Institute of Computational Mathematics and Mathematical Geophysics

Abstracts

Mathematical problems in geophysical investigations of solid Earth

Some differential identities are known in the vector analysis. For example, the divergence of the rotor is equal to zero. The new differential identity for a scalar function of two independent variables is established by means of the group approach. This identity contains the Laplacian and modulus of the gradient of a function. The following rezults are obtained by means of this identity.

1.Some integral identity is obtained. Using this identity,we can calculate some functionals in the inverse problems for the wave equation, the eiconal equation, the Helmholtz equation, the heat conductivity equation and other equations.

2. By means of a certain change of variables the nonlinear equation of characteristics of the wave equation can be reduced to a certain ordinary linear second-order differential equation.

3. The analog of this result is obtained for the certain family of nonlinear first-order differential equations, containing and generalizing the eiconal equation.

4. A certain property of the harmonic function and of the derivative of the conformal mapping is established.

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