Modeling Tools for Environmental Engineers and Scientists

and so on up tof n (x 1 , x 2 , . . . x n ) = 0 (3.11)There are no direct methods for solving simultaneous nonlinear equations.The most popular method for solving nonlinear equations is the Newton’sIteration Method, which is based on Taylor’s expansion of each of the nequations. For example, the first of the above equations can be expressedas follows:∂f f 1 (x 1 ∆x 1 , . . . x n ∆x n ) = f 1 (x 1 , . . . x n ) 1 ∂x ∆x 11+ higher-order terms (3.12)Neglecting higher-order terms,∂f 1 (x 1 ∆x 1 , . . . x n ∆x n ) = f 1 (x 1 , . . . x n ) ∂x ∆x (3.13)1If, from a guessed value of x1, the change ∆x1 would make the left-handside approach 0, then the true root will be x1 ∆x1. Or, in other words, if ∂∂x ∆x = –f (x , . . . x ) (3.14)1 1 1 nf 11then x 1 ∆x 1 will be a root. Thus, extending this argument to the set of equations,this algorithm reduces to the solution of a set of linear equations thatcan be represented in the following form:⎡ ∂f⎢⎢∂x⎢⎢ ∂f⎢∂x⎢⎢∂f⎢⎣⎢∂x1121∂f∂x∂f∂x∂f∂x∂f1⎤⎥∂xn⎥⎥∂f2⎥∂x⎥⎥∂f⎥⎥∂xn⎦⎥n n nf 11(3.15)The algorithms discussed in the previous section for a set of linear equationscan now be applied to the above set of linear equations to find their roots.3.3.2 ORDINARY DIFFERENTIAL EQUATIONS12221 2. . .. . .. . .A vast majority of environmental systems can be described by ODEs. Onlyin a limited number of such cases can these equations be solved analytically,n⎡∆x1⎢⎢∆x2⎢⎢ .⎢⎢ .⎢⎢⎣∆xn⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡−f1⎤⎢ ⎥⎢−f2⎥⎢ ⎥= ⎢ . ⎥⎢ ⎥⎢ . ⎥⎢ ⎥⎢⎣−f ⎥n ⎦© 2002 by CRC Press LLC