## Applied Mathematics - 3 - May 2014

### 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 inverse Laplace transform of $$ \dfrac {S^{2}+5}{(S^{2}+4S+13)^{2}}$$(5 marks)
**1 (b)** IF V=3x^{2}y+6xy-y^{3}, show that the funcltion V is harmonic, find the corresponding analytic function.(5 marks)
**1 (c) ** Evaluate $$ \int_{c}{\bar{z}dz} $$ where C is the upper half of the circle r=1(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)** $$ Evaluate \ \int^{\infty}_{0}\dfrac {\cos at - \cos bt}{t}dt$$(6 marks)
**2 (b)** Obtain complex form of fourier series f(x)=e^{ax} for in (-π, π)(6 marks)
**2 (c) ** Using Crank-Nicholson simplified formula solve ,$$ \dfrac {\partial^{2}u}{\partial x^{2}}- \dfrac {\partial u}{\partial t}=0 \ltbr\gt u(0,t)=0, u(4,t)=0, u(x,o)=\dfrac {x}{3} (16-x^2)$$ for uij i=0,1,2,3,4, and j=0,1,2 (8 marks)
**3 (a)** $$ Evaluate \ \int_{c}\dfrac {\sin^{6}z}{ \left (z-\dfrac {\pi}{6}\right )^{3}}\ where \ C \ is \ |z|=1 $$(6 marks)
**3 (b) ** Find the fourier expansion for f(x)=x-x^{2}-1<x<1 (6 marks)
**3 (c) ** Determine the solution of one dimensional heat equation, $$ \dfrac {\partial u}{\partial t}= C^{2} \dfrac {\partial^{2}u}{\partial x^{2}} $$ under the boundary conditions u(0,t)=0 u(l,t)=0 and u(x,0)=x, (0<x<l), l being length of the rod.(8 marks)
**4 (a) ** Find inverse Laplace transform by using convolution theorem, $$ f(s)= \dfrac {s^{2}}{(s^{2}-a^{2})^{2}} $$(6 marks)
**4 (b) ** Find the image of the region bounded by x=0, x=2, y=0, y=2 in the Z plane under transformation W=(1+i)Z.(6 marks)
**4 (c) ** Find all possible Laurent's expansion of the function $$ f(z)= \dfrac {7z-2}{z(z-2)(z+1)} \ about \ Z=-1 $$(8 marks)
**5 (a) ** $$ Solve\ \dfrac {\partial^{2}u}{\partial x^{2}}-32 \dfrac {\partial u}{\partial t}=0 $$ by Bender-Schmidt method, subject to the conditions u(0,t)=0, u(x,0)=0, u(1,t)=t taking h=0.25, 0 <x <1(6 marks)
**5 (b) ** Obtain half range sine series for f(x) when $$ \begin {align*} f(x)&=x, \ \ &0<x<\dfrac {\pi}{2}\\&=\pi -x, \ &\dfrac {\pi}{2}<x<\pi\end{align*} $$(6 marks)
**5 (c) ** $$ Evaluate \ \int^{\infty}_{-\infty}\dfrac {x^{2}dx}{(x^{2}+a^{2})(x^{2}+b^{2})} $$ by using residues a>0, b>0(8 marks)
**6 (a)** Find the orthogonal trajectory of the family of curves x^{3}y-xy^{3}=c(6 marks)
**6 (b) ** Obtain the fourier expansion of $$ f(x)=\left (\dfrac {\pi - x}{2} \right )^{2} $$ in the interval 0<x<2π, f(x+2π)=f(x) also deduce that $$ \dfrac {\pi}{6}=\dfrac {1}{1^{2}}+\dfrac {1}{2^{2}}+\dfrac {1}{3^{2}}+...... $$(6 marks)
**6 (c) ** Solve using Laplace transform (D^{2}-3D+2)y=4 e^{2t}, with y(0)=-3 y'(0)=5(8 marks)