tailieunhanh - Handbook of mathematics for engineers and scienteists part 86

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 86', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Linear and Quasilinear Equations 563 Figure illustrates the formation of the shock wave described by the generalized solution of Hopf s equation with f w w and generated from a solitary wave with the smooth initial profile . The nonsmooth step curves depicted in Fig. for x and are obtained from the smooth but many-valued curves shown in Fig. by means of Whitham s rule of equal areas. Figure . The formation of a shock wave generated from a solitary wave with the smooth initial profile. . Utilization of integral relations for determining generalized solutions. Generalized solutions which are described by piecewise-smooth piecewise-continuous functions may formally be introduced by considering the following equation written in an integral form f L2É F w Si JJd Sx ay dy dx 0. Here D is an arbitrary rectangle in the yx-plane d d x y is any test function with continuous first derivatives in D that is zero at the boundary of D and the function F w is defined in equation . If w and F w are continuously differentiable then equation is equivalent to the original differential equation . Indeed multiplying equation by d integrating over the domain D and then integrating by parts we obtain equation . Conversely integrating by parts yields d L dw dF w dx dy idy dx 0. Since this equation must be satisfied for any test function d and since F w f w we obtain the original equation . However equation has a wider class of solutions since the admissible functions w x y need not necessarily be differentiable. The functions w x y satisfying the integral relation for all test functions d are referred to as generalized or weak solutions of equation . The use of generalized solutions is convenient for the description of discontinuities since it permits one to obtain jump conditions automatically. Consider a solution of equation