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In the present paper, formulas for solutions and coefficients of second-order differential equations are presented. These formulas can be used for finding solutions of direct and inverse problems encountered in mathematical modeling of various evolutionary processes. Besides, new statements of inverse problems are considered, and results obtained through solution of these problems are reported. In particular, inverse mathematical-analysis problems are formulated pertaining to the Taylor formula and to the integration average theorem. Also, equations with superpositions, and inverse problems for evolutionary and functional equations with a parameter, are considered. Formulas for solutions of inverse problems are given, and existence conditions for solutions of equations with superpositions are formulated.

We consider an inverse problem for a system of isotropic elasticity equations in a sphere domain. The linearized problem of identification of three characteristics of elastic isotropic medium is investigated. It is supposed that the medium density ρ ( r ) depends on the radial variable only and the propagation velocity of longitudinal c ( r , θ ϕ) and transverse a ( r , θ, ϕ) waves can be represented as a 2 (r; θ, ϕ) = a 2 0 + a 1 ( r , θ, ϕ), c 2 ( r , θ, ϕ) = c 2 0 + c 1 ( r , θ, ϕ), where a 2 0 , c 2 0 are some known constants, and unknown functions a 1 ( r , θ, ϕ), c 1 ( r , θ, ϕ) are small in comparison with the constants a 2 0 and c 2 0 correspondingly. The uniqueness theorem is proved and estimates of conditional stability of the inverse problem solution are obtained.

In this paper, the identification of the hydraulic conductivities in a seawater intrusion problem is investigated. The system of two coupled quasilinear partial differential equations describing the direct problem are solved on the basis of an established maximum principle using a fixed point theorem. The existence of an optimal control is proved, even if the conductivities are discontinuous functions, and the necessary conditions for optimality are obtained. Finally, some numerical results are presented for the sake of illustration.

We consider an inverse problem of finding a solution u and a coefficient in the equation where l 0 is an elliptic operator of the second order and l 1 is a first order operator. A function a ( x ) is unknown on some subset G 0 ⊂ G and is given on the set G \ G 0 . The conditions of the first boundary value problem are augmented with the overdetermination condition on the set G 0 , which can be considered as the partial final overdetermination condition. Under some conditions on the data of the problem, the existence and uniqueness questions are studied.