By V. S. Vladimirov (auth.), V. S. Vladimirov (eds.)

The huge software of contemporary mathematical teehniques to theoretical and mathematical physics calls for a clean method of the process equations of mathematical physics. this can be very true just about this sort of primary notion because the 80lution of a boundary worth challenge. the idea that of a generalized answer significantly broadens the sphere of difficulties and permits fixing from a unified place the main attention-grabbing difficulties that can not be solved via utilizing elassical equipment. To this finish new classes were written on the division of upper arithmetic on the Moscow Physics anrl expertise Institute, specifically, "Equations of Mathematical Physics" via V. S. Vladimirov and "Partial Differential Equations" via V. P. Mikhailov (both books were translated into English by means of Mir Publishers, the 1st in 1984 and the second one in 1978). the current choice of difficulties relies on those classes and amplifies them significantly. in addition to the classical boundary worth difficulties, we've got ineluded various boundary price difficulties that experience in simple terms generalized strategies. resolution of those calls for utilizing the tools and result of quite a few branches of recent research. for that reason we now have ineluded difficulties in Lebesgue in tegration, difficulties concerning functionality areas (especially areas of generalized differentiable capabilities) and generalized capabilities (with Fourier and Laplace transforms), and crucial equations.

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**Extra info for A Collection of Problems on the Equations of Mathematical Physics**

**Example text**

67. If f E Hf (a, b), then on Ca, b] f (x) is equivalent to a continuous function. 68. If f (x) E Hf (-00,00), then f (x) tends to zero as I x 1-+ 00. 69. • 0 00 11 f 11'H:'1 (O, 2n) = ~ (aÄ +b~) (k 2+ 1) k=O defines one of the equivalent norms of Hf (0, 231). 70. A function f E L 2 (0, 31) belongs to Hf (0, 31) if and only if the n f series with the general term k2b~ converges, with bk =~ f (x)sinkx dx. o The norm is defined thus: n 00 IIfWol = (' (P+f'2)dx= ~ ~ (k 2 +1)bl. 71. 1 f2 dx ~ a (b ~ a) 2 f /,2 b dx a (the one-dimensional case of the Steklov inequality).

3. No. {b) 1. No. 2. Yes. 3. No. (c) 1. No. 2. Yes. 3. No. 35. a>O, ß>O, 2a; +2if<1. 36. (a) a<1, (b) a>1. 37. a > 7/4. 38. (a) a < n/2, (b) a > n/2, (c) not for a single value of a. 4 Function Spaces A complex (real) linear space is a set M for who'3e elements the operations of addition and multiplication by complex (real) numbers have been defined. t,+ 12) = c/l + c/2' (f) (Cl C2) I = Cl (c 2/), and (g) 11 = I for all /1, 12, 13, and I from M and all complex (real) numbers c, Cl' and c2 • A system of elements /1' ...

The magnetic permeability of the medium, and I (x, t) = (11' 1 2 , 1 3 ) the conduction current. In the case (a) the components of E and of H obey the same telegrapher's equation. e a

0, H 1=0 = Ho ° ° x> , t>O, Hlt=o=O, Htlt=o, sin Qt, t> 0, where c is the speed of light. 34. Ut = a 2u xx , < x< 1, t> 0, u (x, 0) = f (x), 0:::;:; x :::;:; 1; the boundary conditions are (a) u Ix=o = 'PI (t), U Ix=1 =

0, (b) -kSu x Ix=o = ql (t), kSu x Ix=1 = 'P2 (t), t> 0, (c) Ux Ix=o = h Cu (0, t) - 'PI (t)l, Ux Ix=l = -h Cu (l, t) - 'P2 (t»), a 2 = k/cp is the specific heat, 'PI (t) and 'P2 (t) are the temperatures of the ends of the rod in the case (a) 01' the temperatures of the surrounding medium at the two ends in the case (b), and the qi are the heat fluxes at the ends of the rod.