tailieunhanh - Handbook of mathematics for engineers and scienteists part 104

Handbook of mathematics for engineers and scienteists part 104. 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. | . Method of Generalized Separation of Variables 689 2 . At the second stage we successively substitute the YfX and j Y of into all solutions to obtain systems of ordinary differential equations for the unknown functions fp x and v q y . Solving these systems we get generalized separable solutions of the form . Remark 1. It is important that for fixed k the bilinear functional equation used in the splitting method is the same for different classes of original nonlinear mathematical physics equations. Remark 2. For fixed m solution contains m k - m arbitrary constants Ci j. Given k the solutions having the maximum number of arbitrary constants are defined by Solution number Number of arbitrary constants Conditions on k m 2k 4k2 if k is even m 2 k 1 4 k2 - 1 if k is odd. It is these solutions of the bilinear functional equation that most frequently result in nontrivial generalized separable solution in nonlinear partial differential equations. Remark 3. The bilinear functional equation and its solutions play an important role in the method of functional separation of variables. For visualization the main stages of constructing generalized separable solutions by the splitting method are displayed in Fig. . . Solutions of simple functional equations and their application. Below we give solutions to two simple bilinear functional equations of the form that will be used subsequently for solving specific nonlinear partial differential equations. 1 . The functional equation 1 1 2 2 3 3 0 where i are all functions of the same argument and are all functions of another argument has two solutions 1 A1 3 2 A2 3 3 -A1 1 - 2 21 1 A1 3 42 A2 3 3 -A1 1 - A2 2. The arbitrary constants are renamed as follows A1 C1 1 and A2 C2y in the first solution and A1 -1 C1 2 and A2 C1 1 C1 2 in the second solution. The functions on the right-hand sides of the formulas in are assumed