A method for finding coefficients of a quasilinear hyperbolic equation

The inverse problem of finding the coefficients q(s) and p(s) in the equation u _tt = a ² u _xx + q(u)u _t ? p(u)u _x is investigated. As overdetermination required in the inverse setting, two additional conditions are set: a boundary condition and a condition with a fixed value of the timelike variable. An iteration method for solving the inverse problem is proposed based on an equivalent system of integral equations of the second kind. A uniqueness theorem and an existence theorem in a small domain are proved for the inverse problem to substantiate the convergence of the algorithm.     

Inverse coefficient problem for a quasilinear hyperbolic equation with final overdetermination

The inverse problem of recovering a solution-dependent coefficient multiplying the lowest derivative in a hyperbolic equation is investigated. As overdetermination is required in the inverse problem, an additional condition is imposed on the solution to the equation with a fixed value of the timelike variable. Global uniqueness and local existence theorems are proved for the solution to the inverse problem. An iterative method is proposed for solving the inverse problem.     

On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination condition

We study the unique solvability of the inverse problem of determining the righthand side of a parabolic equation whose leading coefficient depends on both the time and the spatial variable under an integral overdetermination condition with respect to time. We obtain two types of condition sufficient for the local solvability of the inverse problem as well as study the so-called Fredholm solvability of the inverse problem under consideration.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 522–534.Original Russian Text Copyright © 2005 by V. L. Kamynin.This revised version was published online in April 2005 with a corrected issue number.     

Existence and uniqueness of an inverse problem for nonlinear Klein‐Gordon equation

In this paper, an initial boundary value problem for nonlinear Klein‐Gordon equation is considered. Giving an additional condition, a time‐dependent coefficient multiplying nonlinear term is determined, and existence and uniqueness theorem for small times is proved. The finite difference method is proposed for solving the inverse problem.     

Determining of three unknown functions of a semilinear ultraparabolic equation

In this paper, we consider the inverse problem for second‐order semilinear ultraparabolic equation. The equation has unknown function of time variable in its minor coefficient and two unknown functions of time and spacial variables in its right‐hand side. Initial, boundary, and integral type overdetermination conditions are posed. By using the properties of the solutions of the corresponding initial‐boundary value problem and the method of successive approximations, the sufficient conditions of the existence, and the uniqueness of the solution for the inverse problem are obtained on some time interval that depends on the coefficients of the equation.     

An inverse coefficient problem for a quasilinear parabolic equation with periodic boundary and integral overdetermination condition

In this paper, the inverse problem of finding the time‐dependent coefficient of heat capacity together with the solution periodic boundary and integral overdetermination conditions is considered. Under some natural regularity and consistency conditions on the input data, the existence, uniqueness, and continuous dependence upon the data of the solution are shown. Some considerations on the numerical solution for this inverse problem are presented with an example. Copyright © 2014 John Wiley & Sons, Ltd.     

Problem of determining the nonstationary potential in a hyperbolic-type equation

We solve the problem of determining the hyperbolic equation coefficient depending on two variables. Some additional information is given by the trace of the direct problem solution on the hyperplane x = 0. We estimate the stability of the solution of the inverse problem under study and prove the uniqueness theorem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 220–225, August, 2008.     

An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions

In this paper, the inverse problem of finding the time‐dependent coefficient of heat capacity together with the solution of heat equation with nonlocal boundary and overdetermination conditions is considered. The existence, uniqueness and continuous dependence upon the data are studied. Some considerations on the numerical solution for this inverse problem are presented with the examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Mixed nonlocal problem for a nonlinear singular hyperbolic equation

In this paper, we study a nonlocal mixed problem for a nonlinear hyperbolic equation. Based on some a priori estimates and some density arguments, we prove the well posedness of the associated linear problem. The existence and uniqueness of the weak solution of the nonlinear problem are then established by applying an iterative process based on the obtained results for the linear problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Determining the conductivity for a nonautonomous hyperbolic operator in a cylindrical domain

This paper is devoted to the reconstruction of the conductivity coefficient for a nonautonomous hyperbolic operator an infinite cylindrical domain. Applying a local Carleman estimate, we prove the uniqueness and a Hölder stability in the determination of the conductivity using a single measurement data on the lateral boundary. Our numerical examples show good reconstruction of the location and contrast of the conductivity function in 3 dimensions.     

Inverse coefficient problem for a second‐order elliptic equation with nonlocal boundary conditions

In this research article, the inverse problem of finding a time‐dependent coefficient in a second‐order elliptic equation is investigated. The existence and the uniqueness of the classical solution of the problem under consideration are established. Numerical tests using the finite‐difference scheme combined with an iteration method are presented, and the sensitivity of this scheme with respect to noisy overdetermination data is illustrated. Copyright © 2015 John Wiley & Sons, Ltd.     

应用拓展双曲函数方法求KP方程的新精确解

本文引入行波解,并应用拓展双曲函数方法,求得（2＋1）维Kadomtsev-Petviashvili（KP）方程的精确解.通过应用拓展双曲函数方法,可以得到关于方程的一类有理函数形式的孤立波,行波以及三角函数周期波的精确解,并且此方法适用于求解一大类非线性偏微分进化方程.     

Local existence and global nonexistence theorems for a damped nonlinear hyperbolic equation

The existence and uniqueness of the local generalized solution to the initial boundary value problem for the three-dimensional damped nonlinear hyperbolic equation

     

On smoothness of a fundamental solution to a second order hyperbolic equation

We consider the problem of constructing a fundamental solution to a second order hyperbolic linear equation with variable coefficients depending on the space variable x ∈ ? ⁿ . Under the assumption of high but finite smoothness of the coefficients, we write out the structure of a fundamental solution, establish smoothness of the coefficients of the expansion of its singular part, and characterize smoothness of the regular part.     

Spline methods for the solution of hyperbolic equation with variable coefficients

In this study, we developed the methods based on nonpolynomial cubic spline for numerical solution of second‐order nonhomogeneous hyperbolic partial differential equation. Using nonpolynomial cubic spline in space and finite difference in time directions, we obtained the implicit three level methods of O(k² + h²) and O(k² + h⁴). The proposed methods are applicable to the problems having singularity at x = 0, too. Stability analysis of the presented methods have been carried out. The presented methods are applied to the nonhomogeneous examples of different types. Numerical comparison with Mohanty's method (Mohanty, Appl Math Comput, 165 (2005), 229–236) shows the superiority of our presented schemes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Recovery of the heat equation from a single boundary measurement

We prove that we can uniquely recover the coefficient of a one-dimensional heat equation from a single boundary measurement and provide a constructive procedure for its recovery. The algorithm is based on the well-known Gelfand–Levitan–Gasymov inverse spectral theory of Sturm–Liouville operators.     

On a first boundary value problem for hyperbolic equations in the plane

In the paper we study a boundary value problem for a hyperbolic equation with two independent variables; this problem is a generalization of the well-known Darboux problem. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 294–306, February, 1999.     

Determining the parameters of a stratified piecewise constant medium for the unknown shape of an impulse source

For a hyperbolic wave equation with some parameter λ, we consider the problem of finding the piecewise constant wave propagation speed and a series of parameters in the conjugation condition. Moreover, the shape is assumed unknown of the impulse point source that excites the oscillation process. We prove that, under certain assumptions on the structure of the medium, its sought parameters are determined uniquely from the displacements of points of the boundary given for two different values of λ. We give an algorithm for solving the problem.