TAILIEUCHUNG - Handbook of mathematics for engineers and scienteists part 109

Handbook of mathematics for engineers and scienteists part 109. 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. | 724 Nonlinear Partial Differential Equations Similarly it can be established that the following special forms of f result in additional operators 1. f e- X4 xdx 2d- 2. f wk k 0 -4 3 -4 X4 kxdx 2wdw 3. f w 4 X4 2xdx - 3wdw X5 x2dx - 3xwdw 4. f w- X4 2xdx - wdw X5 t2dt twdw. The symmetries obtained with the procedure presented can be used to find exact solutions of the differential equations considered see below . . Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions . Using symmetries of equations for constructing one-parameter solutions. Suppose a particular solution w g x y of a given equation is known. Let us show that any symmetry of the equation defined by a transformation of the form generates a one-parameter family of solutions except for the cases where the solution is not mapped into itself by the transformations see Paragraph . Indeed since equation converted to the new variables acquires the same form then the transformed equation has a solution w g x y . in going back to the old variables by formulas we obtain a one-parameter solution of the original equation . Example 1. The two-dimensional heat equation with an exponential source S dw e- has a one-dimensional solution w ln r. x2 Equation admits the operator X3 ydx-xdy see Example 1 in Subsection which defines rotation in the plane. The corresponding transformation is given in Table . Replacing x in by X from Table we obtain a one-parameter solution of equation 2 w ln 7----- --- u x cos e y sin e 2 where e is a free parameter. . Procedure for constructing invariant solutions. Solution of equation is called invariant under transformations if it coincides with solution which must be rewritten in terms of the old variables using formulas . This means that .

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