MORE IN Applied Mathematics - 3
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 $w=\dfrac{3z-4}{z-1}$. Also express it in the normal form $\dfrac{1}{w-\alpha}=\dfrac{1}{z-\alpha}+\lambda \ where\ \lambda$ is a constant and α is the fixed point is this transformation parabolic?
5 M
1 (c) Evaluate $\int_{0}^{1+i}\limits (x^{2}+iy)dz$ along the path i) y=x, ii) y=x2
5 M
1 (d) Prove that $f_{1}(x)=1,f_{2}(x)=x,f_{3}(x)=\dfrac{3x^{2}-1}{2}$ are orthogonal over (-1,1).
5 M

2 (a) Find inverse Laplace transform of $\dfrac{2s}{s^{4}+4}$
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 $f(x)=\left | \cos x \right |in \ (-\pi,\pi)$
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 $\dfrac{\partial^2 u}{\partial x^2}-\dfrac{\partial u}{\partial t}=0\ given\ u(0,t)=0,u(4,t)=0,u(x,0)=\dfrac{x}{3}$ ujj for i=0,1,2,3,4 and j=0,1,2.
6 M
3 (c) Solve the equation $y+\int_{0}^{t}\limits ydt=1-e^{-t}$
8 M

4 (a) Evaluate $\int_{0}^{2x}\limits \dfrac{d\theta}{5+3 \sin\theta}$.
6 M
4 (b) Find half- range cosine series for f(x)=ex,0
6 M
4 (c) Obtain two distinct Laurent's series for $f(z)=\dfrac{2z-3}{z^{2}-4z-3}$ in powers of (z-4) indicating the regions of convergence.
8 M

5 (a) Solve $\dfrac{\partial^2 u}{\partial x^2}-2\dfrac{\partial u}{\partial t}=0$ 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 $e^{-4t}\int_{0}^{t}\limits u\sin3udu$
6 M
5 (c) Evaluate $\int_{c}\limits \dfrac{z+3}{z^{2}+2z+5}dz$ where C is the circle i) |z|=1, ii)|z+1-i|=2
8 M

6 (a) Find inverse Laplace transform of $\dfrac{s}{(s^{2}-a^{2})^{2}}$ by using convolution theorem.
6 M
6 (b) Find an analytic function f(z)=u+iv where u+v=ex(cosy+siny).
6 M
6 (c) Solve the equation $\dfrac{\partial u}{\partial t}=k\dfrac{\partial^2 u}{\partial ^2x^2}$ for the conduction of heat along a rod of length l subjected to following conditions
(i) u is intinity for t→∞
(ii) $\dfrac{\partial u}{\partial x}=0$ 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

More question papers from Applied Mathematics - 3