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Handbook of mathematics for engineers and scienteists part 119. 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. | 794 Nonlinear Partial Differential Equations Hence the velocity vector for the continuous solution u is a right eigenvector of the matrix J for any point u and the corresponding eigenvalue equals the self-similar coordinate Xk u ark. 15.14.4.66 Here Xk Xk u is a root of the algebraic equation det G - XI 0 rk rk u is a solution of the corresponding degenerate linear system of equations G - Xl r 0 and a a u is a positive function that will be defined below. Differentiating both sides of the first equation 15.14.4.66 with respect to yields a v r- k rk r - rn Any n Xn hyperbolic system allows for n continuous solutions of system 15.14.4.65 corresponding to n characteristic velocities X Xk. The continuous solutions are determined by n systems of ordinary differential equations. Each system is represented by a phase portrait in the n-dimensional u-space. A solution trajectory that corresponds to a characteristic velocity Xk is called a kth rarefaction wave. 3 . A trajectory of solution 15.14.4.47 in the space u u1 . un T is called a solution path. The path is parametrized by the self-similar coordinate . The path connects the point u uL with the point u uR. The self-similar coordinate monotonically increases along the path varying from -to at u uL to to at u uR. The path consists of continuous segments representing solutions of the ordinary differential equations 15.14.4.65 rarefaction waves line segments that connect two points u- and u satisfying the Rankine-Hugoniot conditions 15.14.4.54 and evolutionary conditions 15.14.4.59 and rest points u const. Example 8. Consider a solution consisting of two shocks and one rarefaction. The structural formula for the solution path is uL 1 2 uR specifically i ul if -to x t D1 1 u1 if D1 x t A2 u1 u x t u . if A2 u1 x t A2 u2 1 u2 if A2 u2 x t D2 I ur if D2 x t to. The shock speed D1 resp. D2 can be found from the Hugoniot condition by setting u uL and u u1 resp. u- u2 and u uR . Points 1 and 2 are located on the same rarefaction
