移动热源作用下基于分数阶应变的三维弹性体热-机响应

基于分数阶应变理论,研究了移动热源作用下三维弹性体的热-机动态响应.将分数阶应变理论下的控制方程应用于三维半空间模型,通过Laplace积分变换、双重Fourier变换及其数值反变换对控制方程进行求解,得到了不同热源速度和不同分数阶参数下,无量纲温度、应力、应变和位移的分布规律.结果表明,分数阶应变参数对机械波影响显著...     

On some method for solving a nonlinear heat equation

Some exact solutions to a nonlinear heat equation are constructed. An initial-boundary value problem is examined for a nonlinear heat equation. To construct solutions, the problem for a partial differential equation of the second order is reduced to a similar problem for a first order partial differential equation.     

Half Space and Quarter Space Dirichlet Problems for the Partial Differential Equation

A customary, heuristic, method, by which the Poisson integral formula for the Dirichlet problem, for the half space, for Laplace's equation is obtained, involves Green's function, and Kelvin's method of images. Although this heuristic method leads one to guess the correct result, this Poisson formula still has to be verified directly, independently of the method by which it was arrived at, in order to be absolutely certain that a solution of the Dirichlet problem for the half space, for Laplace's equation, has been actually obtained. A similar heuristic method, as seems to be generally known, could be followed in solving the Dirichlet problem, for the half space, for the equation where is a real constant. However, in Part 1, a different, labor-saving, method is used to study Dirichlet problems for the equation. This method is essentially based on what Hadamard called the method of descent. Indeed, it is shown that he who has solved the half space Dirichlet problem for Laplace's equation has already solved the half space Dirichlet problem for the equation In Part 2, the solution formula for the quarter space Dirichlet problem for Laplace's equation is obtained from the Poisson integral formula for the half space Dirichlet problem for Laplace's equation. A representation theorem for harmonic functions in the quarter space is deduced. The method of descent is used, in Part 3, to obtain the solution formula for the quarter space Dirichlet problem for the equation by means of the solution formula for the quarter space Dirichlet problem for Laplace's equation. So that, indeed, it is also shown that he who has solved the quarter space Dirichlet problem for Laplace's equation has already solved the quarter space Dirichlet problem for the " equation" For the sake of completeness and clarity, and for the convenience of the reader, the appendix, at the end of Part 3, contains a detailed proof that the Poisson integral formula solves the half space Dirichlet problem for Laplace's equation. The Bibliography for Parts 1,2, 3 is to be found at the end of Part 1.     

一非线性发展方程的反问题

研究一非线性发展方程的未知源项的反演问题.首先,把所考虑的初边值问题化成一等价非线性发展方程的Cauchy问题;然后,利用半群理论,论证反问题解的存在性和唯一性;最后,利用压缩映射不动点方法,得到反问题的可解性.     

Dirichlet problem for a third-order equation of mixed type in a rectangular domain

For a third-order differential equation of parabolic-hyperbolic type, we suggest a method for studying the first boundary value problem by solving an inverse problem for a second-order equation of mixed type with unknown right-hand side. We obtain a uniqueness criterion for the solution of the inverse problem. The solution of the inverse problem and the Dirichlet problem for the original equation is constructed in the form of the sum of a Fourier series.     

Remarks on a semilinear heat equation with integral boundary conditions

For a semilinear heat equation we consider a nonlocal boundary problem. On the basis of the solution of a Dirichlet problem for a parabolic equation and Volterra integral equation we establish the well-posedness for the nonlocal problem, which generalizes some recent results.     

The eigenvalue problem for solitary waves of a boussinesq equation,via a generalization of the rouché theorem

We consider the study of an eigenvalue problem obtained by linearizing about solitary wave solutions of a Boussinesq equation. Instead of using the technique of Evans functions as done by Pego and Weinstein in [R. Pego and M. Weinstein, Convective Linear Stability of Solitary Waves for Boussinesq equation. AMS, 99, 311–375] for this particular problem, we perform Fourier analysis to characterize solutions of the eigenvalue problem in terms of a multiplier operator and use the strong relationship between the eigenvalue problem for the linearized Boussinesq equation and the eigenvalue problem associated with the linearization about solitary wave solutions of a special form of the KdV equation. By using a generalization of the Rouché Theorem and the asymptotic behavior of the Fourier symbol corresponding to the eigenvalues problem for the Boussinesq equation and the Fourier symbol corresponding to the eigenvalues problem for the KdV equation, we show nonexistence of eigenvalues with respect to weighted space in a planar region containing the right-half plane.     

再论弹性大挠度问题von Kármán方程与量子本征值问题Schrǒdinger方程的关系

本文是文[1]的继续和改善。利用本文的结果,还可以改善文[2～3]中有关弹性大挠度问题的讨论。在本文中,我们再次对弹性大挠度问题的von Kármán方程进行简化,使它最终成为非线性Schr?dinger方程。其次,在本文中我们对AKNS方程在多维条件下进行了更为对称的拓展。由于非线性Schr?dinger方程与AKNS方程即Dirac方程的可积性条件相联系,因此,弹性大挠度问题可以用逆散射方法求得其精确解,也就是说,它完全成了量子本征值问题。对于正交各向异性大挠度问题,本文也作了推论。     

Inverse problem for a quasilinear hyperbolic equation with a nonlocal boundary condition containing a delay argument

We consider a problem for a quasilinear hyperbolic equation with a nonlocal condition that contains a retarded argument. By reducing this problem to a nonlinear integrofunctional equation, we prove the existence and uniqueness theorem for its solution. We pose an inverse problem of finding a solution-dependent coefficient of the equation on the basis of additional information on the solution; the information is given at a fixed point in space and is a function of time. We prove the uniqueness theorem for the solution of the inverse problem. The proof is based on the derivation and analysis of an integro-functional equation for the difference of two solutions of the inverse problem.     

Asymptotic Analysis for a Class of Infinite Systems of First-Order PDE: Nonlinear Parabolic PDE in the Singular Limit

Abstract

We study the asymptotic behavior of solutions of the Cauchy problem for a functional partial differential equation with a small parameter as the parameter tends to zero. We establish a convergence theorem in which the limit problem is identified with the Cauchy problem for a nonlinear parabolic partial differential equation. We also present comparison and existence results for the Cauchy problem for the functional partial differential equation and the limit problem.     

Computational methods of solving the boundary value problems for the loaded differential and Fredholm integro‐differential equations

The article presents a new general solution to a loaded differential equation and describes its properties. Solving a linear boundary value problem for loaded differential equation is reduced to the solving a system of linear algebraic equations with respect to the arbitrary vectors of general solution introduced. The system's coefficients and right sides are computed by solving the Cauchy problems for ordinary differential equations. Algorithms of constructing a new general solution and solving a linear boundary value problem for loaded differential equation are offered. Linear boundary value problem for the Fredholm integro‐differential equation is approximated by the linear boundary value problem for loaded differential equation. A mutual relationship between the qualitative properties of original and approximate problems is obtained, and the estimates for differences between their solutions are given. The paper proposes numerical and approximate methods of solving a linear boundary value problem for the Fredholm integro‐differential equation and examines their convergence, stability, and accuracy.     

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.     

The mixed problem for a nonlinear degenerate elliptic equation of second order

The present paper deals with the mixed boundary value problem for a nonlinear elliptic equation with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the uniqueness and existence of solutions of the above problem for the nonlinear elliptic equation by the extremum principle and the method of parameter extension. The complex method is used to discuss the corresponding problem for degenerate elliptic complex equation of first order and then that of second order.     

Optimal control of solutions of the Showalter-Sidorov-Dirichlet problem for the Boussinesq-Love equation

An optimal control problem for a second-order Sobolev type equation with a relatively polynomially bounded operator pencil is considered. We prove the existence and uniqueness of a strong solution of the Showalter-Sidorov problem for this equation. Necessary and sufficient conditions for the existence and uniqueness of an optimal control of such solutions are obtained. We study the Showalter-Sidorov-Dirichlet problem for the Boussinesq-Love equation.     

Boundary conditions of the second-order differential equation and the Riccati equation

The conversion of a second-order linear ordinary differential equation with variable coefficients into a Riccati equation depends on whether the second-order problem is an initial-value or two-point boundary-value problem. The distinction is critical in determining the initial condition for the Riccati equation. If the second-order problem is an initial-value problem, the choice of the Riccati transformation depends on whether a zero initial condition for the function or its derivative is specified. If the problem is a two-point boundary-value problem, special methods must be introduced as described in the paper.     

Stochastic calculus of variations and mechanics

The Cauchy problem for the heat equation with a potential term (imaginary time analogue of the Schrodinger equation) is considered. After a logarithmic transformation, a positive solution to this Cauchy problem turns into a solution to the dynamic programming equation for a problem of stochastic calculus of variations. The latter problem is one of least average action. The classical principle of least action is obtained as an appropriate parameter tends to zero.     

A penalty approximation method for a semilinear parabolic double obstacle problem

In this work, we present a novel power penalty method for the approximation of a global solution to a double obstacle complementarity problem involving a semilinear parabolic differential operator and a bounded feasible solution set. We first rewrite the double obstacle complementarity problem as a double obstacle variational inequality problem. Then, we construct a semilinear parabolic partial differential equation (penalized equation) for approximating the variational inequality problem. We prove that the solution to the penalized equation converges to that of the variational inequality problem and obtain a convergence rate that is corresponding to the power used in the formulation of the penalized equation. Numerical results are presented to demonstrate the theoretical findings.     

N-SOLITON SOLUTION OF THE KUNDU-TYPE EQUATION VIA RIEMANN-HILBERT APPROACH

In this article, we focus on investigating the Kundu-type equation with zero boundary condition at infinity. Based on the analytical and symmetric properties of eigenfunctions and spectral matrix of its Lax pair, a Riemann-Hilbert problem for the initial value problem of the Kundu-type equation is constructed. Further through solving the regular and nonregular Riemann-Hilbert problem, a kind of general N-soliton solution of the Kundu-type equation are presented. As special cases of this result, ...     

Inverse prediction of frictional heat flux and temperature in sliding contact with a protective strip by iterative regularization method

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent frictional heat flux at the interface of two semi-spaces, one of them is covered by a strip of coating, during a sliding-contact process from the knowledge of temperature measurements taken within one of the semi-space. It is assumed that no prior information is available on the functional form of the unknown heat generation; hence the procedure is classified as the function estimation in inverse calculation. Results show that the relative position between the measured and the estimated quantities is of crucial importance to the accuracy of the inverse algorithm. The current methodology can be applied to the prediction of heat generation in engineering problems involving sliding-contact elements.     

A problem with a nonlocal,with respect to time,condition for multidimensional hyperbolic equations

We study a boundary-value problem for a hyperbolic equation with a nonlocal with respect to time-variable integral condition. We obtain sufficient conditions for unique solvability of the nonlocal problem. The proof is based on reduction of the nonlocal first-type condition to the second-type one. This allows to reduce the nonlocal problem to an operator equation. We show that unique solvability of the operator equation implies the existence of a unique solution to the problem.