MU Applied Mathematics - 3 - May 2015 Exam Question Paper

MU Civil Engineering (Semester 3)
Applied Mathematics - 3
May 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 the Laplace transform of t^-t cosh2t.

5 M

1 (b) Find the fixed points of

w = \frac{3 z - 4}{z - 1}

$w=\dfrac{3z-4}{z-1}$ . Also express it in the normal form

\frac{1}{w - α} = \frac{1}{z - α} + λ w h e r e λ

$\dfrac{1}{w-\alpha}=\dfrac{1}{z-\alpha}+\lambda \ where\ \lambda$ is a constant and α is the fixed point is this transformation parabolic?

5 M

1 (c) Evaluate

\int_{0}^{1 + i} (x^{2} + i y) d z

$\int_{0}^{1+i}\limits (x^{2}+iy)dz$ along the path i) y=x, ii) y=x²

5 M

1 (d) Prove that

f_{1} (x) = 1, f_{2} (x) = x, f_{3} (x) = \frac{3 x^{2} - 1}{2}

$f_{1}(x)=1,f_{2}(x)=x,f_{3}(x)=\dfrac{3x^{2}-1}{2}$ are orthogonal over (-1,1).

5 M

2 (a) Find inverse Laplace transform of

\frac{2 s}{s^{4} + 4}

$\dfrac{2s}{s^{4}+4}$

6 M

2 (b) Find the image of the triangular region whose vertices are i,1+i,1-1 under the transformation w=z+4-2i. Draw the sketch.

6 M

2 (c) Obtain fourier expansion of

f (x) = | \cos x | i n (- π, π)

$f(x)=\left | \cos x \right |in \ (-\pi,\pi)$

8 M

3 (a) Obtain complex form of fourier series for f(x)=cosh 2x+sinh2x in (-2,2).

6 M

3 (b) Using Carnk -Nicholson simplified formula solve

\frac{\partial^{2} u}{\partial x^{2}} - \frac{\partial u}{\partial t} = 0 g i v e n u (0, t) = 0, u (4, t) = 0, u (x, 0) = \frac{x}{3}

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

6 M

3 (c) Solve the equation

y + \int_{0}^{t} y d t = 1 - e^{- t}

$y+\int_{0}^{t}\limits ydt=1-e^{-t}$

8 M

4 (a) Evaluate

\int_{0}^{2 x} \frac{d θ}{5 + 3 \sin θ}

$\int_{0}^{2x}\limits \dfrac{d\theta}{5+3 \sin\theta}$ .

6 M

4 (b) Find half- range cosine series for f(x)=e^x,0

6 M

4 (c) Obtain two distinct Laurent's series for

f (z) = \frac{2 z - 3}{z^{2} - 4 z - 3}

$f(z)=\dfrac{2z-3}{z^{2}-4z-3}$ in powers of (z-4) indicating the regions of convergence.

8 M

5 (a) 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). Assume h=1 and find the value of u upto t=5.

6 M

5 (b) Find the Laplace transform of

e^{- 4 t} \int_{0}^{t} u \sin 3 u d u

$e^{-4t}\int_{0}^{t}\limits u\sin3udu$

6 M

5 (c) Evaluate

\int_{c} \frac{z + 3}{z^{2} + 2 z + 5} d z

$\int_{c}\limits \dfrac{z+3}{z^{2}+2z+5}dz$ where C is the circle i) |z|=1, ii)|z+1-i|=2

8 M

6 (a) Find inverse Laplace transform of

\frac{s}{(s^{2} - a^{2})^{2}}

$\dfrac{s}{(s^{2}-a^{2})^{2}}$ by using convolution theorem.

6 M

6 (b) Find an analytic function f(z)=u+iv where u+v=e^x(cosy+siny).

6 M

6 (c) Solve the equation

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

$\dfrac{\partial u}{\partial t}=k\dfrac{\partial^2 u}{\partial ^2x^2}$ for the conduction of heat along a rod of length l subjected to following conditions
(i) u is intinity for t→∞
(ii)

\frac{\partial u}{\partial x} = 0

$\dfrac{\partial u}{\partial x}=0$ for x=0 and x=l for any time t
(iii) u=lx-x² for t=0 between x=0 and x=l.

8 M

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