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Handbook of mathematics for engineers and scienteists part 108

Handbook of mathematics for engineers and scienteists part 108. 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 717 Remark. In the special case of transformations in the plane the functions p1 and p2 in 15.8.1.1 are independent of w and fi w i.e. w w . The expansion of 15.8.1.1 into truncated Taylor series in e about e 0 to linear terms gives x x eZ x y w y y ey x y w wo w eZ x y w 15.8.1.2 where Az x 9V1 I Z x y w de le 0 z 9fi2 I n x y w de le 0 Az . i l I Z x y w . de le 0 The linear first-order differential operator X Z x y w d n x y w d Z x y w - - dx dy dw 15.8.1.3 which corresponds to the infinitesimal transformation 15.8.1.2 is called an infinitesimal operator. Theorem Lie . Suppose the coordinates Z x y w n x y w Z x y z of the infinitesimal operator 15.8.1.3 are known. Then the transformation 15.8.1.1 having the group property can be completely recovered by solving the Lie equations T2 Z J1 J 2 f j2 fi fi1 J2 f r Z fi1 fi2 f de de de with the initial conditions J1 e 0 X fi2 e 0 y f e 0 W. 15.8.1-2. Invariant of an infinitesimal operator. Transformations in the plane. An invariant of the operator 15.8.1.3 is a function I x y w that satisfies the condition I x y w I x y w . Let us expand this equation in Taylor series in the small parameter e divide the resulting relation by e and the proceed to the limit as e - 0 to obtain a linear partial differential equation for I XI Z x y w n x y w dI Z x y w - 0. dx dy dw 15.8.1.4 Let the associated characteristic system of ordinary differential equations see Paragraph 13.1.1-3 dx dy dw Z x y w n x y w Z x y w 15.8.1.5 In the literature it is also known as an infinitesimal generator or a group generator. In this book we call an operator 15.8.1.3 an infinitesimal operator a group generator or for short an operator. 718 Nonlinear Partial Differential Equations TABLE 15.7 One-parameter transformations in the plane Name Transformation Transformation Invariant Translation in the x-axis x x s y y v d X - dx Ii - y Translation along the straight line ax by 0 x x .