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’ 1