List of top Partial Differential Equations Questions asked in GATE MA

Let \( \Omega = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 < 1\} \) be the open unit disc and \( \partial\Omega \) be its boundary. If \( u(x, y) \) is the solution of the following Dirichlet problem
\[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \quad \text{in } \Omega \]
\[ u(x, y) = x^2 - y^2 \quad \text{on } \partial\Omega, \]
then the value of \( 4\left(u\left(\frac{1}{2}, 0\right) - u\left(0, \frac{1}{2}\right)\right) \) is ______.

Consider the Laplace equation
\[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0, \qquad 0 < x < 1,\; 0 < y < 1, \] with the boundary conditions
\[ T(x,0) = x, \qquad T(0,y) = y \] \[ T(x,1) = 1+x, \qquad T(1,y) = 1+y. \] Then the value of \(T\left(\dfrac{1}{2}, \dfrac{1}{3}\right)\) is equal to

Let \(u(x,t)\) satisfy the wave equation
\[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, \qquad -\infty < x < \infty, \; t > 0, \; c > 0 \] with
\[ u(x,0) = \begin{cases} 1 & \text{if } |x| < 1 \\ 0 & \text{otherwise} \end{cases} \] and
\[ \frac{\partial u}{\partial t}(x,0) = 0. \] By using D'Alembert's formula, the maximum value of \(u(0,t)\) for \(t > 0\) is

Consider the following partial differential equation:
\[ a\frac{\partial^2 f(x,y)}{\partial x^2} + b\frac{\partial^2 f(x,y)}{\partial y^2} = 8f(x,y) \]
where \(a\) and \(b\) are distinct positive real numbers.
The combination(s) of the values of the real parameters \(\xi\) and \(\eta\) for which \(f(x,y) = e^{2\xi x+\eta y}\) is a solution of the given partial differential equation, is/are

Let \( u(x,t) \) be the solution of the initial value problem for the heat equation on the real line:
\[ \frac{\partial u}{\partial t} = k\, \frac{\partial^2 u}{\partial x^2}, \qquad -\infty< x< \infty, \qquad t>0, \qquad k\in\mathbb{R} \]
with the initial condition
\[ u(x,0) = e^{-a|x|}, \qquad a>0. \]
If \( \hat u(w,t) = \displaystyle\int_{-\infty}^{\infty} u(x,t)\, e^{iwx}\, dx \) is the Fourier transform of \( u(x,t) \) with respect to \( x \), then \( \hat u(w,t) \) is equal to