tailieunhanh - Handbook of mathematics for engineers and scienteists part 85

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 85', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 556 First-Order Partial Differential Equations . Cauchy Problem. Existence and Uniqueness Theorem . Cauchy problem. Consider two formulations of the Cauchy problem. 1 . Generalized Cauchy problem. Find a solution w w x y of equation satisfying the initial conditions x h1 y h2 w 3 where is a parameter a 3 and the lik are given functions. Geometric interpretation find an integral surface of equation passing through the line defined parametrically by equation . 2 . Classical Cauchy problem. Find a solution w w x y of equation satisfying the initial condition w p y at x 0 where p y is a given function. It is convenient to represent the classical Cauchy problem as a generalized Cauchy problem by rewriting condition in the parametric form x 0 y w p . . Procedure of solving the Cauchy problem. The procedure of solving the Cauchy problem involves several steps. First two independent integrals of the characteristic system are determined. Then to find the constants of integration C1 and C2 the initial data must be substituted into the integrals to obtain U1 h1 h2 h3 C1 U2 h1 h2 h3 C2. Eliminating C1 and C2 from and yields U1 x y w U1 h1 h2 h3 3 1 2 U2 x y w U2 h1 h2 hs . Formulas are a parametric form of the solution of the Cauchy problem . In some cases one may succeed in eliminating the parameter from relations thus obtaining the solution in an explicit form. Example 1. Consider the Cauchy problem for linear equation W a W bw dx dy subjected to the initial condition . The corresponding characteristic system for equation dx dy dw 1 a bw has two independent integrals y - ax C1 weTbx C2. Represent the initial condition in parametric form and then substitute the data into the integrals . As a result