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

1

2008 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2008

Let \(u(x, t)\) be the solution of \(u_{tt} - u_{xx} = 1, x \in \mathbb{R}, t > 0\), with \(u(x, 0) = 0, u_t(x, 0) = 0, x \in \mathbb{R}\). Then \(u(1/2, 1/2)\) is equal to

A

\(\frac{1}{8}\)

B

\(-\frac{1}{8}\)

C

\(\frac{1}{4}\)

D

\(-\frac{1}{4}\)

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2

2008 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2008

Consider the boundary value problem \(u_{xx} + u_{yy} = 0, x \in (0, \pi), y \in (0, \pi), u(x, 0) = u(x, \pi) = u(0, y) = 0\). Any solution of this boundary value problem is of the form

A

\(\sum_{n=1}^{\infty} a_n \sinh nx \sin ny\)

B

\(\sum_{n=1}^{\infty} a_n \cosh nx \sin ny\)

C

\(\sum_{n=1}^{\infty} a_n \sinh nx \cos ny\)

D

\(\sum_{n=1}^{\infty} a_n \cosh nx \cos ny\)

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3

2008 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2008

If an additional boundary condition \(u_x(\pi, y) = \sin y\) is satisfied, then \(u(x, \pi/2)\) is equal to

A

\(\frac{\pi}{2} (e^x - e^{-x})(e^y + e^{-y})\)

B

\(\frac{\pi(e^x + e^{-x})}{(e^y - e^{-y})}\)

C

\(\frac{\pi(e^x - e^{-x})}{(e^y + e^{-y})}\)

D

\(\frac{\pi}{2} (e^x + e^{-x})(e^y + e^{-y})\)

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4

2009 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2009

For the diffusion problem \( u_{xx} = u_t \ (0 < x < \pi, t > 0), \ u(0,t) = 0, \ u(\pi,t) = 0 \) and \( u(x,0) = 3 \sin 2x \), the solution is given by

A

\(3e^{-4t} \sin 2x \)

B

\(3e^{-4t} \sin 2x \)

C

\(3e^{-9t} \sin 2x \)

D

\(3e^{-2t} \sin 2x \)

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5

2010 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2010

Consider the wave equation \( \frac{\partial^2 u}{\partial t^2} = 4 \frac{\partial^2 u}{\partial x^2} \), \( 0 < x < \pi \), \( t > 0 \), with \( u(0,t) = u(\pi,t) = 0 \), \( u(x,0) = \sin x \) and \( \frac{\partial u}{\partial t} = 0 \) at \( t = 0 \). Then \( u \left( \frac{\pi}{2}, \frac{\pi}{2} \right) \) is

A

2

B

1

C

0

D

-1

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6

2011 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2011

The vertical displacement \( u(x,t) \) of an infinitely long elastic string is governed by the initial value problem \( \frac{\partial^2 u}{\partial t^2} = 4 \frac{\partial^2 u}{\partial x^2}, \quad -\infty < x < \infty, \quad t > 0, \) \( u(x,0) = -x \) and \( \frac{\partial u}{\partial t}(x,0) = 0. \) The value of \( u(x,t) \) at \( x = 2 \) and \( t = 2 \) is equal to

A

2

B

4

C

-2

D

-4

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7

2012 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2012

The function \(u(r,\theta)\) satisfying the Laplace equation \[\frac{\partial^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2}=0,\quad e

A

\(\ln(e/r)\)

B

\(\ln(e/r^2)\)

C

\(\ln(e^2/r)\)

D

\(\sum_{n=1}^{\infty}\left(\frac{r-e^n}{e^n-e^{2n}}\right)\sin n\theta\)

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8

2012 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2012

The diffusion equation
\( \frac{\partial^{2} u}{\partial x^{2}} = \frac{\partial u}{\partial t} , \; u = u(x,t), \; u(0,t) = 0 = u(\pi,t), \; u(x,0) = \cos x \sin 5x \)
admits the solution

A

\(\frac{e^{-3\theta t}}{2} [\sin 6x + e^{3\theta t} \sin 4x] \)

B

\(\frac{e^{-3\theta t}}{2} [\sin 4x + e^{3\theta t} \sin 6x] \)

C

\(\frac{e^{-2\theta t}}{2} [\sin 3x + e^{5\theta t} \sin 5x] \)

D

\(\frac{e^{-3\theta t}}{2} [\sin 5x + e^{2\theta t} \sin x] \)

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9

2013 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2013

Let \(u(x,t)\) be the solution to the wave equation \(\frac{\partial^2 u}{\partial x^2}(x,t) = \frac{\partial^2 u}{\partial t^2}(x,t)\), \(u(x,0) = \cos(5\pi x)\), \(\frac{\partial u}{\partial t}(x,0) = 0\). Then, the value of \(u(1,1)\) is ______

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10

2014 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2014

Consider the heat equation
\[\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad 0 < x < \pi, \quad t > 0,\]
with the boundary conditions \(u(0, t) = 0\), \(u(\pi, t) = 0\) for \(t > 0\), and the initial condition \(u(x, 0) = \sin x\). Then \(u\left(\frac{\pi}{2}, 1\right)\) is ____________

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11

2014 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2014

If \(u(x,t)\) is the D'Alembert's solution to the wave equation \(\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2}\), \(x \in \mathbb{R}\), \(t > 0\), with the condition \(u(x,0) = 0\) and \(\frac{\partial u}{\partial t}(x,0) = \cos x\), then \(u\left(0, \frac{\pi}{4}\right)\) is ______________

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12

2016 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2016

Let u(x,t) be the d'Alembert's solution of the initial value problem for the wave equation
u_tt − c² u_xx = 0
u(x,0) = f(x), u_t(x,0) = g(x),
where c is a positive real number and f, g are smooth odd functions. Then, u(0,1) is equal to __________

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13

2017 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2017

Let \( u(r,\theta) \) be the bounded solution of the following boundary value problem in polar coordinates:
\( r^2 \frac{\partial^2 u}{\partial r^2} + r \frac{\partial u}{\partial r} + \frac{\partial^2 u}{\partial \theta^2} = 0 \), \( 0 < r < 2 \) and \( 0 \le \theta \le 2\pi \),
\( u(2,\theta) = \cos^2 \theta \), \( 0 \le \theta \le 2\pi \).
Then \( u(1, \pi/2) + u(1, \pi/4) \) equals

A

1

B

9/8

C

7/8

D

3/8

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14

2019 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2019

Consider the following heat conduction problem for a finite rod \(\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - xe^t - 2t\), \(t > 0\), \(0 < x < \pi\), with the boundary conditions \(u(0,t) = -t^2\), \(u(\pi,t) = -\pi e^t - t^2\), \(t > 0\), and the initial condition \(u(x,0) = \sin x - \sin^3 x - x\), \(0 \le x \le \pi\). If \(v(x,t) = u(x,t) + xe^t + t^2\), then which one of the following is CORRECT? (A) \(v(x,t) = \frac{1}{4} (e^{-t} \sin x + e^{-9t} \sin 3x)\) (B) \(v(x,t) = \frac{1}{4} (7e^{-t} \sin x - e^{-9t} \sin 3x)\) (C) \(v(x,t) = \frac{1}{4} (e^{-t} \sin x + e^{+9t} \sin 3x)\) (D) \(v(x,t) = \frac{1}{4} (3e^{-t} \sin x - e^{-9t} \sin 3x)\)

A

\(v(x,t) = \frac{1}{4} (e^{-t} \sin x + e^{-9t} \sin 3x)\)

B

\(v(x,t) = \frac{1}{4} (7e^{-t} \sin x - e^{-9t} \sin 3x)\)

C

\(v(x,t) = \frac{1}{4} (e^{-t} \sin x + e^{+9t} \sin 3x)\)

D

\(v(x,t) = \frac{1}{4} (3e^{-t} \sin x - e^{-9t} \sin 3x)\)

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15

2019 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2019

A solution of the Dirichlet problem \(\nabla^2 u(r, \theta) = 0, \quad 0 < r < 1, \; -\pi \leq \theta \leq \pi,\) \(u(1, \theta) = |\theta|, \; -\pi \leq \theta \leq \pi,\) is given by

A

\(u(r, \theta) = \frac{\pi}{2} + \frac{2}{\pi} \sum_{n=1}^{\infty} \left[ \frac{(-1)^n - 1}{n^2} \right] r^n \cos(n\theta)\)

B

\(u(r, \theta) = \frac{2}{\pi} \sum_{n=1}^{\infty} \left[ \frac{(-1)^{n+1}}{n^2} \right] r^n \cos(n\theta)\)

C

\(u(r, \theta) = \frac{\pi}{2} + \frac{2}{\pi} \sum_{n=1}^{\infty} \left[ \frac{(-1)^n - 1}{n^2} \right] r^n \cos(n\theta)\)

D

\(u(r, \theta) = \frac{\pi}{2} - \frac{2}{\pi} \sum_{n=1}^{\infty} \left[ \frac{(-1)^n + 1}{n^2} \right] r^n \cos(n\theta)\)

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16

2020 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2020

Let \(K : \mathbb{R} \times (0, \infty) \to \mathbb{R}\) be a function such that the solution of the initial value problem \(\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, u(x,0) = f(x), x \in \mathbb{R}, t > 0,\) is given by \( u(x,t) = \int_{\mathbb{R}} K(x - y, t) f(y) dy \) for all bounded continuous functions \(f\). Then the value of \(\int_{\mathbb{R}} K(x,t) dx\) is __________

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17

2020 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2020

If \(\{x_{13}, x_{22}, x_{23} = 10, x_{31}, x_{32}, x_{34}\}\) is the set of basic variables of a balanced transportation problem seeking to minimize cost of transportation from origins to destinations, where the cost matrix is,

	\(D_1\)	\(D_2\)	\(D_3\)	\(D_4\)	Availability
\(O_1\)	6	2	-1	0	10
\(O_2\)	4	2	2	3	\(\lambda+5\)
\(O_3\)	3	1	2	1	\(3\lambda\)
Demand	10	\(\mu-5\)	\(\mu+5\)	15

and \(\lambda, \mu \in \mathbb{R}\), then \(x_{32}\) is equal to __________

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18

2020 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2020

Let \(u(x,t)\) be the solution of
\[\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} = 0, \quad u(x,0) = f(x), \quad \frac{\partial u}{\partial t}(x,0) = 0, \quad x \in \mathbb{R}, t > 0,\]
where \(f\) is a twice continuously differentiable function. If \(f(-2) = 4, f(0) = 0,\) and \(u(2,2) = 8\), then the value of \(u(1,3)\) is __________

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19

2021 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2021

The function \(u(x,t)\) satisfies the initial value problem
\[\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2}, \quad x \in \mathbb{R}, \ t > 0,\]
\[u(x,0) = 0, \quad \frac{\partial u}{\partial t}(x,0) = 4xe^{-x^2}.\]
Then \(u(5,5)\) is

A

\(1 - \frac{1}{e^{100}}\)

B

\(1 - e^{100}\)

C

\(1 - \frac{1}{e^{10}}\)

D

\(1 - e^{10}\)

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20

2021 · Mathematics · Partial Differential Equations · Heat, Wave and Laplace Equations

Mathematics (MA) 2021

If \(u(x, t) = A e^{-t} \sin x\) solves the following initial boundary value problem
\[\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad 0 < x < \pi, \quad t > 0,\]
\[u(0, t) = u(\pi, t) = 0, \quad t > 0,\]
\[u(x, 0) = \begin{cases} 60, & 0 < x \leq \frac{\pi}{2}, \\ 40, & \frac{\pi}{2} < x < \pi, \end{cases}\]
then \(\pi A = \) ________.

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