Forward & Inverse Problems for Hyperbolic,
Elliptic & Mixed Type Equations
By A.G. Megrabov
240 pages, Illustrated, 6 1/2" x 9 3/4"
The field of inverse problems is an important and rapidly developing direction in mathematical physics, differential equations and various applied technologies. This volume in the Inverse and Ill-Posed Problems Series focuses on direct and inverse problems for partial differential equations. The type of equations considered are hyperbolic, elliptic and mixed (elliptic-hyperbolic). The direct problems arise as generalizations from the inhomogeneous layer (or from the half-space). The inverse problems are those of determining medium parameters by giving the forms of incident and reflected waves or by giving the vibrations of certain points of the medium. The research method used of all inverse problems is spectral-analytical, consisting in reducing the considered inverse problems to know inverse problems for the Sturm-Liouville equation or for the string equation. Discrete inverse problems are also considered in this volume. In these problems an arbitrary set of point sources (emissive sources, oscillators, point masses) are determined. This volume will be of value and interest to researchers and scientists in mathematical physics, geophysics and biophysics.
CONTENTS: INVERSE PROBLEMS FOR SEMIBOUNDED STRING WITH THE DIRECTIONAL DERIVATIVE CONDITION GIVEN IN THE END Formulation of the direct problem The form of solution of the direct problem convenient for solving the inverse problem The inverse problem with the data and The inverse problem for semibounded string which has no analog in the case x = 0 INVERSE PROBLEMS FOR THE ELLIPTIC EQUATION IN THE HALF-PLANE Formulation of the direct problem The form of solution of the direct problem applied for solution of the inverse problem The setting and solution of the inverse problem INVERSE PROBLEMS OF SCATTERING PLANE WAVES FROM INHOMOGENEOUS TRANSITION LAYERS (HALF-SPACE) The direct problem Determination of properties of inhomogeneous layer by the forms of incident and reflected waves given for a single angle The method of recovery of the density and the speed in the inhomogeneous layer as the functions of the depth given the set of plane waves reflected from the layer at various angles The algorithm of numerical solution of the inverse problem 3.3 Derivation of the speed v(z) and the density in the numerical experiments INVERSE PROBLEMS FOR FINITE STRING WITH THE CONDITION OF DIRECTIONAL DERIVATIVE IN ONE END Formulation of the direct problem Solution of the direct problem The inverse problem with the data in the free end of the string The inverse problem with the data set in the boundary z = 0 Inverse problems for the stringh with the fixed end z = H INVERSE PROBLEMS FOR THE ELLIPTIC EQUATION IN THE STRIP Setting of the direct problem Solution of the direct problem The inverse problem with the data in the boundary z = H The inverse problem with the data in the boundary z = 0 Problems with the condition INVERSE PROBLEMS OF SCATTERING THE PLANE WAVES FROM INHOMOGENEOUS LAYERS WITH A FREE OR FIXED BOUNDARY The direct problem Determination of properties of the inhomogeneous layer given the data for a single angle of incidence Determination of the depth of inhomogeneous layer, the density and the speed v(z) in the layer in the form of incident wave is know Determination of the depth of inhomogeneous layer, the density the density , the speed v(z) in the layer and the form of incident wave DIRECT AND INVERSE PROBLEMS FOR THE EQUATIONS OF MIXED TYPE Formulation and the uniqueness theorem for the direct problem The representation of solution of the direct problem 7.1 The case of The case of Lavrentiev-Bitsadze equation. The formulas for solution of the direct problem 7.1 Inverse problems. The case The general case The other problems The physical content INVERSE PROBLEMS CONNECTED WITH DETERMINATION OF ARBITRARY SET OF POINT SOURCES Direct problem and its solution Some auxiliary geometrical definitions The auxiliary results for the case 1 connected with the T-systems Preliminary remarks on solutions of the inverse problems The static inverse problem with the data on the strait line The nonstationary inverse problem with the data given in the straight line for the case 1 The inverse static and nonstationary problems On zeros of the field u(x, y, z) of form (8.1.13) The zeros of the function u(x, y, z, t) of forms (8.1.4) or (8.1.5) in the plane z = 0 Solution of the nonstationary inverse problem 8.1 in the case 2 for E = E1, E2, E3, E4, E5, E6 Stationary inverse problem Possible applications Bibliography
Inverse and Ill-Posed Problems Series
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