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.     

An efficient computational method for the optimal control problem for the Burgers equation

Pointwise control of the viscous Burgers equation in one spatial dimension is studied with the objective of minimizing the distance between the final state function and target profile along with the energy of the control. An efficient computational method is proposed for solving such problems, which is based on special orthonormal functions that satisfy the associated boundary conditions. Employing these orthonormal functions as a basis of a modal expansion method, the solution space is limited to the smallest lower subspace that is sufficient to describe the original problem. Consequently, the Burgers equation is reduced to a set of a minimal number of ordinary nonlinear differential equations. Thus, by the modal expansion method, the optimal control of a distributed parameter system described by the Burgers equation is converted to the optimal control of lumped parameter dynamical systems in finite dimension. The time-variant control is approximated by a finite term of the Fourier series whose unknown coefficients and frequencies giving an optimal solution are sought, thereby converting the optimal control problem into a mathematical programming problem. The solution space obtained is based on control parameterization by using the Runge–Kutta method. The efficiency of the proposed method is examined using a numerical example for various target functions.     

Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation

An initial–boundary value problem for the two-dimensional heat equation with a source is considered. The source is the sum of two unknown functions of spatial variables multiplied by exponentially decaying functions of time. The inverse problem is stated of determining two unknown functions of spatial variables from additional information on the solution of the initial–boundary value problem, which is a function of time and one of the spatial variables. It is shown that, in the general case, this inverse problem has an infinite set of solutions. It is proved that the solution of the inverse problem is unique in the class of sufficiently smooth compactly supported functions such that the supports of the unknown functions do not intersect. This result is extended to the case of a source involving an arbitrary finite number of unknown functions of spatial variables multiplied by exponentially decaying functions of time.     

The quantum stochastic equation is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation

We prove that the solution of the Hudson-Parthasarathy quantum stochastic differential equation in the Fock space coincides with the solution of a symmetric boundary value problem for the Schrödinger equation in the interaction representation generated by the energy operator of the environment. The boundary conditions describe the jumps in the phase and the amplitude of the Fourier transforms of the Fock vector components as any of its arguments changes the sign. The corresponding Markov evolution equation (the Lindblad equation or the “master equation”) is derived from the boundary value problem for the Schrödinger equation.     

Highly accurate algorithms endowing with boundary functions for solving a nonlinear beam equation involving an integral term

In this paper, the problem of a nonlinear beam equation involving an integral term of the deformation energy, which is unknown before the solution, under different boundary conditions with simply supported, 2‐end fixed, and cantilevered is investigated. We transform the governing equation into an integral equation and then solve it by using the sinusoidal functions, which are chosen both as the test functions and the bases of numerical solution. Because of the orthogonality of the sinusoidal functions, we can find the expansion coefficients of the numerical solution that are given in closed form by using the Drazin inversion formula. Furthermore, we introduce the concept of fourth‐order and fifth‐order boundary functions in the solution bases, which can greatly raise the accuracy over 4 orders than that using the partial boundary functions. The iterative algorithms converge very fast to find the highly accurate numerical solutions of the nonlinear beam equation, which are confirmed by 6 numerical examples.     

Linear B-spline finite element method for the improved Boussinesq equation

In this paper, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the nonlinear partial differential equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem to which many accurate numerical methods are readily applicable. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior.     

Stability of equilibrium solution to inhomogeneous heat equation under a 3-point boundary condition

We consider a one-dimensional heat equation with inhomogeneous term, satisfying three-point boundary conditions, such that the temperature at the end is controlled by a sensor at the point η. We show that the integral solution, in the space of continuous functions satisfying the boundary values, converges to the equilibrium solution. This answers a question posed for nonlinear Laplacians, but in the linear case only.     

On a generalization of the Neumann problem for the Laplace equation outside cuts in a plane

A boundary value problem for harmonic functions outside cuts in a plane is considered. The jump of the normal derivative is specified on the cuts as well as a linear combination of the normal derivative on one side of the cut and the jump of the unknown function. The problem is studied with three different conditions at infinity, which lead to different results on existence and number of solutions. The integral representation for a solution is obtained in the form of potentials density in which satisfies the uniquely solvable Fredholm integral equation of the 2nd kind. Copyright © 2004 John Wiley & Sons, Ltd.     

An approximation method using approximate approximations

The aim of this article is to extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present article we develop an approximation procedure for the solution of the interior Dirichlet problem for the Laplace equation in two dimensions using approximate approximations. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In the first step, the unknown source density in the potential representation of the solution is replaced by approximate approximations. In the second, the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in the third, Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.     

New exact solutions for a generalization of the Korteweg-de Vries equation (KdV6)

The generalized tanh-coth method is used to construct periodic and soliton solutions for a new integrable system, which has been derived from an integrable sixth-order nonlinear wave equation (KdV6). The system is formed by two equations. One of the equations may be considered as a Korteweg-de Vries equation with a source and the second equation is a third-order linear differential equation.     

Sufficient close-to-necessary conditions for the blowup of solutions to a strongly nonlinear generalized Boussinesq equation

An initial boundary value problem for the generalized Boussinesq equation with allowance for linear dissipation and free electron sources is considered. The strong generalized time-local solvability of the problem is proved. Sufficient conditions are obtained for the blowup of the solution and for time-global solvability. Two-sided estimates of the blowup time are derived.     

A mixed boundary value problem for Chaplygin's hodograph equation

In this paper we will prove existence, uniqueness and regularity of a classical solution to a mixed boundary value problem for Chaplygin's hodograph equation, which is degenerate elliptic on a part of the boundary. This problem is derived from the study of detached bow shock ahead of a straight ramp in uniform supersonic flows in the hodograph plane. The proof depends on Perron's method and some techniques from linear elliptic equations.     

On the coupling of BEM and FEM for exterior problems for the Helmholtz equation

This paper deals with the coupled procedure of the boundary element method (BEM) and the finite element method (FEM) for the exterior boundary value problems for the Helmholtz equation. A circle is selected as the common boundary on which the integral equation is set up with Fourier expansion. As a result, the exterior problems are transformed into nonlocal boundary value problems in a bounded domain which is treated with FEM, and the normal derivative of the unknown function at the common boundary does not appear. The solvability of the variational equation and the error estimate are also discussed.

     

A Trefftz/MFS mixed-type method to solve the Cauchy problem of the Laplace equation

By using the multiple-scale Trefftz method (MSTM) to solve the Cauchy problem of the Laplace equation in an arbitrary bounded domain, we may lose the accuracy several orders when the noise being imposed on the specified Cauchy data is quite large. In addition to the linear equations obtained from the MSTM, the fundamental solutions play as the test functions being inserted into a derived boundary integral equation. Therefore, after merely supplementing a few linear equations in the mixed-type method (MTM), which is a well organized combination of the Trefftz method and the method of fundamental solutions (MFS), we can improve the ill-conditioned behavior of the linear equations system and hence increase the accuracy of the solution for the Cauchy problem significantly, as explored by two numerical examples.     

On an exact control problem for a semilinear wave equation with an unknown source

The exact controllability of a semilinear wave equation, with Dirichlet boundary control on a part of the boundary and an unknown source, is shown. The nonlinear term has at most a linear growth, the initial and target spaces are L²(Ω)×H⁻¹(Ω).     

Polynomial spectral collocation method for space fractional advection–diffusion equation

This article discusses the spectral collocation method for numerically solving nonlocal problems: one‐dimensional space fractional advection–diffusion equation; and two‐dimensional linear/nonlinear space fractional advection–diffusion equation. The differentiation matrixes of the left and right Riemann–Liouville and Caputo fractional derivatives are derived for any collocation points within any given bounded interval. Several numerical examples with different boundary conditions are computed to verify the efficiency of the numerical schemes and confirm the exponential convergence; the physical simulations for Lévy–Feller advection–diffusion equation and space fractional Fokker–Planck equation with initial δ‐peak and reflecting boundary conditions are performed; and the eigenvalue distributions of the iterative matrix for a variety of systems are displayed to illustrate the stabilities of the numerical schemes in more general cases. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 514–535, 2014     

一个复合系统边界反馈的Riesz基性质

该文考虑一端固定 ,一端具负荷的梁的振动问题 .证明了线性反馈的闭环系统是一个 Riesz谱系统 ,即系统存在一列广义本征函数列构成状态空间的 Riesz基 .从而系统的谱确定增长条件成立 .在此过程中 ,简单的导出了系统本征值的渐近展开式 .并因此推论出系统的指数稳定性的条件     

Source and coefficient identification problems for the wave equation on graphs

Avdonin and Kurasov proposed a leaf peeling method based on the boundary control to recover a potential for the wave equation on a tree. Avdonin and Nicaise considered a source identification problem for the wave equation on a tree. This paper extends the methodology to the wave equation with unknown potential and source distributed parameters defined on a general tree graph.     

Integral equation methods for scattering by infinite rough surfaces

In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane. These boundary value problems arise in a study of time‐harmonic acoustic scattering of an incident field by a sound‐soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double‐ and single‐layer potential and a Dirichlet half‐plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half‐plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single‐layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd.     

Global existence and nonexistence for a nonlinear wave equation with damping and source terms

In this paper we consider a nonlinear wave equation with damping and source term on the whole space. For linear damping case, we show that the solution blows up in finite time even for vanishing initial energy. The criteria to guarantee blowup of solutions with positive initial energy are established both for linear and nonlinear damping cases. Global existence and large time behavior also are discussed in this work. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)