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solutions to HW #4

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

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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

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(a) Using the explicit Euler’s method with h = 0 . 1. (b) Using the implicit Euler’s method with h = 0 . 1. (c) Using the Euler predictor-corrector with h = 0 . 25.
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• Fall '07
• Christofides
• Partial differential equation, Euler, Numerical ordinary differential equations, fourth-order Runge-Kutta method

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

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