MTH646 Midterm MCQs Oct 2024 by Miss Mehwish

Past Papers for Mids./Finals are also Available · MTH646(Mega File) · Mid Term (Live Quiz)

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1. If the initial conditions for the heat equation

∂u∂t=α2∂2u∂x2,0≤x≤2 with t>0 is u(x,0)=sin⁡(πx2)\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2}, \quad 0 \leq x \leq 2 \text{ with } t > 0 \text{ is } u(x, 0) = \sin\left(\frac{\pi x}{2}\right)∂t∂u=α2∂x2∂2u,0≤x≤2 with t>0 is u(x,0)=sin(2πx) Then what is the value of u(1,0)u(1,0)u(1,0)? ⬜ 1 ⬜ 12\frac{1}{\sqrt{2}}21 ✅ 0 ⬜ 32\frac{\sqrt{3}}{2}23

2. Crank Nicolson scheme is unconditionally stable and second-order accuracy in... ⬜ both space and time ✅ time only ⬜ medium
3. The number of grid points (m) in time direction of a solution domain can be calculated... ✅ m=T−t0km = \frac{T - t_0}{k}m=kT−t0 ⬜ m=T−t0hm = \frac{T - t_0}{h}m=hT−t0 ⬜ m=t0−Thm = \frac{t_0 - T}{h}m=ht0−T ⬜ m=T+t0km = \frac{T + t_0}{k}m=kT+t0

4. The number of grid points (n) for the boundary condition

u(0,t)=u(2,t)=0u(0,t) = u(2,t) = 0u(0,t)=u(2,t)=0 and h=0.1h = 0.1h=0.1 are... ⬜ 10 ✅ 20 ⬜ 0 ⬜ 2

5. The number of grid points in the x-direction of a solution domain can be calculated as: ⬜ n=a+bhn = \frac{a + b}{h}n=ha+b ⬜ n=b−akn = \frac{b - a}{k}n=kb−a ✅ n=b−ahn = \frac{b - a}{h}n=hb−a ⬜ n=a−bkn = \frac{a - b}{k}n=ka−b
6. The implicit schemes are unconditionally stable, allowing the use of increment time steps: ✅ large

⬜ medium