Complex Variables and Partial Differential Equations (3130005)

Summer-2020

BE | Semester 3

Summer - 2020|27-10-2020

Total Marks 70

Q1 (a) Find the analytic function $f (z) = u + iv, if u = x^{3} - 3 {xy}^{2}$

Unit : Complex variable functions | Topic : Harmonic functions

Q1 (b) Find the roots of the equation, $z^{2} - (5 + i) z + (8 + i) = 0$ .

Unit : Complex variable functions | Topic : Powers and Roots

Q1 (c ( i )) Determine & sketch the image $| z | = 1$ of under the transformation $w = z + i$ .

Unit : Complex variable functions | Topic : Conformal mappings

Q1 (c ( ii )) Find the real and imaginary parts of $f (z) = z^{2} + 3 z$ .

Unit : Complex variable functions | Topic : Conformal mappings

Q2 (a)
Evaluate $\int_{C} (x^{2} - i y^{2}) d z$ along the parabola $y = 2 x^{2}$ from $(1, 2)$ to $(2, 8)$ .

Unit : Complex Variable - Integration

Q2 (b) Determine the Mobius transformation which maps $z = \infty, i, 0$ into $w = 0, i, \infty$ .

Unit : Complex variable functions | Topic : Mobius transformations and their properties

Q2 (c(i))
Evaluate $\int_{C} \frac{e^{- z}}{z + 1} d z$ ,where $C : |z| = \frac{1}{2}$ .

Unit : Complex Variable - Integration | Topic : Cauchy-Goursat theorem

Q2 (c(ii))
Find the radius of convergence of $\sum_{1}^{\infty} {(1 + \frac{1}{n})}^{n^{2}} z^{n}$ .

Unit : Complex Variable - Integration | Topic : Power Series

OR

Q2 (c( i ))
Find the fourth roots of -1.

Unit : Complex variable functions | Topic : Powers and Roots

Q2 (c ( ii ))
Find the roots of $\log z = i \frac{π}{2}$ .

Unit : Complex variable functions | Topic : Elementary analytic functions (Exponential, Trigonometric, Logarithm) and their properties

Q3 (a)
Evaluate $\int_{C} \frac{1}{z^{2}} d z$ , Where $C : |z| = 1$ .

Unit : Complex Variable - Integration | Topic : Cauchy-Goursat theorem

Q3 (b)
For, $f (z) = \frac{1}{{(z - 1)}^{2} (z - 3)}$ , find the residue at $z = 1$ .

Unit : Laurent’s series | Topic : Residues

Q3 (c)
Expand $f (z) = \frac{1}{(z + 2) (z + 4)}$ valid for the region $(i) |z| < 2, (ii) 2 < |z| < 4$ and $(i) |z| > 4 .$

Unit : Laurent’s series | Topic : Laurent’s series

OR

Q3 (a)
Evaluate $\int_{C} \frac{z + 4}{z^{2} + 2 z + 5} d z$ , Where $C : |z + 1| = 1$ .

Unit : Complex Variable - Integration | Topic : Cauchy Integral formula

Q3 (b)
Evaluate using Cauchy’s residue theorem $\int_{C} \frac{e^{2 z}}{{(z + 1)}^{3}} d z$ , Where, $C : 4 x^{2} + 9 y^{2} = 16$ .

Unit : Laurent’s series | Topic : Cauchy Residue theorem

Q3 (c)
Expand $f (z) = \frac{1}{z (z - 1) (z - 2)}$ valid for the region $(i) |z| < 1, (ii) 1 < |z| < 2$ and $(i) |z| > 2 .$

Unit : Laurent’s series | Topic : Cauchy Residue theorem

Q4 (a)
Solve $xp + yq = x - y$

Unit : First order partial differential equations | Topic : Solutions of first order linear PDEs

Q4 (b)
Derive partial differential equation by eliminating the arbitrary constants $‘ a ’$ and $‘ b ’$ from $z = ax + by + ab$ .

Unit : First order partial differential equations

Q4 (c ( i ))
Solve the p.d.e. $2 r + 5 s + 2 t = 0$

Unit : HIgher order partial differential equations and applications | Topic : second and higher order by complementary function and particular integral method

Q4 (c ( ii ))
Find the complete integral of $p^{2} = qz$ .

Unit : First order partial differential equations | Topic : Solutions of first order nonlinear PDEs

OR

Q4 (a)
Find the solution of $x^{2} p + y^{2} q = z^{2}$ .

Unit : First order partial differential equations | Topic : Solutions of first order nonlinear PDEs

Q4 (b)
Form the partial differential equation by eliminating arbitrary function $\emptyset$ from $z = \emptyset (\frac{y}{x})$ .

Unit : First order partial differential equations

Q4 (c ( i ))
Solve the PDE $(D^{2} - D'^{2} + D - D') z = 0$ .

Unit : HIgher order partial differential equations and applications | Topic : second and higher order by complementary function and particular integral method

Q4 (c ( ii ))
Solve by Charpit’s method ${yzp}^{2} - q = 0$ .

Unit : First order partial differential equations | Topic : Solutions of first order nonlinear PDEs

Q5 (a)
Solve $(2 D^{2} - 5 DD' + D'^{2}) z = 24 (y - x)$

Unit : HIgher order partial differential equations and applications | Topic : second and higher order by complementary function and particular integral method

Q5 (b)
Solve the p.d.e. $u_{x} + u_{y} = 2 (x + y) u$ using method of separation of variables.

Unit : HIgher order partial differential equations and applications | Topic : Separation of variables method to simple problems in Cartesian coordinates

Q5 (c)
Find the solution of wave equation $u_{tt} = c^{2} u_{xx}, 0 \leq x \leq π$ with initial and boundary condition $u (0, t) = u (π, t) = 0; t > 0, u (x, 0) = k (\sin x - \sin 2 x), u_{t} (x, 0) = 0; 0 \leq x \leq π, (c^{2} = 1)$

Unit : HIgher order partial differential equations and applications | Topic : Modeling and solution of the Wave equations

OR

Q5 (a)
Solve $r + s + q - z = 0$ .

Unit : HIgher order partial differential equations and applications | Topic : Solution to nonhomogeneous linear partial differential equations

Q5 (b)
Using method of separation of variables, solve $2 u_{x} = u_{t} + u, u (x, 0) = 4 e^{- 3 x}$ .

Unit : HIgher order partial differential equations and applications | Topic : Separation of variables method to simple problems in Cartesian coordinates

Q5 (c)
Find the solution of $u_{t} = c^{2} u_{tt}$ together with the initial and boundary conditions $u (0, t) = u (2, t) = 0; t \geq 0, u (x, 0) = 10; 0 \leq x \leq 2 .$

Unit : HIgher order partial differential equations and applications | Topic : Modeling and solution of the Heat equations