tailieunhanh - Handbook of mathematics for engineers and scienteists part 97

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 97', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 640 Linear Partial Differential Equations TABLE The Green s functions of boundary value problems for equations of various types in bounded domains. In all problems the operators Lx and rx are the same x Equation Initial and boundary conditions Green s function Elliptic equation -Lx w x rx w g x for x e S no initial condition required cuk x ufe y X o G x y p IK IP Ak Xk 0 Parabolic equation dtw - Lx w x t w f x at t 0 rx w g x t for x e S G x y t 2 y exp Aki Hyperbolic equation dttw - Lx w x t w f0 x at t 0 w f1 x at t 0 rx w g x t for x e S G x y t H 2 2 siipVxkk k 1 uk pv Ak v 7 . Some remarks and generalizations. Remark 1. Formula can also be used if the domain V is infinite. In this case one should make sure that the integral on the right-hand side is convergent. Remark 2. Suppose the equations given in the first column of Table contain -Lx w - 3w instead of -Lx w with 3 being a free parameter. Then the Xk in the expressions of the Green s function in the third column of Table must be replaced by Xk - 3 just as previously the Xk and uk x were determined by solving the eigenvalue problem - . Remark 3. The formulas for the Green s functions presented in Table will also hold for boundary value problems described by equations of the fourth or higher order in the space variables provided that the eigenvalue problem for equation subject to appropriate boundary conditions is self-adjoint. . Green s Functions Admitting Incomplete Separation of Variables . Boundary value problems for rectangular domains. 1 . Consider the parabolic equation L1 t w Lng w x t where each term Lm t w depends on only one space variable xm and time t Lm t w am xm f . 2 bm xm f . cm xm t w dxm dxm m 1 . n. For equation we set the initial condition of general form w f x at t 0. Consider the domain V am xm 3m m 1 . n which is an n-dimensional parallelepiped. We set the .