The GATE Grind

GATE Partial Differential Equations: Heat and Wave Equations – Previous Year Questions

13 GATE previous year questions on Heat and Wave Equations (Partial Differential Equations, Mathematics) with answers and explanations, from every paper.

  1. GATE 2021 MA Q17 (1 mark, Multiple choice) – For u tt=u xx on x∈ℝ,t>0, u(x,0)=0 and u t(x,0)=4xe(−x²), find u(5,5).
  2. GATE 2021 MA Q61 (2 marks, MTA) – Suppose u(x,t)=Ae(−t)sin x solves u t=u xx on 0<x<π,t>0, with zero endpoint values and initial data u(x,0)=60 for 0<x≤π/2 and 40 for π/2<x<π. Find πA.
  3. GATE 2022 MA Q21 (1 mark, Multiple choice) – For u t−u xx=0,0<x<2,t>0, with u(x,0)=0,u(0,t)=1,u(2,t)=3, find the steady-state value at x=1.
  4. GATE 2022 MA Q46 (2 marks, Multiple choice) – For u tt−u xx=0 on 0<x<π,t>0, u(x,0)=sin x+sin2x+sin3x,u t(x,0)=0 and u(0,t)=u(π,t)=0, find u(π/2,π).
  5. GATE 2023 MA Q26 (1 mark, Numerical answer) – For c>0, u xx−u tt/c²=0 on ℝ×(0,∞), with u(x,0)=sin x and u t(x,0)=cos x. The dependence interval for P=(3,2) has endpoints Q,R on the x-axis. If…
  6. GATE 2023 MA Q64 (2 marks, Numerical answer) – For 0<x<π,t>0 solve u xx−u tt=sin x sin(2t), with u(x,0)=u t(x,0)=0 and u(0,t)=u(π,t)=0. Find u(π/2,3π/2) to two decimals.
  7. GATE 2024 MA Q23 (1 mark, Multiple choice) – u t−4u xx=0 on 0<x<1,t>0, zero endpoint values, u(x,0)=sinπx. Let g(t)=∫₀¹u(x,t)²dx. Choose its behavior.
  8. GATE 2024 MA Q62 (2 marks, Numerical answer) – For u tt−u xx=0 on ℝ,t>0, u(x,0)=0 and initial velocity v(x)=x⁴(1−x)⁴ on 0<x<1, zero otherwise. Find α=inft>0:u(2,t)>0 to two decimals.
  9. GATE 2025 MA Q20 (1 mark, Multiple choice) – u t=u xx on 0<x<π,t>0, with u(0,t)=u(π,t)=0 and u(x,0)=sin4x cos3x. Find u(π/4,t).
  10. GATE 2025 MA Q40 (2 marks, Multiple choice) – u tt−9u xx=0 on ℝ,t>0, with u(x,0)=eˣ and u t(x,0)=sinx. Find u(π/2,π/6).
  11. GATE 2025 MA Q52 (2 marks, Multiple select) – u t=2u xx on 0<x<1,t>0 with zero endpoint values and u(x,0)=2x(1−x). Which statements hold?
  12. GATE 2026 MA Q15 (1 mark, Multiple choice) – On the real line, u t=ku xx, t > 0, k ∈ ℝ, with u(x,0)=e-a x , a > 0. For Fourier convention u(w,t)= -u(x,t)eiwxdx, find the transform.
  13. GATE 2026 MA Q43 (2 marks, Multiple choice) – u tt=c2u xx on ℝ, t > 0,c > 0, with initial displacement 1 for x < 1 and zero otherwise, and initial velocity zero. Find maximum u(0,t) for t > 0.