List of top Partial Differential Equations Questions

The steady state solution for the heat equation \[ \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0, \quad 0<x<2, \, t>0, \] with the initial condition \( u(x, 0) = 0, \, 0<x<2 \) and the boundary conditions \( u(0, t) = 1 \) and \( u(2, t) = 3, \, t>0 \) at \( x = 1 \) is

If eigenfunctions corresponding to distinct eigenvalues \( \lambda \) of the Sturm-Liouville problem
\[ \frac{d^2y}{dx^2} - 3 \frac{dy}{dx} = \lambda y, \quad 0<x<\pi,
y(0) = y(\pi) = 0 \] are orthogonal with respect to the weight function \( w(x) \), then \( w(x) \) is

If the Laplace equation \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \quad 1<x<2, \quad 1<y<2 \] with the boundary conditions \[ \frac{\partial u}{\partial x}(1, y) = y, \quad \frac{\partial u}{\partial x}(2, y) = 5, \quad 1<y<2 \] and \[ \frac{\partial u}{\partial y}(x, 1) = \frac{\alpha x^2}{7}, \quad \frac{\partial u}{\partial y}(x, 2) = x, \quad 1<x<2 \] has a solution, then the constant \( \alpha \) is __________.

Let \( u(x, y) \) be the solution of the first order partial differential equation \[ x \frac{\partial u}{\partial x} + (x^2 + y) \frac{\partial u}{\partial y} = u, \quad \text{for all} \, x, y \in \mathbb{R} \] satisfying \( u(2, y) = y - 4 \), \( y \in \mathbb{R} \). Then, the value of \( u(1, 2) \) is __________.

Let \( u(x, t) \) be the solution of the wave equation \[ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2}, \quad 0<x<\pi, \, t>0, \] with the initial conditions \[ u(x, 0) = \sin x + \sin 2x + \sin 3x, \quad \frac{\partial u}{\partial t}(x, 0) = 0, \quad 0<x<\pi, \] and the boundary conditions \[ u(0, t) = u(\pi, t) = 0, \quad t \geq 0. \] Then, the value of \( u \left( \frac{\pi}{2}, \pi \right) \) is