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.     

Generalized differential equation arising in the solution of the inverse scattering problem in a layered medium

We consider a nonclassical ordinary differential equation containing not only an unknown function but also an unknown coefficient depending on the unknown function. We show that if the desired solution is assumed to have bounded variation and be a.e. constant on the interval where the equation is considered, then the problem of finding the solution and the unknown coefficient does not have a unique solution in terms of the classical derivative. We prove that if the derivative is understood as a distribution, than this problem has a unique solution. These results are used to show that the acoustic impedance and the damping factor in the inverse scattering problem in a layered dissipative medium can be determined simultaneously.     

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.     

Monotone-iterative method for mixed boundary value problems for generalized difference equations with “maxima”

In this paper the authors investigate special type of difference equations which involve both delays and the maximum value of the unknown function over a past time interval. This type of equations is used to model a real process which present state depends significantly on its maximal value over a past time interval. An appropriate mixed boundary value problem for the given nonlinear difference equation is set up. An algorithm, namely, the monotone iterative technique is suggested to solve this problem approximately. An important feature of our algorithm is that each successive approximation of the unknown solution is equal to the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima”, and a formula for its explicit form is given. Also, each approximation is a lower/upper solution of the given nonlinear boundary value problem. Several numerical examples are considered to illustrate the practical application of the suggested algorithm.     

A semi-analytical method of solving problems of the high-frequency diffraction of elastic waves by cracks

A semi-analytical approach is proposed for constructing an effective solution of the problem of high-frequency diffraction of elastic waves by a crack in an isotropic plane. It essentially consists of separating the strongly oscillating solution of the main integral equation of the problem, which holds uniformly over the whole crack length for high oscillation frequencies. The solution is sought in the form of the product of a strongly oscillating function, corresponding to the qualitative behaviour of the solution, and a certain slowly varying unknown modulating function, which also becomes the main unknown in the initial equation. It is shown that, to find this new unknown function correctly, it is sufficient to take an order of magnitude smaller number of collocation points than for the direct approach.     

System of nonlinear integral equations for the unknown functions in a functional-differential hyperbolic equation

We consider the inverse problem for a functional-differential equation in which the delay function and a function occurring in the source are unknown. The values of the solution and its derivative at x = 0 are given as additional information. We derive a system of nonlinear integral equations for the unknown functions. This system is used to prove a uniqueness theorem for the inverse problem.     

A Class of Quadratic Integral-Differential Equations

A nonlinear integro-ordinary differential equation built up by a linear ordinary differential operator of n th order with constant coefficients and a quadratic integral term is dealt with. The integral term represents the so-called autocorrelation of the unknown function. Applying the Fourier cosine transformation, the integral-differential equation is reduced to a quadratic boundary value problem for the complex Fourier transform of the solution in the upper half-plane. This problem in turn is reduced to a linear boundary value problem which can be solved in closed form. There are infinitely many solutions of the integral-differential equation depending on the prescribed zeros of a function related to the complex Fourier transform.     

The Sinc‐Galerkin method for solving an inverse parabolic problem with unknown source term

The inverse problem of determining an unknown source term depending on space variable in a parabolic equation is considered. A numerical algorithm is presented for recovering the unknown function and obtaining a solution of the problem. As this inverse problem is ill‐posed, Tikhonov regularization is used for finding a stable solution. For solving the direct problem, a Galerkin method with the Sinc basis functions in both the space and time domains is presented. This approximate solution displays an exponential convergence rate and is valid on the infinite time interval. Finally, some examples are presented to illustrate the ability and efficiency of this numerical method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Existence of solutions to Hamilton-Jacobi functional differential equations

This paper deals with the Cauchy problem for nonlinear first order partial functional differential equations. The unknown function is the functional variable in the equation and the partial derivatives appear in a classical sense. A theorem on the local existence of a generalized solution is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions of this system is proved by using a method of successive approximations. A method of bicharacteristics and integral inequalities are applied.     

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.     

一类抛物型方程系数反问题的分裂算法

考虑利用终端时刻的温度u(x,T)=Z_T(x)反演热传导方程u_t-a~2u_(xx) q(x)u=0,x∈(0,1)中的未知系数q(x)的反问题.通过引进变换v(x,t)=(u_t(x,t)/u(x,t))将此非线性不适定问题的求解分解为两步.首先利用输入数据迭代求解一个非线性的正问题(该过程独立于未知系数),得到其迭代解v~(k)(x,t).其次利用q(x)与v(x,t)的关系式求出q(x)的近似解.对提出的反演方法,证明了采用的变换的可行性,得到了原反问题与由变换后的非线性正问题反演q(x)的等价性并且证明了迭代解的收敛性,给出了收敛速度.数值结果表明了该方法的有效性.     

A mathematical modeling approach to the formation of urban and rural areas: Convergence of global solutions of the mixed problem for the master equation in sociodynamics

Urban and rural areas are formed by human migration from thinly populated areas to densely populated areas. It is known in sociodynamics that human migration is described by a nonlinear integro-partial differential equation whose unknown function denotes the population density. This equation is called the master equation. The master equation has its origin in statistical physics, and is regarded as one of the most fundamental equations in natural sciences, as its name suggests. We describe the formation of urban and rural areas by making use of global solutions of the mixed problem for this equation. In this paper we prove sufficient conditions for the mixed problem to have a unique global solution that converges to a two-tier step function as the time variable tends to infinity. This step function is a stationary solution of the master equation, and the higher (lower, respectively) step represents a stationary urban (rural, respectively) area. This result mathematically describes the formation of urban and rural areas in the real world.     

Numerical solution method for the electric impedance tomography problem in the case of piecewise constant conductivity and several unknown boundaries

We study the electrical impedance tomography problem with piecewise constant electric conductivity coefficient, whose values are assumed to be known. The problem is to find the unknown boundaries of domains with distinct conductivities. The input information for the solution of this problem includes several pairs of Dirichlet and Neumann data on the known external boundary of the domain, i.e., several cases of specification of the potential and its normal derivative. We suggest a numerical solution method for this problem on the basis of the derivation of a nonlinear operator equation for the functions that define the unknown boundaries and an iterative solution method for this equation with the use of the Tikhonov regularization method. The results of numerical experiments are presented.     

Two regularization methods for inverse source problem on the Poisson equation

This paper is devoted to discuss an inverse problem of determining an unknown source on the Poisson equation. This is a mildly ill-posed problem. Two regularization methods, one based on the mollification of the data and the other based on the modification of the ‘kernel’ of the solution, are proposed to solve this problem. The convergence estimates between the exact solution and the regularization solution are presented using a priori regularization parameter choice rule. Numerical results are presented to illustrate the accuracy and efficiency of the proposed methods.     

海洋波导中三维可穿透目标的快速声学成像

研究了用声传播远场分布信息来成像海洋波导环境中三维可穿透目标的反问题.建立了求解这类反问题的远场方程,基于内透射边界值问题的分析,讨论了远场方程解的唯一性和可解性,证明了总能找到远场方程的一个在最小平方意义下的近似解,其模在可穿透目标内部的取值是小的,而在外部的取值是大的,进而发展了一种快速成像可穿透目标的一种指示器样本方法.数值试验表明了这种方法是有效的,即使在有限孔径测量方式的情况,也能够得到未知目标的一个理想成像,而且不需要先验知道可穿透目标的任何几何与物理信息.     

The method of central difference for the inverse time-dependent heat source problem

This paper investigates the inverse problem of determining a heat source in a parabolic equation where data are given at some fixed location. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. A central difference regularization method is given and an order optimal stability estimate is obtained. Numerical results for several benchmark test problems indicate that the central difference regularization method is an accurate and flexible method to determine the unknown time-dependent heat source.     

The convection-diffusion equation for a finite domain with time varying boundaries

A solution is developed for a convection-diffusion equation describing chemical transport with sorption, decay, and production. The problem is formulated in a finite domain where the appropriate conservation law yields Robin conditions at the ends. When the input concentration is arbitrary, the problem is underdetermined because of an unknown exit concentration. We resolve this by defining the exit concentration as a solution to a similar diffusion equation which satisfies a Dirichlet condition at the left end of the half line. This problem does not appear to have been solved in the literature, and the resulting representation should be useful for problems of practical interest.

Authors of previous works on problems of this type have eliminated the unknown exit concentration by assuming a continuous concentration at the outflow boundary. This yields a well-posed problem by forcing a homogeneous Neumann exit, widely known as Danckwerts condition. We provide a solution to that problem and use it to produce an estimate which demonstrates that Danckwerts condition implies a zero concentration at the outflow boundary, even for a long flow domain and a large time.     

Inverse source problem for a diffusion equation involving the fractional spectral Laplacian

In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach.     

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.     

Classical solutions of boundary value problems for differential-difference equations

For a differential-difference equation of the second order on the interval [0, d], we study the existence of a classical solution for arbitrary continuous right-hand sides. We show that a necessary and sufficient condition for the existence of a classical solution of the problem in the case where there exists a generalized solution is the absence of argument shifts in the derivatives of the unknown function occurring in the equation.