tailieunhanh - Handbook of mathematics for engineers and scienteists part 100

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 100', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Transformations of Equations of Mathematical Physics 661 With - we find the second derivatives d2w _ d2u d2w _ d2u d2w _ d2u dx2 dy2 dxdy d dy dy2 dd 2 where d2w d2w d2w 2 1 d2u d2u d2u 2 dx2 dy2 dxdy J J d 2 dy2 d dy The Legendre transformation with J 0 allows us to rewrite a general second-order equation with two independent variables dw dw d2w d2w d2w y dx dy dx2 dxdy dy2 in the form i du du A du du A d2u d2u 0 dn d ndï u n Jdn2 J d dn 0. Sometimes equation may be simpler than . Let u u n be a solution of equation . Then the formulas define the corresponding solution of equation in parametric form. Remark. The Legendre transformation may result in the loss of solutions for which J 0. Example 1. The equation of steady-state transonic gas flow ÈS 0 is reduced by the Legendre transformation to the linear equation with variable coefficients d1 u T-u aSdn2 df 0 Example 2. The Legendre transformation reduces the nonlinear equation f dw dw d2 w dw dw d2 w dw dw cdw o dx dy J dx2 g dx dy J dxdy dx dy J dy2 to the following linear equation with variable coefficients f 5 n - g S n - h . n o. ôq d oq d r . Euler transformation. The Euler transformation belongs to the class of contact transformations and is defined by the relations x y n w x - u. 9 Differentiating the last relation in with respect to x and y and taking into account the other two relations we find that dw dw -du. dx dy dy 662 Nonlinear Partial Differential Equations Differentiating these expressions in x and y we find the second derivatives .2 wxx ----- wxy - L Wyy JF1--------- LFL. u u uK The subscripts indicate the corresponding partial derivatives. The Euler transformation is employed in finding solutions and linearization of certain nonlinear partial differential equations. The Euler transformation allows us to reduce a general .