TAILIEUCHUNG - Handbook of mathematics for engineers and scienteists part 107

Handbook of mathematics for engineers and scienteists part 107. 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. | 710 Nonlinear Partial Differential Equations Equating the functional coefficient of u to zero and taking into account we obtain g --12 x Qt 2. 2 Substituting the expressions and into we arrive at the relation fiaau uu u 2 2 x tt Qtt u 2ipipttu - y 2 tx Qt 2 tt 6z fi .k x Qt 2 0. Let us perform the double differentiation of the expression in square brackets and then divide all terms by y6. Excluding x with the help of the relation x z - Q we get au 1 uu u 2 zAzUz Qtt - Q tt u 2y 5ipttu 0. Let us require that the functional coefficient of u be a function of only one variable z . x ttz ipQtt - Q tt g -yttz 5 Qtt - Q tt Az B where A and B are arbitrary constants. Hence we obtain the following system of ordinary differential equations for the functions ip and Q Ptt Af Qtt AQ B 4. Let us eliminate the second and the third derivatives of the functions p and Q from . As a result we arrive at the following ordinary differential equation for the function u z au uu u Az B u 2Au - 2 Az B 2 0. Formulas and together with equations - describe an exact solution of the Boussinesq equation . . Description of the Clarkson-Kruskal direct method. 1 . The basic idea of the method is the following for an equation with the unknown function w w x t an exact solution is sought in the form w x t f x t u z g x t z z x t . The functions f x t g x t and z x t are determined in the subsequent analysis so that ultimately one obtains a single ordinary differential equation for the function u z . 2 . Inserting into a nonlinear partial differential equation with a quadratic or a power nonlinearity we obtain 1 x t H1 u 2 x Wu m x t Hm u 0. Here n u are differential forms that are the products of nonnegative integer powers of the function u and its derivatives u z u zz etc. and k x t depend on the functions f x t g

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