tailieunhanh - Handbook of mathematics for engineers and scienteists part 98

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 98', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Duhamel s Principles in Nonstationary Problems 647 . Hyperbolic equations with two independent variables. Consider the problem for the homogeneous linear hyperbolic equation - x j a x b x c x w dt2 dt dx2 dx with the homogeneous initial conditions w 0 at t 0 dtw 0 at t 0 and the boundary conditions and . The solution of problem with the nonstationary boundary condition at x x1 can be expressed by formula in terms of the solution u x t of the auxiliary problem for equation with the initial conditions and boundary condition for u instead of w and the simpler stationary boundary condition at x x1. In this case the remark made in Paragraph remains valid. . Second-order equations with several independent variables. Duhamel s first principle can also be used to solve homogeneous linear equations of the parabolic or hyperbolic type with many space variables dk w dtk z z lJ w dw Saj x dx-dX- E bi x dx c x w i j 1 J i 1 where k 1 2 and x x1 . xn . Let V be some bounded domain in R with a sufficiently smooth surface S dV. The solution of the boundary value problem for equation in V with the homogeneous initial conditions if k 1 or if k 2 and the nonhomogeneous linear boundary condition Tx w g t for xe S is given by z d f . ft du. w x t J u x t - t g r dr J x t - t g r dr. Here u x t is the solution of the auxiliary problem for equation with the same initial conditions or for u instead of w and the simpler stationary boundary condition rx u 1 for x g S. Note that can represent a boundary condition of the first second or third kind the coefficients of the operator Tx are assumed to be independent of t. 648 Linear Partial Differential Equations . Problems for Nonhomogeneous Linear Equations .