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Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 87', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 570 First-Order Partial Differential Equations Here n and are arbitrary numbers and t 0. We assume that problem 13.1.4.10 13.1.4.12 has a unique bounded solution. The stable generalized solution of the Cauchy problem 13.1.4.8 13.1.4.9 is given by w x y - 0 W x x y _ x y 14 13 w x y 0 W x x y x y where - x y and x y denote respectively the greatest lower bound and the least upper bound of the set of values n for which the function I x y f v n _ W 0 x y n dy 13.1.4.14 Jo takes the minimum value for fixed x and y x 0 . If function 13.1.4.14 takes the minimum value for a single 1 then _ and relation 13.1.4.14 describes the classical smooth solution. 13.2. Nonlinear Equations 13.2.1. Solution Methods 13.2.1-1. Complete general and singular integrals. A nonlinear first-order partial differential equation with two independent variables has the general form F x y w p q 0 where p q . dx dy 13.2.1.1 Such equations are encountered in analytical mechanics calculus of variations optimal control differential games dynamic programming geometric optics differential geometry and other fields. In this subsection we consider only smooth solutions w w x y of equation 13.2.1.1 which are continuously differentiable with respect to both arguments Subsection 13.2.3 deals with nonsmooth solutions . 1 . Let a particular solution of equation 13.2.1.1 w E x y C1 C2 13.2.1.2 depending on two parameters C1 and C2 be known. The two-parameter family of solutions 13.2.1.2 is called a complete integral of equation 13.2.1.1 if the rank of the matrix 13.2.1.3 M E1 Ex1 E 1 E2 Ex2 Ey2 is equal to two in the domain being considered for example this is valid if Ex1Ey2 -Ex2Ey1 0 . In equation 13.2.1.3 En denotes the partial derivative of E with respect to Cn n 1 2 Exn is the second partial derivative with respect to x and Cn and Eyn is the second partial derivative with respect to y and Cn . In some cases a complete integral can be found using the method of undetermined coefficients by presetting an .
