TAILIEUCHUNG - Handbook of mathematics for engineers and scienteists part 115

Handbook of mathematics for engineers and scienteists part 115. 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. | 766 Nonlinear Partial Differential Equations . Conservation Laws and Integrals of Motion . Basic Definitionsand Examples . General form of conservation laws. Consider a partial differential equation with two independent variables dw dw d2w d2w d2w V W dx dt dx2 dxdt dx2 A conservation law for this equation has the form dT dX dt dx where T t x t w dx dt X x x t w dx dt The left-hand side of the conservation law must vanish for all sufficiently smooth solutions of equation . In simplest cases the substitution of relations into the conservation law followed by differentiation and elementary transformations leads to a relation that coincides with up to a functional factor. The quantities T and X in are called a density and a flow respectively. The density and the flow in the conservation law are not uniquely determined they can be changed according to the rules T T p x X X fl t where p x and fl t are arbitrary functions or in general by the rules T aT X aX - dx dt where x t is an arbitrary function of two variables and a 0 is any number. For nonstationary equations with n spatial variables xi . xn conservation laws have the form 1 dk 0 Partial differential equations can have several sometimes infinitely many conservation laws or none at all. . Integrals of motion. If the total variation of the quantity X on the interval a x b is equal to zero which means that the relation X a X b holds then the following integral of motion takes place i T dx const for all t . J a For many specific equations relations of the form have a clear physical meaning and are used for approximate analytical solution of the corresponding problems as well as for the verification of results obtained by numerical methods. . Conservation Laws and Integrals of Motion 767 . Conservation laws for some nonlinear equations of mathematical .

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