MU Applied Mathematics - 3 - December 2015 Exam Question Paper

MU Civil Engineering (Semester 3)
Applied Mathematics - 3
December 2015

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) Find Laplace transform of tsin³t.

5 M

1 (b) Find half range sine series in (0, π) for x(π-x).

5 M

1 (c) Find the image of the rectangular region bounded by
x=0, x=3, y=0, y=2 under the transformation ω=z+(1+i).

5 M

1 (d) Evaluate ∫f(z)dz along the parabola y=2x², z=0 to z=3+18i where f(z)=x²-2iy.

5 M

2 (a) Find two Laurent's series of

f (z) = \frac{1}{z^{3} (z - 1) (z + 2)}

$f(z) = \dfrac {1}{z^3 (z-1)(z+2)}$ about z=0 for
i) |z|<1 ii) 1<|z|<2.

8 M

2 (b) Find complex form of Fourier series for f(x)=cos h2x+sin h2x in (-2,2).

6 M

2 (c) Find bilinear transformation that maps 0,1,&infty; of the z plane into -5, -1, 3 of ω plane.

6 M

3 (a) Solve by using Laplace transformation
(D² + 2D+5) y=e^-1 sint when y(0) and y¹(0)=1.

8 M

3 (b) Solve

\frac{\partial^{2} u}{\partial x^{2}} - 2 \frac{\partial u}{\partial t} = 0

$\dfrac {\partial^2 u}{\partial x^2} - 2 \dfrac {\partial u}{\partial t} =0$ by Bender Schmidt method given u(0,t)=0, u(4,t)=0, u(x,0)-x(4-x)

6 M

3 (c) Expand f(x)=lx-x² 0

6 M

4 (a) Evaluate \[ \int^{2\pi}_{0} \dfrac {2\theta}{(2+cos \theta)^2}

8 M

4 (b) Evaluate

\int_{0}^{\infty} e^{- 2 t} \frac{\cos 2 t \sin 3 t}{t} d t

$\int^{\infty}_0 e^{-2t} \dfrac {\cos 2t \sin 3t}{t}dt$

6 M

4 (c) Using Crank Nicholson method solve.

\frac{\partial^{2} u}{\partial x^{2}} - \frac{\partial u}{\partial t} = 0 u (0, t) = 0, u (4, t) = 0 u (x, 0) = \frac{x}{3} (16 - x^{2})

$\dfrac {\partial ^2 u}{\partial x^2}- \dfrac {\partial u}{\partial t} =0 \\ u(0,t)=0, \ u(4,t)=0 \\ u(x,0) = \dfrac {x}{3} (16-x^2)$ Find u₈ for i=0,1,2,3,4 and j=0,1,2.

6 M

5 (a) Find analytic function whose real part is

\frac{\sin 2 x}{\cosh 2 y + \cos 2 x}

$\dfrac{\sin 2x}{\cosh 2y+\cos 2x}$

8 M

5 (b) Find

i) L^{- 1} [\frac{e^{- π 3}}{s^{2} - 2 s + 2}] i i) L^{- 1} [\tan^{- 1} (\frac{s + a}{b})]

$i) \ L^{-1}\left [ \dfrac {e^{-\pi 3}}{s^2 -2s+2} \right ] \\ ii) \ L^{-1}\left [ \tan^{-1} \left ( \dfrac {s+a}{b} \right ) \right ]$

6 M

5 (c) Find the solution of one dimensional heat equation

\frac{\partial u}{\partial t} = e^{2} \frac{\partial^{2}}{\partial x^{2}}

$\dfrac {\partial u}{\partial t} =e^2 \dfrac {\partial ^2}{\partial x^2}$ under the boundary conditions u(0, t)=0
u(l,t)=0 and u(x,0)=x
0

6 M

6 (a) A string is stretched and fastened to two points distance l apart. Motion is started by displacing the string in the form y=a sin (πx/l) which it is released at time t=0. Show that the displacement of a point at a distance x from one end at time t is given by

y_{x, t} = a \sin (\frac{π x}{l}) \cos (\frac{π c t}{l})

$y_{x, t}=a \sin \left ( \dfrac {\pi x}{l} \right ) \cos \left ( \dfrac {\pi ct} { l}\right )$

8 M

6 (b) Find the residue of

\frac{\sin π z^{2} + \cos π z^{2}}{(z - 1) (z - 2)^{2}}

$\dfrac {\sin \pi z^2 + \cos \pi z^2}{(z-1)(z-2)^2}$ at its poles.

6 M

6 (c) Find Fourier series of xcosx in (-π, π).

6 M

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