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Handbook of mathematics for engineers and scienteists part 109

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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 . 15.8.3. Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions 15.8.3-1. Using symmetries of equations for constructing one-parameter solutions. Suppose a particular solution w g x y 15.8.3.1 of a given equation is known. Let us show that any symmetry of the equation defined by a transformation of the form 15.8.1.1 generates a one-parameter family of solutions except for the cases where the solution is not mapped into itself by the transformations see Paragraph 15.8.3-2 . Indeed since equation 15.8.2.1 converted to the new variables 15.8.1.1 acquires the same form 15.8.2.2 then the transformed equation 15.8.2.2 has a solution w g x y . 15.8.3.2 in 15.8.3.2 going back to the old variables by formulas 15.8.1.1 we obtain a one-parameter solution of the original equation 15.8.2.1 . Example 1. The two-dimensional heat equation with an exponential source S dw e- 15.8.3.3 has a one-dimensional solution 15.8.3.4 w ln r. x2 Equation 15.8.3.3 admits the operator X3 ydx-xdy see Example 1 in Subsection 15.8.2 which defines rotation in the plane. The corresponding transformation is given in Table 15.7. Replacing x in 15.8.3.4 by X from Table 15.7 we obtain a one-parameter solution of equation 15.8.3.3 2 w ln 7----- --- u x cos e y sin e 2 where e is a free parameter. 15.8.3-2. Procedure for constructing invariant solutions. Solution 15.8.3.1 of equation 15.8.2.1 is called invariant under transformations 15.8.1.1 if it coincides with solution 15.8.3.2 which must be rewritten in terms of the old variables using formulas 15.8.1.1 . This means that .

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