TAILIEUCHUNG - 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. | . Classical Method of Studying Symmetries of Differential Equations 731 . Group invariants. Local transformations of derivatives. A universal invariant or for short an invariant of a group and a operator 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 XI Zi x w - Z x w 0. dxi dw From the theory of first-order partial differential equations it follows that the group and the operator have n functionally independent universal invariants. On the other side this means that any function F x w that is invariant under the group can be written as a function of n invariants. In the new variable the derivatives are transformed as follows dw dw d2w d2w d3w d3w dxi dx E i dxidVCj dxidxj E ij dxidxjdxk dxidxjdxk E ijk 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 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 . . 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 where i j k 1 . n. The group analysis of equation consists of several stages. At the first stage let us require that equation be invariant must preserve its form under transformations so that d 3w w drCi gXidXj dXidXjdxk 0 d2w 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 .

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