## Applied Mathematics - 3 - May 2015

### Mechanical 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 te^{-4} 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 λ is a constant and α is the fixed point. Is this transformation parabolic? (5 marks)

** 1 (c) ** Evaluate $$ \int^{1+i}_0 (X^2 - iy)dz $$ along the path

i) y=x ii) y=x^2 (5 marks)

** 1 (d) ** Prove that $$ f_i(x)=1, f_2(x)=x, f_3(x) = \dfrac {3x^2 -1} {2} $$ 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-i under the transformation w=z+4-2i. Draw the sketch. (6 marks)

** 2 (c) ** Obtain fourier expansion of f(x)=|cosx|in (-π,π). (8 marks)

** 3 (a) ** Obtain complex form of fourier series for f(x)=cosh 2x+sinh2x in (-2,2). (6 marks)

** 3 (b) ** Using Crank Nicholson simplified formula solve $$ \dfrac { \partial^2 u} {\partial x^2} - \dfrac {\partial u} {\partial t} = 0 $$ given $$ u(0,t)=0, \ u(4,t)=0, \ u(x,0)= \dfrac {x} {3} (16-x^2) $$ find uij for i=0, 1, 2, 3, 4 and j=0, 1, 2. (6 marks)

** 3 (c) ** Solve the equation $$ y + \int^1_0 ydt = 1-e\ltsup\gt-1\lt/sup\gt. $$ (8 marks)

** 4 (a) ** Evaluate $$ \int^{2\pi} {0} \dfrac {d \theta } {5+3\sin \theta } $$ (6 marks)

** 4 (b) ** Find half-range cosine series for f(x)=e^{x}, 0<x<1. <="" a="">

</x<1.><>(6 marks)

** 5 (b) ** Find the Laplace transform of $$e^{-4t} \int^1_0 u\sin 3 \ udu. $$ (6 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 $$ \dfrac {\partial u} {\partial t} = k \dfrac {\partial ^2 u} {\partial ^2 x^2} $$ for the conduction of heat along a rod of length l subject to following conditions.

i) u is not infinity 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)
**4 (c)** Obtain two district 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 values of u upto t=5.(6 marks)