tailieunhanh - Handbook of mathematics for engineers and scienteists part 90

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 90', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Basic Problems of Mathematical Physics 591 many particular solutions. The specific solution that describes the physical phenomenon under study is separated from the set of particular solutions of the given differential equation by means of the initial and boundary conditions. Throughout this section we consider linear equations in the n-dimensional Euclidean space R or in an open domain V e R exclusive of the boundary with a sufficiently smooth boundary S dV. . Parabolic equations. Initial and boundary conditions. In general a linear second-order partial differential equation of the parabolic type with n independent variables can be written as ơw - Lx t w x t where _ _ _ . d2w . . dw . . Lx t w s 2 aij x t bi x t Ox- c x t w i j 1 j i 1 x x1 . xn 2 aij x f i j ơ 2 2 ơ 0. Parabolic equations govern unsteady thermal diffusion and other phenomena dependent on time t. Equation is called homogeneous if x t 0. Cauchy problem t 0 x e Rra . Find a function w that satisfies equation for t 0 and the initial condition w f x at t 0. Boundary value problem t 0 x e V . Find a function w that satisfies equation for t 0 the initial condition and the boundary condition rx t w g x t at xe S t 0 . In general rx t is a first-order linear differential operator in the space variables x with coefficient dependent on x and t. The basic types of boundary conditions are described in Subsection . The initial condition is called homogeneous if f x 0. The boundary condition is called homogeneous if g x t 0. . Hyperbolic equations. Initial and boundary conditions. Consider a second-order linear partial differential equation of the hyperbolic type with n independent variables of the general form - - x t - Lx t w x t ƠI- dt Boundary value problems for parabolic and hyperbolic equations are sometimes called mixed or initialboundary value problems. 592 Linear Partial