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=x²(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.<></span>(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-x² for t=0 between x=0 and x=l.(8 marks)