Inverse Problems For Evolution Equations Research Articles

A class of inverse problems for evolution equations with the Riemann‐Liouville derivative in the sectorial case

We investigate the unique solvability of a class of linear inverse problems with a time‐independent unknown coefficient in an evolution equation in Banach space, which is resolved with respect to the fractional Riemann‐Liouville derivative. We assume that the operator in the right‐hand side of the equation generates a family of resolving operators for the corresponding homogeneous equation, which is exponentially bounded and analytic in a sector containing the positive semiaxis. It is shown that the inverse problem is well‐posed with respect to the graph norm of the generating operator only. A well‐posedness criterion is found. The obtained abstract results are applied to the unique solvability study of an inverse problem for a class of time‐fractional partial differential equations. An example, in particular, shows that in the case of an unbounded generating operator, the inverse problem can be ill‐posed with respect to the norm of the whole space.

(REVIEW ARTICLE) A Unified Approach to Solving Some Inverse Problems for Evolution Equations by Using Observability Inequalities

(REVIEW ARTICLE) A Unified Approach to Solving Some Inverse Problems for Evolution Equations by Using Observability Inequalities

Operator Formulas in Inverse Problems for Evolution Equations

We present general representations of solutions to the inverse problems for evolution equations and write out the second kind equations related to construction of special solutions to nonlinear equations generated by mappings of the Euclidean spaces.

Recovery of a Rapidly Oscillating Absolute Term in the Multidimensional Hyperbolic Equation

The paper is devoted to the development of the theory of inverse problems for evolution equations with summands rapidly oscillating in time. A new approach to setting such problems is developed for the case in which additional constraints (overdetermination conditions) are imposed only on several first terms of the asymptotics of the solution rather that on the whole solution. This approach is realized in the case of a multidimensional hyperbolic equation with unknown absolute term.

Inverse problems for evolution equations with time dependent operator-coefficients

In this paper we study an inverse problem with time dependent operator-coefficients. Weindicate sufficient conditions for the existence and theuniqueness of a solution to this problem. A number of concreteapplications to partial differential equations is described.

Inverse problems for evolution equations with fractional integrals at boundary-value conditions

The following problem is considered: to find a solution and a term of a first-order differential equation in a Banach space if the initial-value condition and an excessive condition containing the fractional Riemann–Liouville integral are given. We show that the solvability of the considered problem depends on the distributions of zeroes of the Mittag-Leffler function.

Constructive methods of studying the inverse problems for evolution equations

The article is devoted to constructive methods of studying the multidimensional inverse problems for evolution equations. The formulas and relations are given for solutions and coefficients of evolution equations and some inverse problems.

Selected formulas of the theory of inverse problems

The work is devoted to constructive methods of investigating multi-dimensional inverse problems for evolution equations. We present formulas and relations for solutions and coefficients of evolution equations and inverse problems.

Inverse problems for evolution equations and matrix Fourier transform

Inverse problems for evolution equations and matrix Fourier transform

Estimation of the spectral radius of an operator and the solvability of inverse problems for evolution equations

Estimation of the spectral radius of an operator and the solvability of inverse problems for evolution equations

Inverse problems for evolution equations of the integrodifferential type

The problem of determining the relaxation kernel using a solution of the inverse problem for an evolution equation which describes the distortion of the profile of an individual wave is analysed. The general case when the kernel is specified in parametric form, and the special case of an exponential kernel, are considered. The non-linear inverse problem is solved using the gradient method. For a linear equation the possibility of using the Laplace transformation method is pointed out.