Teaching Haryana (HPSC) Assistant Professor Mock Test 2025 Mathematical Science Partial Differential Equations Laplace Equation, Heat and Wave Equations
Let u be the unique solution of
\(\left.\begin{array}{ll} \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} \text { where }(x, t) \in(0,1) \times(0, \infty) \\ u(x, 0)=\sin \pi x, & x \in(0,1) \\ u(0, t)=u(1, t)=0, & t \in(0, \infty) \end{array}\right\}\)
Then which of the following is true?
1
There exists (x, t) ∈ (0, 1) × (0, ∞) such that u(x, t) = 0
2
There exists (x, t) ∈ (0, 1) × (0, ∞) such that \(\frac{\partial u}{\partial t}\)(x, t) = 0
3
The function etu(x, t) is bounded for (x, t) ∈ (0, 1) × (0, ∞)
4
There exists (x, t) ∈ (0, 1) × (0, ∞) such that u(x, t) > 1
5
Question Not Attempted