Biju Patnaik University of Technology 2007-3rd Sem B.Tech ,,Mathematics-3 - Question Paper
Thursday, 23 May 2013 04:10Web
1.
a)Solve the partial differential formula
Uxy-Uy=0
b)Solve the partial differential formula Uxx-Uyy =0, by separation of variable.
c)Write the D'Alemberts Solution of the wave formula Utt=C 2Uxx U(x,0)=f(x)
And Ut(x, 0) =g(x).
d)Classically the partial differential formula Uxx+ 4Uxy-Uyy=0 as elliptic, parabolic and hyperbolic.
e)Write the transform in general form which can transfer the partial differential formula Uxx+4Uxy+4Uyy=0 into canonical form.
f)Is the function
F (z) =xy/x2+y2 z=! 0
0, z=0
Analytic at z=0? Why?
g)How many conditions are needed to determine all the unknown current in the linear fractional transformation T(z)=az+b/az+d
Where ad-bc ?0 and why?
h)How many fixed points does the linear fractional transformation have and why?
i)Is there any analytic function f (z) whose imaginary part is v(x, y) =x3+2y and why?
j)Find the poles of the function f (z) =sec (z).
2
An elastic string of length 'L' has its ends at x=0 and x=L fixed. If the point s=L/3 is drawn aside a small distance h and released at time t=0, them obtain the displacement U(x, t) of the string at any following time t>0.
3
An insulated rod of length L has its ends A and B maintained at 0 and 100 respectively until steady state conditions prevail. If the end B is suddenly decreased to 0C and maintained at 0C, then obtain the temperature distribution U(x, t) at a distance x from the end A at any following time t>0.
4
obtain the steady state temperature distribution U(x, y) in the uniform unit square 0=x=1 and 0=y=1 when edge y=1 maintained at the temperature x (1-x) and the other 3 edges being thermally insulated.
5
Write the ans according to the instruction:
obtain the harmonic conjugate of U(x,y)=cos(x)sinh(y)
obtain the linear fractional transformation which maps lower half plane into unit disk.
6
Evaluate the subsequent integrations over the prescribed simple closed path C:
where z={z:|z|=3}.(3)
where z={z|z|=1}
And f (z) is analytic over boundary and interior of the simple closed path C.
where c=(z:|z|=3).
7
Evaluate as per the instruction:
obtain the Laurent series of the function f(z)= which is valid in the region 0.5<|z+1|<0.75.
obtain the residue of the function f(z)= at z=1.
8
Evaluate the subsequent integrations using residue theorem
a)integration dx/1+x2 0
(b)integration dx/2+cosx 0
| Earning: Approval pending. |
