STQM6064
1. RMN
2. RMN
3.
a) Show how the expansion x(󰂃) = 1 + 󰂃x1 + 󰂃2 x2 + · · · will fail when applied to
(1 − 󰂃)x2 − 2x + 1 = 0,
󰂃 ≪ 1.
Then suggest the correct expansion.
b) Consider the following nonlinear system of two-point boundary-value problem:
f ′′ − θ′ = 0,
λ+1 ′
θ′′ +
f θ − λf ′ θ = 0,
2
(1)
(2)
with boundary conditions
f (0) = fw ,
θ(0) = 1,
f ′ (∞) = 0,
θ(∞) = 0,
(3)
(4)
where the primes denote differentiation with respect to η and both fw and λ
are constants.
First reduce (1)–(4) to a problem in terms of f only. Then obtain the Adomian
recursive (analytical) algorithm.
[25 marks]
4.
a) Give two example problems which can be solved by the method of calculus of
variations.
b) Derive the Euler differential equation corresponding to
I=
󰁝
x1
x0
F (x, y1 , y2 , . . . , yn , y1′ , y2′ , . . . , yn′ ) dx,
where the continuously differentiable functions yi (x) (i = 1, . . . , n) and their
derivatives are prescribed at x = x0 and x = x1 .
[25 marks]
‘SELAMAT MAJU JAYA’
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