Inverse problems for partial differential equations
This monograph is devoted to identification problems of coefficients in equations of mathematical physics. Differential properties of the solutions for the original direct problems and their behavior under great values of time are studied on the basis of solution properties for direct problems.
VIII, 211 S. 25 cm
9789067643580, 9067643580
248450720
Ch. 1. Auxiliary information from functional analysis and theory of differential equations. 1.1. The basic notions and notations. 1.2. Inequalities. 1.3. Some concepts and theorems of functional analysis. 1.4. Linear partial differential equation of the first order. 1.5. The maximum principle for parabolic equations of second order
Ch. 2. The weak approximation method. 2.1. Examples reducing to the concept of the weak approximation method. 2.2. General formulation of the weak approximation method. 2.3. Two theorems. 2.4. An example. The linear partial differential equation
Ch. 3. Identification problems for parabolic equations with Cauchy data. 3.1. The unknown source function. 3.2. The unknown lowest coefficient. 3.3. The unknown coefficient to the first order derivative. 3.4. An unknown coefficient to the time derivative. 3.5. Inverse problem for a semilinear parabolic equation. 3.6. Equations of Burgers type. 3.7. The splitting of one many-dimensional inverse problem into problems of lower dimension
Ch. 4. The identification of the source function for a system of composite type and parabolic equation. The behavior of the problem's solution under t [approaches +infinity]. 4.1. The statement of the problem. 4.2. Theorems of existence and uniqueness "on the whole" 4.3. The behavior of solution by t [approaches +infinity]. 4.4. Stationary problem. 4.5. Convergence to the stationary problem solution. 4.6. Unique solvability of the problem of identifying the source function for a parabolic equation. 4.7. The stabilization of the solution to the inverse problem
Ch. 5. The problem of determining the coefficient in a parabolic equation and some properties of its solution. 5.1. Statement of the problem. 5.2. Theorems of existence and uniqueness "on the whole" 5.3. Properties of solution under t[approaches +infinity]
Ch. 6. Two unknown coefficients of a parabolic type equation. 6.1. Uniform boundary conditions. 6.2. Inhomogeneous conditions of over-determination. 6.3. Input data of the special form
Ch. 7. Some inverse boundary value problems. 7.1. Unknown source function. 7.2. Nonlinear heat equation. 7.3. Hyperbolic equation with small parameter
Literaturverz. S. 201 - 211