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.

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