The Inverse Problem for Numerical Determination of the Nonlinear Right-Hand Side in a Parabolic Equation

The article considers the inverse problem of determining the nonlinear right-hand side of a quasi-linear parabolic equation and proves a uniqueness theorem. A method is proposed for numerical solution of the inverse problem based on parametric representation of the sought coefficient. The inverse problem thus reduces to finding the error-minimizing vector of unknown coefficients of the parametric representation of the sought function.     

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.     

On the Unique Solvability of an Inverse Problem for Parabolic Equations under a Final Overdetermination Condition

We consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation with the leading coefficient depending on time and space variables under a final overdetermination condition. We obtain two types of conditions that are sufficient for the local solvability of the inverse problem and also prove the so-called Fredholm solvability of the inverse problem under study.     

The inverse problem of determining the source in the stationary transport equation on a riemannian manifold

The transport equation with an unknown right-hand side is considered on a compact Riemannian manifold. The right-hand side of this equation is recovered from values of the outcoming flow. Assumptions under which the solution of the inverse problem is unique are formulated.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 236–242.     

A nonlocal inverse problem for a mixed-type equation

We establish a criterion for the unique solvability of the inverse problem for the Lavrent’ev-Bitsadze equation with unknown right-hand side. We construct a solution as the sum of a series over a system of bi-orthogonal root functions of the corresponding adjoint problems on eigenvalues.     

On the stabilization to zero of the solutions of the inverse problem for a degenerate parabolic equation with two independent variables

Mathematical Notes - Theorems on the stabilization to zero as t → +∞ of solutions of the inverse problem of determining the unknown right-hand side of a degenerate parabolic equation...     

Localization of the discontinuity line of the right-hand side of a differential equation

We propose a new approach to studying the inverse problems for differential equations with constant coefficients. Its application is illustrated by an example of some partial differential equation with three independent variables. The right-hand side of the equation is assumed to be a function discontinuous in spatial variables. In the inverse problem, it is required to find some hull containing the discontinuity line of the right-hand side. An algorithm for constructing such a hull is obtained: It is a square whose sides are tangent to the discontinuity line.     

The inverse problem for a mixed-type parabolic-hyperbolic equation in a rectangular domain

In this paper with the help of the spectral method we obtain a criterion for the unique solvability of the inverse problem for a mixed-type parabolic-hyperbolic equation in a rectangular domain. This problem implies the search of the unknown right-hand side.     

On determining the source function in heat and mass transfer problems under integral overdetermination conditions

We examine an inverse problem of determining the right-hand side (the source function) in a parabolic equation from integral overdetermination data. By a solution to a parabolic equation we mean a weak solution, and the right-hand side in this equation can be a distribution of a certain class. Under some conditions on the data of the problem, we demonstrate that this inverse problem is well posed and, in particular, some stability estimates hold.     

On a class of problems of determining the temperature and density of heat sources given initial and final temperature

We consider a class of problems modeling the process of determining the temperature and density of heat sources given initial and finite temperature. Their mathematical statements involve inverse problems for the heat equation in which, solving the equation, we have to find the unknown right-hand side depending only on the space variable. We prove the existence and uniqueness of classical solutions to the problem, solving the problem independently of whether the corresponding spectral problem (for the operator of multiple differentiation with not strongly regular boundary conditions) has a basis of generalized eigenfunctions.     

The inverse problem for an equation of mixed parabolic-hyperbolic type

We study the inverse problem related to finding the unknown right-hand side of the equation of mixed parabolic-hyperbolic type in a rectangular domain. A uniqueness criterion for the solution is established. The solution is constructed as the sum of the series in eigenfunctions of the corresponding one-dimensional spectral 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 Some Inverse Problems for First Order Operator-Differential Equations

Siberian Mathematical Journal - We study solvability of the inverse problems of recovering an unknown function on the nonlinear right-hand side of a first order operator-differential equation in...     

An iterative method for solving an inverse problem for a first-order nonlinear partial differential equation with estimates of guaranteed accuracy and the number of steps

For a partial differential equation simulating population dynamics, the inverse problem of determining its nonlinear right-hand side from an additional boundary condition is studied. This inverse problem is reduced to a functional equation, for which the existence and uniqueness of a solution is proven. An iterative method for solving this inverse problem is proposed. The accuracy of the method is estimated, and restrictions on the number of steps are obtained.     

Uniqueness in One Inverse Problem for the Elasticity System

We consider an inverse problem for the stationary elasticity system with constant Lame coefficients and a variable matrix coefficient depending on the spatial variables and frequency. The right-hand side contains a delta-function whose support (source) varies in some domain disjoint from the support of the variable coefficient. The inverse problem is to find the coefficient from the scattered wave measured at the same point at which the perturbation originates. A uniqueness theorem is proven. The proof bases on reduction of the inverse problem to a family of equations with the M. Riesz potential.     

Asymptotic Behavior of Solutions of Inverse Problems for Degenerate Parabolic Equations

We obtain theorems on the proximity as t → +∞ between the solution of the inverse problem for a second-order degenerate parabolic equation with one spatial variable and the solution of the inverse problem for a second-order degenerate ordinary differential equation under an additional integral observation condition. The conditions imposed on the input data admit oscillations of the functions on the right-hand side in the parabolic equation under study.     

Differentiation of the functional in a parametric optimization problem for a coefficient of a semilinear elliptic equation

We study parametric optimization with respect to an integral criterion of the higher coefficient and the right-hand side of a second-order semilinear elliptic equation with the Dirichlet boundary condition. We obtain formulas for the first partial derivatives of the objective functional with respect to the control parameters. The total preservation (preservation for the entire set of control parameters) of the unique solvability of the boundary value problem for this equation is proved based on the theory of monotone operators.     

Interpolating solutions of the Poisson equation in the disk based on Radon projections

We consider an algebraic method for reconstruction of a function satisfying the Poisson equation with a polynomial right-hand side in the unit disk. The given data, besides the right-hand side, is assumed to be in the form of a finite number of values of Radon projections of the unknown function. We first homogenize the problem by finding a polynomial which satisfies the given Poisson equation. This leads to an interpolation problem for a harmonic function, which we solve in the space of harmonic polynomials using a previously established method. For the special case where the Radon projections are taken along chords that form a regular convex polygon, we extend the error estimates from the harmonic case to this Poisson problem. Finally we give some numerical examples.     

Algorithm for shadowing the solution of a parabolic equation on an infinite time interval

For a parabolic equation, we consider the problem of constructing a feedback control synthesis algorithm ensuring that the solution of a given equation shadows the solution of another equation with unknown right-hand side. We suggest a noise-immune algorithm based on the extremal shift method well known in guaranteed control theory.     

On a linear inverse problem for a second-order parabolic equation

The question is studied of existence and uniqueness of a solution to the inverse problem of finding the right-hand side of a special kind for a second-order parabolic equation.