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
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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 predictorcorrector with
h
= 0
.
25.
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 Fall '07
 Christofides
 Partial differential equation, Euler, Numerical ordinary differential equations, fourthorder RungeKutta method

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