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Handbook of mathematics for engineers and scienteists part 111

Handbook of mathematics for engineers and scienteists part 111. 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. | 738 Nonlinear Partial Differential Equations Substituting the derivatives of the function w from 15.10.1.5 and 15.10.1.6 into 15.10.1.7 we obtain the following third-order ordinary differential equation for Vy - W yy aVyyy 15.10.1.8 which represents the compatibility condition for equations 15.10.1.4 and 15.10.1.5 . In order to construct an exact solution we integrate equation 15.10.1.5 to obtain w ip y x Q y . 15.10.1.9 The function Q y is found by substituting 15.10.1.9 into 15.10.1.4 and taking into account the condition 15.10.1.8 . As a result we arrive at the ordinary differential equation yQy - PQyy aQ yy. 15.10.1.10 Finally we obtain an exact solution of the form 15.10.1.9 with the functions ip and Q described by equations 15.10.1.8 and 15.10.1.10 . Remark. It is easier to obtain the above solution by directly substituting expression 15.10.1.9 into the original equation 15.10.1.4 . 15.10.1-2. General description of the differential constraints method. The procedure of the construction of exact solutions to nonlinear equations of mathematical physics by the differential constraints method consists of several steps described below. 1 . In the general case the identification of particular solutions of the equation dw dw d2w d2w d2 w V W dx dy dx2 dxdy dy2 15.10.1.11 is performed by supplementing this equation with an additional differential constraint i dw dw d2w d2w d2w y w dx dy dx2 dxdy dy2 15.10.1.12 The form of the differential constraint 15.10.1.12 may be prescribed on the basis of i a priori considerations for instance it may be required that the constraint should represent a solvable equation ii certain properties of the equation under consideration for instance it may be required that the constraint should follow from symmetries of the equation or the corresponding conservation laws . 2 . In general the thus obtained overdetermined system 15.10.1.11 - 15.10.1.12 requires a compatibility analysis. If the differential constraint 15.10.1.12 is specified .