By V. S. Vladimirov (auth.), V. S. Vladimirov (eds.)
The broad software of recent mathematical teehniques to theoretical and mathematical physics calls for a clean method of the process equations of mathematical physics. this can be very true on the subject of this kind of primary suggestion because the 80lution of a boundary price challenge. the concept that of a generalized answer significantly broadens the sector of difficulties and permits fixing from a unified place the main fascinating difficulties that can't be solved through utilizing elassical equipment. To this finish new classes were written on the division of upper arithmetic on the Moscow Physics anrl expertise Institute, particularly, "Equations of Mathematical Physics" through V. S. Vladimirov and "Partial Differential Equations" through V. P. Mikhailov (both books were translated into English through 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. along with the classical boundary price difficulties, now we have ineluded a number of boundary price difficulties that experience purely generalized ideas. resolution of those calls for utilizing the tools and result of numerous branches of contemporary research. for that reason we've got ineluded difficulties in Lebesgue in tegration, difficulties related to functionality areas (especially areas of generalized differentiable features) and generalized services (with Fourier and Laplace transforms), and necessary equations.
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Extra resources for A Collection of Problems on the Equations of Mathematical Physics
Ue~-uTlTI- -U~t=O; S=x, 'YJ=Y-X, s ~= ; x--}·y+-}z. 6. Utt+unn+ + ucc + Un = 0; = x, 'YJ = y - x, I; = x - 11 + z, 't = 2x - 2y + z + t. 7. : + u n = 0; S = x+ y, 'YJ = = y - x, 1; = z, 't = Y + z + t. 8. Ust - UT]T] + ucc - u n = 0; S = x + y. 'YJ = x - y, ~ = -2y + z + t, 't = Z - t. 9. Ust- UT]n + ut:c = 0; = x, 'YJ = y - x, I; = 2x - y+ z, T = x+ + z+ 10. t. uss+unTl=O; s s=x, n 11. 2J k=1 'Yj=y, 1;= -x-y+z, n k USkSk 0, Sk = = k 2J1=1Xl; k = 1, 2, ... , n. 12. =0, Sk= ~ XI, k= 1,2, ... , n.
X 2u xx - 2xu xy Uyy = O. + + + Suppose the coefficients of Eq. 1) are continuous in a region D. The function U (x, y) is said to be a solution of Eq. 1) if it belongs to the dass C 2 (D) and satisfies Eq. 1) in D. The collection of all the solutions of Eq. 1) is said to be the general solution of Eq. 1). 3. Find the general solution of each of the equations with constant coefficients given below: 1. u xy = O. 2. Uxx - a2uyy = O. 2u xy - 3uyy = O. 4. u xy + au x = O. 5u xy - 2uyy + 3u x + uy = 2.
Prove that a function that is nondecreasing (nonincreasing) in [a, b] is measurahle. 15. Prove that if I (a:) is measurable in Q, then there is a sequence of polynomials that converges to I (x) almost everywhere in Q. We will assume that a function I (x) defined in a region Q belongs to the class L+ (Q) if there is a nondecreasing sequence of finite functions In (x), n = 1, 2, ... , that are continuous in Q such that it converges to I (x) alm ost everywhere in Q and such that the sequence of the (Riemann) integrals In (x) da: is bounded from above.
A Collection of Problems on the Equations of Mathematical Physics by V. S. Vladimirov (auth.), V. S. Vladimirov (eds.)