solutions to HW #4

solutions to HW #4 - ChemE 109 - Numerical and Mathematical...

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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) Fourth-order Runge-Kutta 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) Fourth-order Runge-Kutta 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 predictor-corrector 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.

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solutions to HW #4 - ChemE 109 - Numerical and Mathematical...

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