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Handbook of mathematics for engineers and scienteists part 110

Handbook of mathematics for engineers and scienteists part 110. 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. | 15.8. Classical Method of Studying Symmetries of Differential Equations 731 15.8.4-2. Group invariants. Local transformations of derivatives. A universal invariant or for short an invariant of a group 15.8.4.1 and a operator 15.8.4.3 is a function I x w that satisfies the condition I x w I x w . The expansion in a series in powers of the small parameter e gives rise to the linear partial differential equation for I 15.8.4.4 XI Zi x w - Z x w 0. dxi dw From the theory of first-order partial differential equations it follows that the group 15.8.4.1 and the operator 15.8.4.3 have n functionally independent universal invariants. On the other side this means that any function F x w that is invariant under the group 15.8.4.1 can be written as a function of n invariants. In the new variable 15.8.4.1 the derivatives are transformed as follows dw dw d2w d2w d3w d3w dxi dx E i dxidVCj dxidxj E ij dxidxjdxk dxidxjdxk E ijk 15.8.4.5 The coordinates Zi Zij and Zijk of the first three prolongations are expressed as Zi Di Z - PsDi Zs Zij Dj Zi - i D Zs 15.8.4.6 Zijk Dk Zij - rjsDk Zs where summation is assumed over the repeated index s and the following short notation partial derivatives are used dw d2w d3w pi qij t e rijk t e e dxi dxidxj dxidxj dxk Di d Pw Fj- rijk- dxi dw dpj dqjk with Di being the total differential operator with respect to xi . 15.8.4-3. Invariant condition. Splitting procedure. Invariant solutions. We will consider partial differential equations of order m in n independent variables Z dw d2w d3w F I x w nZ dxi dxidxj dxidxjdxk 15.8.4.7 where i j k 1 . n. The group analysis of equation 15.8.4.7 consists of several stages. At the first stage let us require that equation 15.8.4.7 be invariant must preserve its form under transformations 15.8.4.1 so that d 3w w drCi gXidXj dXidXjdxk 0 d2w 15.8.4.8 732 Nonlinear Partial Differential Equations Let us expand this expression into a series in powers of e about e 0 taking into account that the leading term 15.8.4.7 .