GATE Overflow for GATE XE - Recent activity in Partial Differential Equations

Edited: GATE XE 2026 | Question: 9

Sat, 28 Mar 2026 10:28:23 +0000

Consider the following partial differential equation (PDE)

$$(y-1) \frac{\partial^{2} u}{\partial x^{2}}-(x-3)^{2} \frac{\partial^{2} u}{\partial y^{2}}+y^{2} \frac{\partial u}{\partial x}+x^{2} \frac{\partial u}{\partial y}+(x-y) u=0 .$$

Then, which of the following statements is/are true?

In the region $\left\{(x, y) \in \mathbb{R}^{2}: y>1, x>3\right\}$, the PDE is hyperbolic
In the region $\left\{(x, y) \in \mathbb{R}^{2}: y>1, x<3\right\}$, the PDE is elliptic
In the region $\left\{(x, y) \in \mathbb{R}^{2}: y<1, x>3\right\}$, the PDE is elliptic
In the region $\left\{(x, y) \in \mathbb{R}^{2}: y<1, x<3\right\}$, the PDE is hyperbolic

Edited: GATE XE 2025 | Question: 21

Sun, 29 Jun 2025 12:26:41 +0000

Let $u(x, t)$ be the solution of the initial boundary value problem

$$ \begin{array}{c}
\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}-u=0, \quad 0< x < \pi, t>0, \\
u(x, 0)=2 \sin \left(\frac{3 x}{2}\right) \cos \left(\frac{x}{2}\right), \quad 0<x<\pi, \\ u(0, t)=u(\pi, t)=0, \quad t>0. \end{array}$$

Then the value of $\displaystyle \lim _{ t \rightarrow \infty} u\left(\frac{3 \pi}{4}, t\right)$ is equal to (rounded off to two decimal places) $\_\_\_\_\_\_\_\_$

Edited: GATE XE 2025 | Question: 12

Sun, 29 Jun 2025 12:07:34 +0000

Consider the second order Partial Differential Equation $\textsf{(PDE)}$

$$4 x^{2} \frac{\partial^{2} u}{\partial x^{2}}+4(x+y) \frac{\partial^{2} u}{\partial x \partial y}+\left(x^{2}+y^{2}\right) \frac{\partial^{2} u}{\partial y^{2}}-u=0$$

Then which one of the following statements is correct?

The $\textsf{PDE}$ is hyperbolic in the region $ \{(x, y) \in \mathbb{R}^{2}:-1 < x < 0, y < 0 \}$
The $\textsf{PDE}$ is hyperbolic in the region $ \{(x, y) \in \mathbb{R}^{2}:-1 < x < \infty, y < 0 \}$
The $\textsf{PDE}$ is elliptic in the region $ \{(x, y) \in \mathbb{R}^{2}:0 < x < 1, y > 0 \}$
The $\textsf{PDE}$ is parabolic in the region $ \{(x, y) \in \mathbb{R}^{2}:1 < x < \infty, y < 0 \}$

Recategorized: GATE XE 2022 | Question: 3

Mon, 05 May 2025 12:44:24 +0000

If the partial differential equation
$$
(x+2) \frac{\partial^2 u}{\partial x^2}+2(x+y) \frac{\partial^2 u}{\partial x \partial y}+2(y-1) \frac{\partial^2 u}{\partial y^2}-3 y^2 \frac{\partial u}{\partial y}=0
$$
is parabolic on the circle $(x-a)^2+(y-b)^2=r^2$, then the values of $a, b$ and $r$ are given by

$a=1, b=2, r=1$
$a=-1, b=2, r=1$
$a=1, b=-2, r=1$
$a=-1, b=-2, r=1$

Recategorized: GATE XE 2023 | Question: 11

Mon, 05 May 2025 12:21:52 +0000

Let $u(x, t)$ be the solution of the initial boundary value problem

$\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}=0, \quad x \in(0,2), t>0$

$u(x, 0)=\sin (\pi x), \quad x \in(0,2)$

$u(0, t)=u(2, t)=0$

Then the value of $e^{\pi^{2}}\left(u\left(\frac{1}{2}, 1\right)-u\left(\frac{3}{2}, 1\right)\right)$ is ________$\text{(in integer}$).