tailieunhanh - Handbook of mathematics for engineers and scienteists part 103

Handbook of mathematics for engineers and scienteists part 103. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 682 Nonlinear Partial Differential Equations Linear separable equations of mathematical physics admit exact solutions in the form w x y 1 X 1 y 2 x 2 y Pn x n y where wi pi x i y are particular solutions the functions p-i x as well as the functions i y with different numbers i are not related to one another. Many nonlinear partial differential equations with quadratic or power nonlinearities f1 x g1 y H1 w f2 x g2 y w fm x gm y nm w 0 also have exact solutions of the form . In the ni w are differential forms that are the products of nonnegative integer powers of the function w and its partial derivatives dx w dy w dxxw dxy w dyy w dxxxw etc. We will refer to solutions of nonlinear equations as generalized separable solutions. Unlike linear equations in nonlinear equations the functions pi x with different subscripts i are usually related to one another and to functions j y . In general the functions pi x and j y in are not known in advance and are to be identified. Subsection gives simple examples of exact solutions of the form with n 1 and n 2 for 1 p2 1 to some nonlinear equations. Note that most common of the generalized separable solutions are solutions of the special form w x y p x y x x the independent variables on the right-hand side can be swapped. In the special case of X x 0 this is a multiplicative separable solution and if p x 1 this is an additive separable solution. Remark. Expressions of the form are often used in applied and computational mathematics for constructing approximate solutions to differential equations by the Galerkin method and its modifications . . General form of functional differential equations. In general on substituting expression into the differential equation one arrives at a functional differential equation 1 X tf 1 Y 2 X 2 Y k Wfc Y 0 for the pi x and y . The functionals dj X and j Y depend only on x and y