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Handbook of mathematics for engineers and scienteists part 94

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 94', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 14.7. Boundary Value Problems for Parabolic Equations with One Space Variable 619 where 14.7.1.2 Lx t w a x t - b x t W c x t w a x t 0. dx2 dx Consider the nonstationary boundary value problem for equation 14.7.1.1 with an initial condition of general form w f x at t 0 14.7.1.3 and arbitrary nonhomogeneous linear boundary conditions dW n 01 P1W gi t dx dw n 02 P2W g2 t dx at x x1 at x x2. 14.7.1.4 14.7.1.5 By appropriately choosing the coefficients o1 02 P1 and P2 in 14.7.1.4 and 14.7.1.5 we obtain the first second third and mixed boundary value problems for equation 14.7.1.1 . 14.7.1-2. Representation of the problem solution in terms of the Green s function. The solution of the nonhomogeneous linear boundary value problem 14.7.1.1 - 14.7.1.5 can be represented as ft fX2 fX2 w x t y T G x y t t dydr i f y G x y t 0 dy J0 JX1 JX1 g1 T a x1 t A1 x t t dT g2 T a x2 t A2 x t t dT. 14.7.1.6 00 Here G x y t t is the Green s function that satisfies for t t 0 the homogeneous equation dG t - Lx t G 0 14.7.1.7 with the nonhomogeneous initial condition of special form G ô x - y at t t 14.7.1.8 and the homogeneous boundary conditions dG n t O - p1G 0 at x x1 dx 14.7.1.9 dG n t o2 p2G 0 at x x2. dx 14.7.1.10 The quantities y and t appear in problem 14.7.1.7 - 14.7.1.10 as free parameters with x1 y x2 and ô x is the Dirac delta function. The initial condition 14.7.1.8 implies the limit relation c X2 f x lim f y G x y t t dy t T.Jx1 for any continuous function f f x . 620 Linear Partial Differential Equations TABLE 14.6 Expressions of the functions A1 x t t and A2 x t t involved in the integrands of the last two terms in solution 14.7.1.6 Type of problem Form of boundary conditions Functions Am x t t First boundary value problem ai a2 0 3i 32 1 w g1 t at x x1 w g2 t at x x2 Ai x t t dyG x y t t 1 y x1 A2 x t t -dy G x y t t 1 y x2 Second boundary value problem ai 02 1 3i 32 0 dxw g1 t at x x1 dxW g2 t at x x A1 x t t -G x x1 t t A2 x t t G x x2 t t Third boundary value problem .