Heat, Wave and Laplace Equations - Partial Differential Equations - Engineering Sciences Previous Year Questions

1

2007 · Engineering Sciences · Partial Differential Equations · Heat, Wave and Laplace Equations

Engineering Sciences (XE) 2007

Let \( u(x,t) \) be the solution of the initial value problem
\( \frac{\partial^2 u}{\partial t^2} = 9 \frac{\partial^2 u}{\partial x^2} \), \( t > 0 \), \( -\infty < x < \infty \),
\( u(x,0) = x + 5 \),
\( \frac{\partial u}{\partial t}(x,0) = 0 \).
Then \( u(2,2) \) is

A

7

B

13

C

14

D

26

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2

2007 · Engineering Sciences · Partial Differential Equations · Heat, Wave and Laplace Equations

Engineering Sciences (XE) 2007

The potential \(u(x, y)\) satisfies the equation \(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\) in the square \(0 \leq x \leq \pi\), \(0 \leq y \leq \pi\). Three of the edges \(x = 0\), \(x = \pi\) and \(y = 0\) of the square are kept at zero potential and the edge \(y = \pi\) is kept at nonzero potential. The potential \(u(x, y)\) is given by

A

\(u(x, y) = \sum_{n=1}^{\infty} A_n \cosh nx \sin ny\)

B

\(u(x, y) = \sum_{n=1}^{\infty} A_n \sin nx \cosh ny\)

C

\(u(x, y) = \sum_{n=1}^{\infty} A_n \sinh nx \sin ny\)

D

\(u(x, y) = \sum_{n=1}^{\infty} A_n \sin nx \sinh ny\)

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3

2007 · Engineering Sciences · Partial Differential Equations · Heat, Wave and Laplace Equations

Engineering Sciences (XE) 2007

If the edge \(y = \pi\) is kept at the potential \(\sin x\), then the potential \(u(x, y)\) is given by

A

\(u(x, y) = \sum_{n=1}^{\infty} \frac{\sin nx \sinh ny}{\sinh n\pi}\)

B

\(u(x, y) = \frac{\sin x \sinh y}{\sinh \pi}\)

C

\(u(x, y) = \frac{\sin x \cosh y}{\cosh \pi}\)

D

\(u(x, y) = \sum_{n=1}^{\infty} \frac{\cosh nx \sin ny}{\cosh n\pi}\)

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4

2008 · Engineering Sciences · Partial Differential Equations · Heat, Wave and Laplace Equations

Engineering Sciences (XE) 2008

If \( u = u(x, t) \) is such that \[ \frac{\partial^2 u}{\partial t^2} = 4 \frac{\partial^2 u}{\partial x^2}, \quad 0 \leq x \leq \pi, \quad t \geq 0, \] \( u(0, t) = u(\pi, t) = 0, \) \( u(x, 0) = 0, \) \( \frac{\partial u}{\partial t}(x, 0) = \sin x, \) then \( u \left( \frac{\pi}{3}, \frac{\pi}{6} \right) \) is

A

\(\frac{3}{4} \)

B

\(\frac{3}{8} \)

C

\(\frac{\sqrt{3}}{4} \)

D

\(\frac{\sqrt{3}}{8} \)

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5

2009 · Engineering Sciences · Partial Differential Equations · Heat, Wave and Laplace Equations

Engineering Sciences (XE) 2009

The solution u(x,t) of the one-dimensional heat equation, \[ \frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2}, x \in \mathbb{R} \] with a Gaussian initial condition,

A

travels with finite constant wave-speed

B

travels with finite variable wave-speed

C

spreads in both directions, with the magnitude of the peak increasing with time

D

spreads in both directions, with the magnitude of the peak decreasing with time

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6

2010 · Engineering Sciences · Partial Differential Equations · Heat, Wave and Laplace Equations

Engineering Sciences (XE) 2010

Which one of the following is a possible solution to the partial differential equation \(\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} = 0\) with boundary conditions \(u(0,t)=0\), \(\frac{\partial u(\pi,t)}{\partial x}=0\), for \(t \geq 0\), \(u(x,0)=0\), \(\frac{\partial u(x,0)}{\partial t}=\pi\), for \(0 \leq x \leq \pi\) ?

A

\(u(x,t) = \sum_{n=0}^{\infty} a_n \sin\left((n+\frac{1}{2})t\right) \sin\left((n+\frac{1}{2})x\right)\)

B

\(u(x,t) = \sum_{n=0}^{\infty} a_n \cos\left((n+\frac{1}{2})t\right) \sin\left((n+\frac{1}{2})x\right)\)

C

\(u(x,t) = \sum_{n=0}^{\infty} a_n \sin\left((n+\frac{1}{2})t\right) \cos\left((n+\frac{1}{2})x\right)\)

D

\(u(x,t) = \sum_{n=0}^{\infty} a_n \cos\left((n+\frac{1}{2})t\right) \cos\left((n+\frac{1}{2})x\right)\)

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7

2011 · Engineering Sciences · Partial Differential Equations · Heat, Wave and Laplace Equations

Engineering Sciences (XE) 2011

The solution of the initial boundary value problem \(\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}\), \(0 < x < \pi\), \(t > 0\), with boundary and initial conditions \(\frac{\partial u}{\partial x}(0,t) = 0 = u(\pi,t)\), \(t > 0\) and \(u(x,0) = f(x)\), \(0 < x < \pi\), is

A

\(u(x,t) = \sum_{n=0}^{\infty} A_n \exp\left(-\left(\frac{2n+1}{2}\right)^2 t\right) \cos\left(\frac{2n+1}{2}x\right)\), with \(A_n = \frac{2}{\pi} \int_0^{\pi} f(x) \cos\left(\frac{2n+1}{2}x\right) dx\)

B

\(u(x,t) = \sum_{n=0}^{\infty} A_n \exp(-n^2 t) \cos(nx)\), with \(A_n = \frac{2}{\pi} \int_0^{\pi} f(x) \cos(nx) dx\)

C

\(u(x,t) = \sum_{n=1}^{\infty} A_n \exp\left(-\left(\frac{2n+1}{2}\right)^2 t\right) \sin\left(\frac{2n+1}{2}x\right)\), with \(A_n = \frac{2}{\pi} \int_0^{\pi} f(x) \sin\left(\frac{2n+1}{2}x\right) dx\)

D

\(u(x,t) = \sum_{n=1}^{\infty} A_n \exp(-n^2 t) \sin(nx)\), with \(A_n = \frac{2}{\pi} \int_0^{\pi} f(x) \sin(nx) dx\)

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8

2012 · Engineering Sciences · Partial Differential Equations · Heat, Wave and Laplace Equations

Engineering Sciences (XE) 2012

For the solution of \(\nabla^2 u = 0\), the domain and boundary conditions are shown below.

Which of the following statements is TRUE?

A

The solution cannot be obtained using separation of variables because the governing equation is non-separable.

B

The solution cannot be obtained using separation of variables because all the boundary values are non-zero.

C

The solution cannot be obtained using separation of variables because not all the boundaries are along constant coordinate lines.

D

The solution can be obtained by separation of variables.

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9

2017 · Engineering Sciences · Partial Differential Equations · Heat, Wave and Laplace Equations

Engineering Sciences (XE) 2017

If \(u(x, t) = g(t) \sin x\) is the solution of the wave equation
\(u_{tt} = u_{xx}, \quad t > 0, \quad 0 < x < \pi,\)
with the initial conditions
\(u(x, 0) = 2 \sin x, \quad u_t(x, 0) = 0, \quad 0 \le x \le \pi,\)
and the boundary conditions
\(u(0, t) = u(\pi, t) = 0, \quad t \ge 0,\)
then the value of \(g\left(\frac{\pi}{3}\right)\) is ______________.

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10

2018 · Engineering Sciences · Partial Differential Equations · Heat, Wave and Laplace Equations

Engineering Sciences (XE) 2018

Let \( u(x,t) \) satisfy the initial and boundary value problem \( \frac{\partial u}{\partial t} = 2 \frac{\partial^2 u}{\partial x^2}, \ 0 < x < \pi, \ t > 0, \ u(0,t) = 0 = u(\pi,t), \ t > 0, \ u(x,0) = \sin x + 2\sin 4x, \ 0 < x < \pi. \) Then the value of \( u\left( \frac{\pi}{2}, \ln(5) \right) \) is __________

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11

2021 · Engineering Sciences · Partial Differential Equations · Heat, Wave and Laplace Equations

Engineering Sciences (XE) 2021

Let \( u(x, y) = (x^2 - y^2)v(x, y) \) be such that both \( u(x, y) \) and \( v(x, y) \) satisfy the Laplace equation in a domain \( \Omega \) of the xy-plane. Then, which one of the following is TRUE in \( \Omega \) ?

A

\(x \frac{\partial v}{\partial x} - y \frac{\partial v}{\partial y} = 0 \)

B

\(x \frac{\partial v}{\partial x} + y \frac{\partial v}{\partial y} = 0 \)

C

\(x \frac{\partial v}{\partial y} - y \frac{\partial v}{\partial x} = 0 \)

D

\(x \frac{\partial v}{\partial y} + y \frac{\partial v}{\partial x} = 0 \)

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12

2023 · Engineering Sciences · Partial Differential Equations · Heat, Wave and Laplace Equations

Engineering Sciences (XE) 2023

Let \( u(x, t) \) be the solution of the initial boundary value problem \[ \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0, \quad x \in (0, 2), \ t > 0 \] \[ u(x, 0) = \sin(\pi x), \ x \in (0, 2) \] \[ u(0, t) = u(2, t) = 0. \] Then the value of \( e^{\pi^2} \left( u\left( \frac{1}{2}, 1 \right) - u\left( \frac{3}{2}, 1 \right) \right) \) is __________ (in integer).

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13

2025 · Engineering Sciences · Partial Differential Equations · Heat, Wave and Laplace Equations

Engineering Sciences (XE) 2025

Let \(u(x,t)\) be the solution of the initial boundary value problem
\[ \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} - u = 0, \quad 0 < x < \pi, \ t > 0, \]
\[ u(x,0) = 2 \sin\left(\frac{3x}{2}\right) \cos\left(\frac{x}{2}\right), \quad 0 < x < \pi, \]
\[ u(0,t) = u(\pi,t) = 0, \quad t > 0. \]
Then the value of \(\lim_{t \to \infty} u\left(\frac{3\pi}{4}, t\right)\) is equal to (rounded off to two decimal places)________

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