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0800MAT201122102

Module 2

Solve the boundary value problem described by uy-C7Uxx=0, 0 2 x > ۸, <0

०८0, t) = ०८९.) =0, (20, u(x, 0) = 10 sin (=) ۱ ‏نے‎ 050

Find the temperature u(x, t) in a homogenous bar heat conducting material
of length / whose ends kept at 0°c and whose initial temperature is given by
u(x, 0) = lx — x?.

Derive one dimensional wave equation.

The ends A and B of a rod 10 cm in length are kept at temperatures 0°C and
100°C until the steady state condition prevails. If B is Suddenly reduced to

0°C and kept so . Find the temperature distribution in the rod at time t.
Module 3

Show that an analytic function f(z) = u + iv is constant if its modulus is
constant.

Find the image of 1 < |z| < 2, = 29 > 3 पतल the mapping w = 2
Verify whether u = x*?—3xy? is harmonic and find its conjugate
harmonic function v.

Find the image of the region between real axis and a line parallel to real axis
at y ಎ under the mapping W= ९९

Module 4

Evaluate ‏,ل‎ |z|? dz where C is the circle [ச] = 2.

2
Evaluate ழ்‌. ன க dz where C is the circle |z| = 4 using Cauchy’s integral
formula
Evaluate फ. ‏ہے‎ dz, where c is |z| = 2 using Cauchy’s integral
formula

2
Evaluate f= a over (a) |2| = 1.5 (0) |2 + (| = 1
Module 5

Find the Laurent series expansion for f(z) = गातो valid in

(a) 1 ಆ |z| <2 (0) ) |z| <2

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