TAILIEUCHUNG - 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 and into we obtain the following third-order ordinary differential equation for Vy - W yy aVyyy which represents the compatibility condition for equations and . In order to construct an exact solution we integrate equation to obtain w ip y x Q y . The function Q y is found by substituting into and taking into account the condition . As a result we arrive at the ordinary differential equation yQy - PQyy aQ yy. Finally we obtain an exact solution of the form with the functions ip and Q described by equations and . Remark. It is easier to obtain the above solution by directly substituting expression into the original equation . . 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 is performed by supplementing this equation with an additional differential constraint i dw dw d2w d2w d2w y w dx dy dx2 dxdy dy2 The form of the differential constraint 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 - requires a compatibility analysis. If the differential constraint is specified .

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