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Unformatted text preview: ChemE 109  Numerical and Mathematical Methods in Chemical and Biological Engineering Fall 2007 Solution to Homework Set 4 1. Consider the following nonlinear ordinary differential equation: dx dt = t (1 x 2 ) e 2 x Show the exact form of the equations needed to calculate x i +1 from x i , with an integration step equal to h , using the: (a) Explicit Euler method (b) Implicit Euler method (c) Fourthorder RungeKutta method Solution: (a) Explicit Euler method x i +1 = x i t i (1 x 2 i ) e 2 x i h (b) Implicit Euler method x i +1 = x i ( t i + h )(1 x 2 i +1 ) e 2 x i +1 h (c) Fourthorder RungeKutta method x i +1 = x i + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) h k 1 = t i (1 x 2 i ) e 2 x i k 2 = ( t i + 1 2 h )(1 ( x i + 1 2 k 1 h ) 2 ) e 2( x i + 1 2 k 1 h ) k 3 = ( t i + 1 2 h )(1 ( x i + 1 2 k 2 h ) 2 ) e 2( x i + 1 2 k 2 h ) k 4 = ( t i + h )(1 ( x i + k 3 h ) 2 ) e 2( x i + k 3 h ) 2. Consider the following nonlinear ordinary differential equation: dx dt = x + t 2 t with x (1) = 1. Estimate x (1 . 5) 1 (a) Using the explicit Eulers method with h = 0 . 1. (b) Using the implicit Eulers method with h = 0 . 1. (c) Using the Euler predictorcorrector with h = 0 . 25....
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This homework help was uploaded on 04/07/2008 for the course CBE 109 taught by Professor Christofides during the Fall '07 term at UCLA.
 Fall '07
 Christofides

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