M.
Sc Mathematics – Partial Differential Equations
Unit 4: MCQs (1–40)
Multiple Choice Questions (1–40)
1. Which of the following PDEs is best solved using the method of separation of variables?
(a) ux + uy = 0
(b) uxx + uyy = 0
(c) ut + uux = 0
(d) ux = sin y
Answer: (b)
2. In separation of variables, the solution u(x, t) = X(x)T (t) assumes:
(a) A product of independent solutions in each variable
(b) An integral transform
(c) A Fourier series
(d) An inverse Laplace transform
Answer: (a)
3. For the heat equation ut = α2 uxx , the separation of variables gives:
(a) Two linear ODEs
(b) Two nonlinear PDEs
(c) One ODE and one PDE
(d) A single integral equation
Answer: (a)
4. If u(x, t) = X(x)T (t) solves ut = 4uxx , and X(x) = sin(πx), find the form of T (t).
2t
(a) T (t) = e−4π
2t
(b) T (t) = e−16π
1
2t
(c) T (t) = e−π
(d) T (t) = sin(4π 2 t)
Answer: (a)
5. A similarity variable for the PDE ut = αuxx is:
(a) x
x
(b) √
t
x
(c) t2
(d) xt
Answer: (b)
6. Which of the following is a similarity solution for the linear heat equation?
2 /4t
(a) u(x, t) = e−x
(b) u(x, t) = x2 + t2
(c) u(x, t) = sin(x + t)
(d) u(x, t) = x + t
Answer: (a)
7. In traveling wave solutions, the function u(x, t) = f (x − ct) satisfies which form of
PDE?
(a) Linear parabolic
(b) Hyperbolic
(c) Elliptic
(d) Time-independent
Answer: (b)
2
8. Consider u(x, t) = e−(x−2t) . What is the wave speed?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b)
9. The Fourier transform of f (x) = e−a|x| is:
2a
(a) a2 +ω 2
2
a
(b) a2 +ω 2
1
(c) a2 +ω 2
2
(d) a2 +ω 2
Answer: (a)
10. The Laplace transform of f (t) = sin(at) is:
a
(a) s2 +a2
s
(b) s2 +a2
1
(c) s2 +a2
s
(d) s+a2
Answer: (a)
11. The Hopf-Cole transformation linearizes which nonlinear PDE?
(a) Burgers’ equation
(b) Wave equation
(c) Laplace equation
(d) Korteweg–de Vries equation
Answer: (a)
12. If u(x, t) = −2ν ϕϕx , and ϕ satisfies the heat equation, then u satisfies:
(a) Wave equation
(b) Burgers’ equation
(c) Laplace equation
(d) Hamilton-Jacobi equation
Answer: (b)
13. The hodograph method is useful for:
(a) Linear PDEs in two variables
(b) First-order nonlinear PDEs
(c) Integral equations
(d) Elliptic PDEs
Answer: (b)
14. The Legendre transform is applied to convert a function of:
(a) One variable to two variables
3
(b) Derivatives into variables
(c) Variables into derivatives
(d) Variables into momentum coordinates
Answer: (d)
15. The Legendre transform of f (x) = x2 is:
(a) f ∗ (p) = p2
p2
(b) f ∗ (p) = 4
p2
(c) f ∗ (p) = 2
2
(d) f ∗ (p) = − p4
Answer: (c)
16. A potential function ϕ satisfies ∇ϕ = F⃗ if:
(a) F⃗ is divergence-free
(b) F⃗ is irrotational
(c) F⃗ is solenoidal
(d) F⃗ is conservative
Answer: (d)
17. If F⃗ = ∇ϕ, then ∇ × F⃗ = ?
(a) ∇ϕ
(b) 0
(c) 1
(d) ϕ
Answer: (b)
18. For the function ϕ(x, y) = x2 + y 2 , compute F⃗ = ∇ϕ.
(a) (x, y)
(b) (2x, 2y)
(c) (−2x, −2y)
(d) (x2 , y 2 )
Answer: (b)
19. Which transform is best suited to solve PDEs with boundary conditions at infinity?
4
(a) Fourier transform
(b) Laplace transform
(c) Hodograph transform
(d) Legendre transform
Answer: (a)
20. The general traveling wave solution of ut + cux = 0 is:
(a) u(x, t) = f (x + ct)
(b) u(x, t) = f (x − ct)
(c) u(x, t) = e−ct
(d) u(x, t) = ct
Answer: (b)
21. In separation of variables, boundary conditions determine:
(a) Time dependence
(b) Spatial eigenfunctions
(c) Initial data
(d) Transform domain
Answer: (b)
