tailieunhanh - Handbook of mathematics for engineers and scienteists part 99

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 99', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 654 Nonlinear Partial Differential Equations Denote a dr- b 1 dF c df. where p f-f q 2 r . dp 2 dq dr dx2 dxdy dy2 Let us select a specific solution w w x y of equation and calculate a b and c by formulas at some point x y and substitute the resulting expressions into . Depending on the sign of the discriminant 6 the type of nonlinear equation at the point x y is determined according to if 6 0 the equation is parabolic if 6 0 it is hyperbolic and if 6 0 it is elliptic. In general the coefficients a b and c of the nonlinear equation depend not only on the selection of the point x y but also on the selection of the specific solution. Therefore it is impossible to determine the sign of 6 without knowing the solution w x y . To put it differently the type of a nonlinear equation can be different for different solutions at the same point x y . A line Q x y C is called a characteristic of the nonlinear equation if it is an integral curve of the characteristic equation a dy 2 - 2b dxdy c dx 2 0. The form of characteristics depends on the selection of a specific solution. In individual special cases the type of a nonlinear equation other than the semilinear equation may be independent of the selection of solutions. Example. Consider the nonhomogeneous Monge-Ampere equation d2 w y dxdy It is a special case of equation with d2w d2w J. f x y . dx2 dy2 d2 w d2 w d2w P dx2 q dxdy r dy2 Using formulas and we find the discriminant 6 q2 - pr f x y . F x y p q r q2 - pr - f x y 0 Here the relation between the highest derivatives and f x y defined by equation has been taken into account. From and it follows that the type of the nonhomogeneous equation Monge-Ampere at a point x y depends solely on the sign of f x y and is independent of the selection of a particular solution. At the points where f x y 0 the equation