tailieunhanh - Handbook of mathematics for engineers and scienteists part 101

Handbook of mathematics for engineers and scienteists part 101. 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. | 668 Nonlinear Partial Differential Equations where A k plays the role of the wave propagation velocity the sign of A can be arbitrary the value A 0 corresponds to a stationary solution and the value k 0 corresponds to a space-homogeneous solution . Traveling-wave solutions are characterized by the fact that the profiles of these solutions at different time instants are obtained from one another by appropriate shifts translations along the æ-axis. Consequently a Cartesian coordinate system moving with a constant speed can be introduced in which the profile of the desired quantity is stationary. For k 0 and A 0 the wave travels along the æ-axis to the right in the direction of increasing æ . A traveling-wave solution is found by directly substituting the representation into the original equation and taking into account the relations wx kW wt -AW etc. the prime denotes a derivative with respect to z . Traveling-wave solutions occur for equations that do not explicitly involve independent variables f w ÎW . . 2 0. dæ dt dæ2 dædt dt2 J Substituting into we obtain an autonomous ordinary differential equation for the function W z F W kW -AW k2 W -kAW A2W . 0 where k and A are arbitrary constants. Example 1. The nonlinear heat equation dw d I dw at dX f w dX admits a traveling-wave solution. Substituting into we arrive at the ordinary differential equation fc2 f W W XW 0. Integrating this equation twice yields its solution in implicit form 0 WCW -z where C1 and C2 are arbitrary constants. Example 2. Consider the homogeneous Monge-Ampere equation d2w A2 d2w d2w dxdt J dx2 dt2 Inserting into this equation we obtain an identity. Therefore equation admits solutions of the form w W kx - At where W z is an arbitrary function and k and A are arbitrary constants. . Invariance of solutions and equations under translation transformations. Traveling-wave solutions are .