the parameter but are not necessarily linear in the parameter.     

Inverse Problem for a Fourth-Order Differential Operator with Nonseparated Boundary Conditions

Uniqueness theorems are proved for two inverse problems for a fourth-order differential operator with nonseparated boundary conditions. The first of the problems, which has technical applications, is the problem of identification of a differential equation and two boundary conditions, and the second problem is the problem of identification of a differential equation and four boundary conditions. One of two data sets is used as the spectral data of the problem. The first data set is the spectrum of the problem itself (or three of its eigenvalues) and the spectral data of a system of three problems, and the second data set is the spectrum of the problem itself (or three of its eigenvalues) and the spectra of ten boundary value problems.     

Irregular Boundary Value Problems with Discontinuous Coefficients and the Eigenvalue Parameter

In this study, a Birkhoff-irregular boundary value problem for linear ordinary differential equations of the second order with discontinuous coefficients and the spectral parameter has been considered. Therefore, at the discontinuous point, two additional boundary conditions (called transmission conditions) have been added to the boundary conditions. The eigenvalue parameter is of the second degree in the differential equation and of the first degree in a boundary condition. The equation contains an abstract linear operator which is (usually) unbounded in the space Lq(−1, 1). Isomorphism and coerciveness with defects 1 and 2 are proved for this problem. The case of the biharmonic equation is also studied.     

分数阶微分方程边值问题解的存在性

应用Gteen函数将分数阶微分方程边值问题可转化为等价的积分方程．近来此方法被应用于讨论非线性分数阶微分方程边值问题解的存在性．讨论非线性分数阶微分方程边值问题,应用Green函数,将其转化为等价的积分方程,并设非线性项满足Caratheodory条件,利用非紧性测度的性质和M6nch’s不动点定理证明解的存在性．     

Boundary element method and wave equation

In this paper the boundary integral expression for a one-dimensional wave equation with homogeneous boundary conditions is developed. This is done using the time dependent fundamental solution of the corresponding hyperbolic partial differential equation. The integral expression developed is a generalized function with the same form as the well-known D'Alembert formula. The derivatives of the solution and some useful invariants on the characteristics of the partial differential equation are also calculated.The boundary element method is applied to find the numerical solution. The results show excellent agreement with analytical solutions.A multi-step procedure for large time steps which can be used in the boundary element method is also described.In addition, the way in which boundary conditions are introduced during the time dependent process is explained in detail. In the Appendix the main properties of Dirac's delta function and the Heaviside unit step function are described.     

Adjoint and self-adjoint boundary value problems on a geometric graph

We consider boundary value problems of arbitrary order for linear differential equations on a geometric graph. Solutions of boundary value problems are coordinated at the interior vertices of the graph and satisfy given conditions at the boundary vertices. For considered boundary value problems, we construct adjoint boundary value problems and obtain a self-adjointness criterion. We describe the structure of the solution set of homogeneous self-adjoint boundary value problems with alternating coefficients of a differential equation and obtain nondegeneracy conditions for these boundary value problems.     

The method of lines for Poisson's equation with nonlinear or free boundary conditions

Summary The method of lines is used to solve Poisson's equation on an irregular domain with nonlinear or free boundary conditions. The partial differential equation is approximated by a system of second order ordinary differential equations subject to multi-point boundary conditions. The system is solved with an SOR iteration which employs invariant imbedding for each one dimensional problem. An application of the method to a boundary control problem and to a free surface problem arising in electrochemical machining is described. Finally, some theoretical convergence results are presented for a model problem with radiative boundary conditions on fixed boundaries.This work was supported by the U.S. Army Research Office under Grant DA-AG29-76-G-0261     

一类非线性发展方程组的定解问题

将不可压缩的广义neo-Hookean材料组成的超弹性圆柱壳径向对称运动的数学模型归结为一类非线性发展方程组的初边值问题.利用材料的不可压缩条件和边界条件求得了描述圆柱壳内表面径向运动的二阶非线性常微分方程.给出了微分方程的周期解(即圆柱壳的内表面产生非线性周期振动)的存在条件,讨论了材料参数和结构参数对方程的周期解的影响,并给出了相应的数值模拟.     

Solving two-point boundary value problems using combined homotopy perturbation method and Green’s function method

In this paper, an algorithm is proposed for the solution of second-order boundary value problems with two-point boundary conditions. The Green’s function method is applied first to transform the ordinary differential equation into an equivalent integral one, which has already satisfied the boundary conditions. And then, the homotopy perturbation method is used to the resulting equation to construct the numerical solution for such problems. Numerical examples demonstrate the efficiency and reliability of the algorithm developed, it is quite accurate and readily implemented for both linear and nonlinear differential equations with homogeneous and nonhomogeneous boundary conditions. Furthermore, the lower order approximation is of higher accuracy for most cases. Some other extended applications of this algorithm are also exhibited.     

一类非线性抛物型方程组解的整体存在及爆破

研究了一类带有非线性边界条件的非线性抛物型方程组解的整体存在及解在有限时刻爆破问题.通过构造方程组的上、下解.得到了解整体存在及解在有限时刻爆破的充分条件.对指数型反应项和边界流采用了常微分方程方法构造其上下解,而其它例如第一特征值等方法运用于该方程就比较困难.     

Group method solution for solving nonlinear heat diffusion problems

The linear transformation group approach is developed to simulate heat diffusion problems in a media with the thermal conductivity and the heat capacity are nonlinear and obeyed a striking power law relation, subject to nonlinear boundary conditions due to radiation exchange at the interface according to the fourth power law. The application of a one-parameter transformation group reduces the number of independent variables by one so that the governing partial differential equation with the boundary conditions reduces to an ordinary differential equation with appropriate corresponding conditions. The Runge–Kutta shooting method is used to solve the nonlinear ordinary differential equation. Different parametric studies are worked out and plotted to study the effect of heat transfer coefficient, density and radiation number on the surface temperature.     

Numerical solution of the nonlinear Schrodinger equation by feedforward neural networks

We present a method to solve boundary value problems using artificial neural networks (ANN). A trial solution of the differential equation is written as a feed-forward neural network containing adjustable parameters (the weights and biases). From the differential equation and its boundary conditions we prepare the energy function which is used in the back-propagation method with momentum term to update the network parameters. We improved energy function of ANN which is derived from Schrodinger equation and the boundary conditions. With this improvement of energy function we can use unsupervised training method in the ANN for solving the equation. Unsupervised training aims to minimize a non-negative energy function. We used the ANN method to solve Schrodinger equation for few quantum systems. Eigenfunctions and energy eigenvalues are calculated. Our numerical results are in agreement with their corresponding analytical solution and show the efficiency of ANN method for solving eigenvalue problems.     

Wellposedness of the boundary value formulation of a fixed strike Asian option

This work is the follow up to [J. Hugger, Numerical Mathematics and Advanced Applications—Enumath 2001, Springer, Italy, 2003] where a partial differential equation equivalent to the stochastic formulation for a fixed strike Asian option was derived.In the present work the differential equation is complemented with boundary value conditions that are derived from financial conditions.With the complete boundary value formulation thus recovered, wellposedness of the problem is adressed. It turns out that the problem takes the form of a degenerated parabolic boundary value problem with a second-order, linear, time-dependent PDE with non-negative characteristic form. Apart from the degeneracy in the PDE, also the boundary conditions (derived from the financial understanding) are “the wrong ones” or at least are non-standard. There are conditions on boundaries where none are expected to be needed bacause of the degeneracy and there are boundaries where conditions are expected to be needed but none can be found.     

Solution of Poisson's equation with global,local and nonlocal boundary conditions

Boundary value problems (BVPs) for partial differential equations are common in mathematical physics. The differential equation is often considered in simple and symmetric regions, such as a circle, cube, cylinder, etc., with global and separable boundary conditions. In this paper and other works of the authors, a general method is used for the investigation of BVPs which is more powerful than existing methods, so that BVPs investigated by the method can be considered in anti-symmetric and arbitrary regions surrounded by smooth curves and surfaces. Moreover boundary conditions can be local, non-local and global. The BVP is expanded in a convex and bounded region D in a plane. First, by generalized solution of the adjoint of the Poisson equation, the necessary boundary conditions are obtained. The BVP is then reduced to the second kind of Fredholm integral equation with regularized singularities.     

Boundary value problems with operator right-hand side

For a second-order boundary value problem with operator right-hand side and with functional boundary conditions, we prove solvability theorems with mixed and Dirichlet boundary conditions assuming the existence of a lower and an upper function. These theorems are analogs of theorems for the corresponding boundary value problems for an ordinary second-order differential equation with right-hand side satisfying the Carathéodory conditions.