## Applied Mathematics - 3 - May 2015

### Civil Engineering (Semester 3)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.
**1 (a)** Find the Laplace transform of t^{-t} cosh2t.(5 marks)
**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 marks)
**1 (c)** Evaluate $$\int_{0}^{1+i}\limits (x^{2}+iy)dz$$ along the path i) y=x, ii) y=x^{2}(5 marks)
**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 marks)
**2 (a)** Find inverse Laplace transform of $$\dfrac{2s}{s^{4}+4}$$(6 marks)
**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 marks)
**2 (c)** Obtain fourier expansion of $$f(x)=\left | \cos x \right |in \ (-\pi,\pi)$$(8 marks)
**3 (a)** Obtain complex form of fourier series for f(x)=cosh 2x+sinh2x in (-2,2).(6 marks)
**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 marks)
**3 (c)** Solve the equation $$y+\int_{0}^{t}\limits ydt=1-e^{-t}$$(8 marks)
**4 (a)** Evaluate $$\int_{0}^{2x}\limits \dfrac{d\theta}{5+3 \sin\theta}$$.(6 marks)
**4 (b)** Find half- range cosine series for f(x)=e^{x},0<x<1.< a="">

</x<1.<><>(6 marks)
**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 marks)
**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 marks)
**5 (b)** Find the Laplace transform of $$e^{-4t}\int_{0}^{t}\limits u\sin3udu$$(6 marks)
**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 marks)
**6 (a)** Find inverse Laplace transform of $$\dfrac{s}{(s^{2}-a^{2})^{2}}$$ by using convolution theorem.(6 marks)
**6 (b)** Find an analytic function f(z)=u+iv where u+v=e^{x}(cosy+siny).(6 marks)
**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-x^{2} for t=0 between x=0 and x=l.(8 marks)