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By S. L. Sobolev

S.L. Sobolev (1908-1989) was once a very good mathematician of the 20th century. His chosen works integrated during this quantity laid the rules for extensive improvement of the fashionable concept of partial differential equations and equations of mathematical physics, they usually have been a gold mine for brand spanking new instructions of practical research and computational mathematics.The issues lined during this quantity comprise Sobolev's basic works on equations of mathematical physics, computational arithmetic, and cubature formulation. a number of the articles are in most cases unknown to mathematicians simply because they have been released in journals which are tough to entry.

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Lamb’s integral, and replace ϑ by π − ϑ1 and θ by −θ1 . Renaming and taking the half-sum of the results, we obtain π/2 b −1 dϑ πµ 1 w0 = 2π a 0 − 1 B πµ √ ′ b2 θ θ2 − a2 Q∗t (t − ̺θ cos ϑ) dθ √ √ (2θ2 − b2 )2 − 4θ2 θ2 − a2 θ2 − b2 ∞ b + 1 πµ −a −b √ ′ b2 θ(2θ2 − b2 )2 θ2 − a2 Q∗t (t − ̺θ cos ϑ) dθ (2θ2 − b2 )4 + 16θ4 (θ2 − a2 )(b2 − θ2 ) √ ′ b2 θ θ2 − a2 Q∗t (t − ̺θ cos ϑ) dθ √ √ (2θ2 − b2 )2 − 4θ2 θ2 − a2 θ2 − b2 −b 1 + B πµ −∞ 1 + 2π π b −1 dϑ πµ a π/2 1 − B πµ ∞ b 1 + πµ −a −b + 1 B πµ √ ′ b2 θ(2θ2 − b2 ) θ2 − a2 Q∗t (t − ̺θ cos ϑ) dθ (2θ2 − b2 )4 + 16θ4 (θ2 − a2 )(b2 − θ2 ) √ ′ b2 θ(2θ2 − b2 ) θ2 − a2 Q∗t (t − ̺θ cos ϑ) dθ (2θ2 − b2 )4 + 16θ4 (θ2 − a2 )(b2 − θ2 ) √ ′ b2 θ θ2 − a2 Q∗t (t − ̺θ cos ϑ) dθ √ √ (2θ2 − b2 )2 − 4θ2 θ2 − a2 θ2 − b2 √ ′ b2 θ(2θ2 − b2 )2 θ2 − a2 Q∗t (t − ̺θ cos ϑ) dθ (2θ2 − b2 )4 + 16θ4 (θ2 − a2 )(b2 − θ2 ) −b −∞ √ ′ b2 θ θ2 − a2 Q∗t (t − ̺θ cos ϑ) dθ √ √ .

8. In conclusion, let us point out several simple physical consequences following from our formulas. θ= Ξx − y 32 S. L. Sobolev It is interesting to trace moments of appearance of different phases of vibrations inside the medium in the case when initial conditions are zero for t < t0 , and the force begins to act for t ≥ t0 . For simplification, we assume that the force acting on the surface of the medium is the short and very intensive force of impulse type. This is reduced to the assumption that we have to take the function Q(t) under the integral sign, extremely large in a small interval of time and vanishing outside this interval.

25) From these formulas we immediately obtain the integral representation for u and v at any point inside the medium. 1) √ 1 i(2c2 − b2 ) c2 − a2 Qt (t + cx + i c2 − a2 y) E−c v = Re F (θ) µ √ 1 i(2c2 − b2 ) c2 − a2 Qt (t − cx + i c2 − a2 y) Ec − F (θ) µ 1 − πµi −a −b √ √ (θ2 − a2 )4θ2 (2θ2 − b2 ) b2 − θ2 Qt (t − θx + i θ2 − a2 y) dθ (2θ2 − b2 )4 − 16θ4 (θ2 − a2 )(θ2 − b2 ) 1 − 2πµi a −a ×[Qt (t − θx − √ (2θ2 − b2 ) a2 − θ2 √ √ (2θ2 − b2 )2 + 4θ2 a2 − θ2 b2 − θ2 a2 − θ2 y) + Qt (t − θx + a2 − θ2 y)] dθ 20 S.

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