By V. S. Vladimirov (auth.), V. S. Vladimirov (eds.)
The broad program 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 with reference to this kind of primary proposal because the 80lution of a boundary worth challenge. the concept that of a generalized resolution significantly broadens the sector of difficulties and permits fixing from a unified place the main fascinating difficulties that can not be solved through making use of 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" by means of V. S. Vladimirov and "Partial Differential Equations" by way of V. P. Mikhailov (both books were translated into English through Mir Publishers, the 1st in 1984 and the second one in 1978). the current number of difficulties is predicated on those classes and amplifies them significantly. in addition to the classical boundary price difficulties, we now have ineluded a number of boundary price difficulties that experience basically generalized strategies. answer of those calls for utilizing the equipment and result of numerous branches of recent research. as a result now we have ineluded difficulties in Lebesgue in tegration, difficulties related to functionality areas (especially areas of generalized differentiable services) and generalized features (with Fourier and Laplace transforms), and crucial equations.
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Additional resources for A Collection of Problems on the Equations of Mathematical Physics
46. If fE L 2 (at b) and ~ x"f (x) dx = f (x) = ° a ° for k = 0, 1, ... , then almost everywhere in (a, b). 47. If fEL 2 (Q) and ~ xrt,f(x)dx=O for all a,lal=O, Q 1, ... , then f(x)=O almost everywhere in Q. 48. If fk and gk' k = 1, 2, ... 49. Every orthonormal system el' e 2 , ••• , pendent. 50. PI' ... Pn belonging to L 2 (Q) is linearly independent if and only if Gram's determinant det i, j = 1, ... , n, is nonzero. PI' ... Pn constitutes a linearly independent system of functions belonging to L 2 (Q) (or L 2 • P (Q».
U;y 2xyuy - 2xu 7. u xy + Ux + yu u + (y - 1) U = O. 8. u xy + XU x + 2yuy + 2xyu = O. 3. + + + y2 uuu = = O. O. Answers to Problems of Sec. 1. 1. 1 1 u~s+unn+u~c=O; S=x, 'YJ=y-x, ~=x-2Y+T z. 1 1 1 2. uGS-UTlTI+u~~+UTj=O; S=T x , 'YJ= TX+Y' ~= - 2 x-y+z. 3. UH-unTj+2u~=0; S=x+y, 'YJ=y-x, I;=y+z. 4. U;, +uTITI = 0; S= x, 'YJ = y-x, ~= 2x- y + z. 5. 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.
23. , n=O, 1, ... , eonstitute a system of polynomials (the Hermite polynomials) orthogonal in L z•e-x2 ( - 00, 00). 24. Suppose the operator -d2/dx 2 is defined on funetions that beCi ([0, 11) with the boundary conditions (hu long to ca «0, 1» - ux ) 1=0 = U 1=1 = (h a eonstant). Show that the eigenfunctions of this operator corresponding to different eigenvalues eonstitute an orthogonal system in La (0, 1). 25. Suppose the operator - \72 is defined on functions that nCi (Q) belong to C2 (Q) + g (x) u) Ir = ° or uIr = 0, wi th the boundary condi tion ( :: + where gE C (r).