Selected Works of S.L. Sobolev: Volume I: Equations of by Gennadii V. Demidenko, Vladimir L. Vaskevich PDF

By Gennadii V. Demidenko, Vladimir L. Vaskevich

ISBN-10: 038734148X

ISBN-13: 9780387341484

ISBN-10: 0387341498

ISBN-13: 9780387341491

The subjects coated during this quantity comprise Sobolev’s primary works on equations of mathematical physics, computational arithmetic, and cubature formulation. the various articles are typically unknown to mathematicians simply because they have been released in journals which are tricky to entry. this can be the 1st visual appeal in English of many works via this crucial Russian mathematician.

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Extra info for Selected Works of S.L. Sobolev: Volume I: Equations of Mathematical Physics, Computational Mathematics, and Cubature Formulas

Example text

Finally, in the third case, when |θ| > b, both equations are elliptic. We discuss all three cases separately. √ 2 − θ 2 y = η into the first equation, and a In the first case, substituting 1 √ b2 − θ2 y = η2 into the second equation, we reduce both equations to the form ∂2ψ ∂2ψ ∂2ϕ ∂2ϕ = 0. − = 0, − 2 2 ∂η22 ∂ξ 2 ∂η1 ∂ξ The general solution of these equations has the form ϕ = ϕ1 (ξ + η1 ) + ϕ2 (ξ − η1 ) = ϕ1 (t − θx + √ a2 − θ2 y) + ϕ2 (t − θx − ψ = ψ1 (ξ + η2 ) + ψ2 (ξ − η2 ) = ψ1 (t − θx + √ b2 − θ2 y) + ψ2 (t − θx − √ √ a2 − θ2 y), (4) b2 − θ2 y).

Ed. 12 S. L. 2) where f 1 and f 2 are functions of complex variables, bounded and regular in the upper half-plane. For this purpose, it is necessary to make an agreement to assign either positive real or positive imaginary value to the radicals in this formula. Obviously, the first case is obtained immediately, since the arguments of f1 and f2 are real, and their factors are real as well. As is known, the real part of a function of a complex variable on the real axis can take completely arbitrary value.

L. Sobolev As in any method of representing a solution as a definite integral, for solving a problem we need to define so-called density of spectrum in the representation. For this purpose, we identify integrals obtained by H. Lamb with those obtained by us. After defining in such way the spectral function, we transform the obtained results to a form more convenient for calculation. Now we move on to the presentation of our method. 2. In our note [2] we had already studied the reflection of longitudinal and transverse elastic plane waves falling at different angles on the plane.

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Selected Works of S.L. Sobolev: Volume I: Equations of Mathematical Physics, Computational Mathematics, and Cubature Formulas by Gennadii V. Demidenko, Vladimir L. Vaskevich

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