tailieunhanh - Handbook of mathematics for engineers and scienteists part 102

Handbook of mathematics for engineers and scienteists part 102. 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. | . Traveling-Wave Self-Similar and Other Simple Solutions. Similarity Method 675 Exponential self-similar solutions exist if equation is invariant under transformations of the form t i ln C x Ck x w Cmw where C 0 is an arbitrary constant for some k and m. Transformation is a combination of a translation transformation in t and scaling transformations in x and w. It should be emphasized that these transformations contain an arbitrary parameter C while the equation concerned is independent of C . Let us find the relation between the parameters a 3 in solution and the parameters k m in the scaling transformation . Let w x t be a solution of equation . Then the function w ai t is a solution of equation . In view of the explicit form of solution we have w eatV xe3t . Going back to the original variables using we obtain w C m-aeatV C k-3 xe3t . Let us require that this solution coincide with which means that the uniqueness condition for the solution must be satisfied for any C 0. To this end we set a m 3 -k. in practice exponential self-similar solutions are sought using the above existence criterion if k and m in are known then the new variables have the form with parameters . Remark. Sometimes solutions of the form are also called limiting self-similar solutions. Example 1. Let us show that the nonlinear heat equation 3 W dt dx dx admits an exponential self-similar solution. Inserting into yields Cm -33 aC mn m-2k Xfn Equating the powers of C gives one linear equation m mn m - 2k. It follows that k 2mn where m is any number. Further using formulas and and setting without loss of generality m 2 this is equivalent to scaling in t we find the new variables w e2tV xe nt. Substituting these expressions into we arrive at an ordinary differential equation for V a Vn V j n Vg - 2