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.     

Initial boundary and inverse problems for the inhomogeneous equation of a mixed parabolic-hyperbolic equation

A problem with inhomogeneous boundary and initial conditions is studied for an inhomogeneous equation of mixed parabolic-hyperbolic type in a rectangular domain. The solution is constructed as the sum of an orthogonal series. A criterion for the uniqueness of the solution is established. It is shown that the uniqueness of the solution and the convergence of the series depend on the ratio of the sides of the rectangle from the hyperbolic part of the mixed domain. On the basis of this problem, inverse problems for finding the factors of the time-dependent right-hand sides of the original equation of mixed type are stated and studied for the first time. The corresponding uniqueness theorems and the existence of solutions are proved using the theory of integral equations for inverse problems.     

一类非线性色散型发展方程反问题的整体解

将一类非线性色散型发展方程反问题转化为抽象空间非线性发展方程Cauchy问题。利用半群方法和赋等价范数技巧,建立了该类抽象发展方程整体解的存在唯一性定理,并应用于所论反问题,得到了该类非线性色散型发展方程反问题整体解的存在唯一性定理,本质地改进了袁忠信得出的解的局部存在唯一性结果。     

Inverse problem for an equation of parabolic-hyperbolic type with a nonlocal boundary condition

For an equation of the parabolic-hyperbolic type, we consider an inverse problem with a nonlocal condition relating solution derivatives that belong to different types of the equation in question. We justify a uniqueness criterion and prove the existence of a solution of the problem by the spectral analysis method. We prove the stability of the solution with respect to the nonlocal boundary condition.     

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.     

波动方程系数反问题的求解方法

讨论了一维波动方程系数反演的一种求解方法,将解进行一阶渐进展开,得到相应的反问题,将其转化为第二类Volttera型积分方程组,证明了反问题解的存在唯一性.     

Inverse problems for the Sturm-Liouville equation with the discontinuous coefficient

In this study we derive the Gelfand-Levitan-Marchenko type main integral equation of the inverse problem for the boundary value problem $L$ and prove the uniquely solvability of the main integral equation. Further, we give the solution of the inverse problem by the spectral data and by two spectrum.     

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.     

Inverse problem with a time‐integral condition for a fractional diffusion equation

We find the conditions for the unique solvability of the inverse problem for a time‐fractional diffusion equation with Schwarz‐type distributions in the right‐hand sides. This problem is to find a generalized solution of the Cauchy problem and an unknown space‐dependent part of an equation's right‐hand side under a time‐integral overdetermination condition.     

Integral-Functional Equation Arising in the Study of an Inverse Problem for a Quasilinear Hyperbolic Equation

A problem with data on the characteristics is considered for a quasilinear hyperbolic equation. The inverse problem of determining two unknown coefficients of the equation from some additional information about the solution is posed. One of the unknown coefficients depends on the independent variable, and the other, on the solution of the equation. Uniqueness theorems are proved for the solution of the inverse problem. The proof is based on the derivation of the integro-functional equation and the analysis of the uniqueness of its solution.     

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.     

Inverse problems for a quasilinear hyperbolic equation in the case of moving observation point

We consider two inverse coefficient problems for a quasilinear hyperbolic equation, where the additional information used for finding the coefficients is the values of the solution on some curve. (This corresponds to measurements performed at a moving observation point.) The unknown coefficient depends on the space variable in the first inverse problem and on the solution of the equation in the second inverse problem. We prove theorems of uniqueness of solution to the inverse problems.     

Inverse problem for a pseudoparabolic equation with integral overdetermination conditions

We consider inverse problems of finding an unknown coefficient in the leading term of a linear pseudoparabolic equation of filtration type on the basis of integral data over the entire boundary or its part under the assumption that the unknown coefficient depends on time. We derive conditions for the time-global solvability and uniqueness of the solution of the inverse problem.     

Iterative method for solving an inverse coefficient problem for a hyperbolic equation

For a hyperbolic equation, we consider an inverse coefficient problem in which the unknown coefficient occurs in both the equation and the initial condition. The solution values on a given curve serve as additional information for determining the unknown coefficient. We suggest an iterative method for solving the inverse problem based on reduction to a nonlinear operator equation for the unknown coefficient and prove the uniform convergence of the iterations to a function that is a solution of the inverse problem.     

On the coefficient inverse problem of heat conduction in a degenerating domain

ABSTRACT

In the paper, we consider a coefficient inverse problem for the heat equation in a degenerating angular domain. It has been shown that the inverse problem for the homogeneous heat equation with homogeneous boundary conditions has a nontrivial solution up to a constant factor consistent with the integral condition. Moreover, the solution of the considered inverse problem is found in explicit form. In conclusion, statements of possible generalizations and the results of numerical calculations are given.     

Identifying an unknown source term in a spherically symmetric parabolic equation

The aim of this work is to solve the inverse problem of determining an unknown source term in a spherically symmetric parabolic equation. The problem is ill-posed: the solution (if it exists) does not depend continuously on the final data. A spectral method is applied to formulate a regularized solution, and a Hölder type estimate of the error between the approximate solution and the exact solution is obtained with a suitable choice of regularization parameter.     

Inverse problem for an equation of mixed type with the Lavrent’ev-Bitsadze operator and with a nonlocal boundary condition

For an equation of the mixed elliptic-hyperbolic type, we study the inverse problem with a nonlocal condition relating the derivatives of the solution on the elliptic and hyperbolic parts of the boundary. We prove a uniqueness criterion and construct the solution in the form of a Fourier series.     

Poisson方程热源识别反问题

探讨了半带状区域上二维Poisson方程只含有一个空间变量的热源识别反问题.这类问题是不适定的,即问题的解(如果存在的话)不连续依赖于测量数据.利用Carasso-Tikhonov正则化方法,得到了问题的一个正则近似解,并且给出了正则解和精确解之间具有Holder型误差估计.数值实验表明Carasso-Tikhonov正则化方法对于这种热源识别是非常有效的.     

On the uniqueness of the solution of dual equation of a singular Sturm–Liouville problem

In this paper we consider differential systems having a singularity and one turning point. First, by a replacement, we transform the system to a linear second-order equation of Sturm–Liouville type with a singularity. Using the infinite product representation of solutions provided in [8], we obtain the dual equation, then we investigate the uniqueness of the solution for the dual equation of the inverse spectral problem of Sturm–Liouville equation. This result is necessary for expressing inverse problem of indefinite Sturm–Liouville equation.     

On a method of solution of a Cauchy problem for singular parabolic equations

A system of parabolic type with singular coefficients on hyperplane boundaries is considered. Solution of a Cauchy problem is reduced to an integral equation and a fundamental solution is determined as the kernel of an inverse operator of the Cauchy problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 1, pp. 135–138, January, 1992.