# Partial Differential Equation

 Question 1
The function $f(x, y)$ satisfies the Laplace equation
$\triangledown ^2f(x,y)=0$
on a circular domain of radius $r = 1$ with its center at point P with coordinates $x = 0, y = 0$. The value of this function on the circular boundary of this domain is equal to 3.
The numerical value of $f(0, 0)$ is:
 A 0 B 2 C 3 D 1
GATE CE 2022 SET-2   Engineering Mathematics
Question 1 Explanation:
According to given condition given function f(x,y) is nothing but constant function i.e. f(x,y)=3 because this is the only function whose value is 3 at any point on the boundary of unit circle and it is also satisfying Laplace equation, so
f(0,0)=3
 Question 2
The Fourier cosine series of a function is given by:
$f(x)=\sum_{n=0}^{\infty }f_n\cos nx$
For $f(x)=\cos ^4x$, the numerical value of $f_4+f_5$ is ______ . (round off to three decimal places)
 A 2.255 B 0.652 C 0.125 D 1.585
GATE CE 2022 SET-1   Engineering Mathematics
Question 2 Explanation:
\begin{aligned} \cos ^4 x&=\cos ^2 x\cos ^2 x\\ &=\left ( \frac{1+\cos 2x}{2} \right )\left ( \frac{1+\cos 2x}{2} \right )\\ &=\frac{1}{4}(1+2 \cos 2x+ \cos ^2 2x)\\ &=\frac{1}{4}(1+2 \cos 2x+ \frac{(1+\cos 4x)}{2})\\ &=\frac{1}{4}+\frac{\cos 2x}{2}+\frac{1}{8}+\frac{\cos 4x}{8}\\ &=\frac{3}{8}+\frac{\cos 2x}{2}+\frac{\cos 4x}{8}\\ f_4&=\frac{1}{8}=0.125\\ f_5&=0\\ \rightarrow f_4+f_5&=0.125 \end{aligned}
 Question 3
Consider the following expression:
$z=\sin(y+it)+\cos(y-it)$
where $z, y,$ and $t$ are variables, and $i=\sqrt{-1}$ is a complex number. The partial differential equation derived from the above expression is
 A $\frac{\partial^2 z}{\partial t^2}+\frac{\partial^2 z}{\partial y^2}=0$ B $\frac{\partial^2 z}{\partial t^2}-\frac{\partial^2 z}{\partial y^2}=0$ C $\frac{\partial z}{\partial t}-i\frac{\partial z}{\partial y}=0$ D $\frac{\partial z}{\partial t}+i\frac{\partial z}{\partial y}=0$
GATE CE 2022 SET-1   Engineering Mathematics
Question 3 Explanation:
\begin{aligned} z&=\sin(y+it)+ \cos (y-it)\\ \frac{\partial z}{\partial y}&=\cos (y+it)-\sin (y-it)\\ \frac{\partial ^2 z}{\partial ^2 y^2}&=-\sin(y+it)- \cos (y-it)\\ \frac{\partial ^2 z}{\partial ^2 y^2}&=-z \;\;...(i)\\ \frac{\partial z}{\partial t}&=i \cos (y+it)+i\sin (y-it)\\ \frac{\partial ^2 z}{\partial ^2 t^2}&=+\sin(y+it)+ \cos (y-it)\\ \frac{\partial ^2 z}{\partial ^2 t^2}&=z\;\;...(ii)\\ &\text{Adding (i) and (ii)}\\ &\frac{\partial ^2 z}{\partial ^2 y^2}+\frac{\partial ^2 z}{\partial ^2 t^2}=0 \end{aligned}
 Question 4
The Fourier series to represent $x-x^2$ for $- \pi \leq x\leq \pi$ is given by

$x-x^2=\frac{a_0}{2}+\sum_{n=1}^{\infty }a_n \cos nx +\sum_{n=1}^{\infty }b_n \sin nx$

The value of $a_0$ (round off to two decimal places), is _______.
 A -1.52 B 2.12 C -6.58 D 5.23
GATE CE 2020 SET-2   Engineering Mathematics
Question 4 Explanation:
\begin{aligned} a_0&=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)dx\\ &=\frac{1}{\pi}\int_{-\pi}^{\pi} (x-x^2)dx\\ &=\frac{-1}{\pi}\int_{0}^{\pi}2x^2 dx\\ &=\frac{-1}{\pi}\left ( \frac{2x^3}{3} \right )_0^{\pi}\\ &=-\frac{2}{3 \pi}[\pi^3]=\frac{-2 \pi^2}{3}\\ &=-6.58 \end{aligned}
 Question 5
The Laplace transform of $\sinh (at)$ is
 A $\frac{a}{s^2-a^2}$ B $\frac{a}{s^2+a^2}$ C $\frac{s}{s^2-a^2}$ D $\frac{s}{s^2+a^2}$
GATE CE 2019 SET-2   Engineering Mathematics
Question 5 Explanation:
$L(\sinh (at))=\frac{a}{s^2-a^2}$
 Question 6
Consider a two-dimensional flow through isotropic soil along x direction and z direction. If h is the hydraulic head, the Laplace's equation of continuity is expressed as
 A $\frac{\partial h}{\partial x}+\frac{\partial h}{\partial z}=0$ B $\frac{\partial h}{\partial x}+\frac{\partial h}{\partial x}\frac{\partial h}{\partial z}+\frac{\partial h}{\partial z}=0$ C $\frac{\partial^2 h}{\partial x^2}+\frac{\partial^2 h}{\partial z^2}=0$ D $\frac{\partial^2 h}{\partial x^2}+\frac{\partial^2 h}{\partial x \partial z}+\frac{\partial^2 h}{\partial z^2}=0$
GATE CE 2019 SET-1   Engineering Mathematics
Question 6 Explanation:
The Laplace's equation of continuity for two dimensional flow in a soil is expressed as:
$k_x\frac{\partial^2 h}{\partial x^2}+k_z\frac{\partial^2 h}{\partial z^2}=0$... for anisotropic soil $[k_x\neq k_z]$
:
$\frac{\partial^2 h}{\partial x^2}+\frac{\partial^2 h}{\partial z^2}=0$... for isotropic soil $[k_x = k_z]$
 Question 7
The laplace transform F(s) of the exponential function ,$f(t)=e^{at}\; when \; t\geq 0$, where a is a constant and (s-a)$\gt$0, is
 A $\frac{1}{s+a}$ B $\frac{1}{s-a}$ C $\frac{1}{a-s}$ D $\infty$
GATE CE 2018 SET-2   Engineering Mathematics
Question 7 Explanation:
\begin{aligned} L(e^{at}) &=\frac{1}{s-a} \\ L(e^{at}) &=\int_{0}^{\infty }e^{-st}e^{at}dt \\ &= \int_{0}^{\infty }e^{-(s-a)t}dt\\ &= \left.\begin{matrix} \frac{e^{-(s-a)t}}{-(s-a)} \end{matrix}\right|_{0}^{\infty }\\ &=-\frac{1}{s-a}(0-1)\\ &=\frac{1}{s-a} \end{aligned}
 Question 8
The Fourier series of the function,
$\begin{matrix} f(x) & =0 & -\pi \lt x \leq 0 \\ f(x) &=\pi-x & 0 \lt x \lt \pi \end{matrix}$

in the interval $[-\pi ,\pi ]$ is

$f(x)=\frac{\pi }{4}+\frac{2}{\pi }\left [ \frac{\cos x}{1^{2}}+\frac{\cos 3x}{3^{2}}+\cdots\: \cdots \ \cdots \right ]$ $+ \left [ \frac{\sin x}{1}+\frac{\sin 2x}{2}+\frac{\sin 3x}{3}+\cdots \: \cdots\: \cdot \right ]$

The convergence of the above Fourier series at x = 0 gives
 A $\sum_{n=1}^{\infty }\frac{1}{n^{2}}=\frac{\pi ^{2}}{6}$ B $\sum_{n=1}^{\infty }\frac{(-1)^{n+1}}{n^{2}}=\frac{\pi ^{2}}{12}$ C $\sum_{n=1}^{\infty }\frac{1}{(2n-1)^{2}}=\frac{\pi ^{2}}{8}$ D $\sum_{n=1}^{\infty }\frac{(-1)^{n+1}}{(2n-1)}=\frac{\pi}{4}$
GATE CE 2016 SET-2   Engineering Mathematics
Question 8 Explanation:
The function is $f(x)=0$
$-p\lt x\leq 0$
$=p-x,\, 0 \lt x \lt \pi$
And Fourier series is,
$f\left ( x \right )=\frac{\pi }{4}+\frac{2}{\pi }\left [ \frac{\cos x}{1^{2}}+\frac{\cos 3x}{3^{2}}+\frac{\cos 5x}{5^{2}}+... \right ]+\left [ \frac{\sin x}{1}+\frac{\sin 2x}{2}+\frac{\sin 3x}{3}+... \right ] ...\left ( i \right )$
At x=0, (a point of discontinuity), the fourier series converges to $\frac{1}{2}\left [ f\left ( 0^{-1} \right )+f\left ( 0^{+} \right ) \right ]$
where $f\left ( 0^{-} \right )=\lim_{x\rightarrow 0}\left ( \pi -x \right )=\pi$
Hence, eq. (i), we get,
$\frac{\pi }{2}=\frac{\pi }{4}+\frac{2}{\pi }\left [ \frac{1}{1^{2}}+\frac{1}{3^{2}}+... \right ]$
$\Rightarrow \;\; \frac{1}{1}+\frac{1}{3^{2}}+\frac{1}{5^{2}}+...\frac{\pi ^{2}}{8}$
 Question 9
The infinite series $1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+...$ corresponds to
 A sec x B $e^{x}$ C cos x D $1+sin^{2}x$
GATE CE 2012   Engineering Mathematics
Question 9 Explanation:
$e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}...$
(By McLaurin's series expansion)
 Question 10
Laplace transform for the function $f (x) = cosh(ax)$ is
 A $\frac{a}{s^{2}-a^{2}}$ B $\frac{s}{s^{2}-a^{2}}$ C $\frac{a}{s^{2}+a^{2}}$ D $\frac{s}{s^{2}+a^{2}}$
GATE CE 2009   Engineering Mathematics
Question 10 Explanation:
It is a standard result that
$L\left ( \cosh at \right )=\frac{s}{s^{2}-a^{2}}$
There are 10 questions to complete.