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Handbook of mathematics for engineers and scienteists part 117. 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. | 780 Nonlinear Partial Differential Equations The functions w1 . wn-1 are thus assumed to be expressible in terms of wn. From 15.14.3.24 in view of 15.14.3.25 we find the following system of n - 1 ordinary differential equations dwk Fk w1 . wn j i id o - ----------------- k 1 . n - 1. 15.14.3.26 dwn n w1 . wn Further assuming that a solution of system 15.14.3.26 has been obtained and the functions 15.14.3.25 are known we substitute them into 15.14.3.22 - 15.14.3.23 to arrive at the system of two equations dr du dt g M x - 0 15.14.3.27 dwn f . fn wn U dt where g wn Gw1 wn . . . wn-1 wn w fn wn Fra w1 wra . . . wn-1 wn wn . Introducing the new dependent variable w g wn s Gw1 wn . n-1 wn wn in 15.14.3.27 we obtain system 15.14.3.11 in which the function f f w is defined parametrically f g wn fn wn w g wn with wn being the parameter. Remark 1. Solutions of the above special form arise for example in problems with initial and boundary conditions w1 wn u 0 at t 0 u 1 at x 0. The initial conditions correspond to the absence of particle in the reservoir at the initial time and the boundary condition corresponds to a given suspended concentration in the injected fluid. In this case the system ordinary differential equations 15.14.3.26 is solved under the following initial conditions w1 wn-1 0 at wn 0. Remark 2. In problems on flows of n-component fluids through porous media the mass balance for retained and suspended particles is governed by equation 15.14.3.22 in which the function G is the sum of individual components n G w1.wn wk equations 15.14.3.23 define the capture kinetics. 15.14.4. First-Order Hyperbolic Systems of Quasilinear Equations. Systems of Conservation Laws of Gas Dynamic Type 15.14.4-1. Systems of two equations. Systems in the form of conservation laws. 1 . Consider the system of two quasilinear equations of the form f1 u w g1 u w h1 u w k1 u w 0 dt dt dx dx 15 1441 r- i r- i r- i O l .l T. T.-L f2 u w g2 u w - h2 u w - k2 u w 0 dt dt dx dx Subsection .
