Government Engineering College, ModasaB.E Semester IV : Tutorial of Numerical Method

Tutorial A : Solution of non-linear equation

1. Write brief note on Error. Discuss types of Error.2. Find a real root to the following equations using the Bisection Method correct upto mentioned decimal places (i) (upto 3 decimal places). (ii) ( upto 3 decimal places). (iii) (upto 3 decimal places). (iv) (upto 3 decimal places). (v) (upto 4 decimal places). (vi) (upto 2 decimal places). (vii) (upto 2 decimal places). (viii) (upto 3 decimal places).3. Find a negative root to the following equations using the Bisection method correct upto mentioned decimal places (i) (upto 4 decimal places). (ii) (upto 2 decimal places).4. Find a real root to the following equations using the Newton-Raphsonm ethod correct upto mentioned decimal places (i) (upto 4 decimal places). (ii) ( upto 4 decimal places). (iii) (upto 4 decimal places). (iv) (upto 4 decimal places). (v) (upto 5 decimal places).

(vi) (upto 5 decimal places).5. Set up a Newton iteration for computing the square root of x for a given positive number c and apply it to c = 2.6. Derive Newton-Raphson iteration Formula for finding k th root of the positive number x. Use it to compute 7. Derive Newton-Raphson iteration Formula for finding reciprocal of the

positive number x. Use it to compute

8. Solve the given problem by the Secant Method, using and as indicated (i) (ii) cos x cosh x = 1 (iii) (iv) (v) (vi)

Tutorial B : Finite difference and Interpolation.

:Finite Difference:

1. Construct the forward difference table for ; x = 0,1,2,3,4. Also find and

:Interpolation with Equal Space Suninterval:

1. Find the cubic polynomial which takes the following values: y(1) = 24, y(3) = 120, y(5) = 336, y(7) = 720. Hence obtain the value of y(8).2. What is the degree of the interpolation polynomial for the following data (1,5) , (2,18) , (3,37) , (4,62) , (5,93) ? .Find the polynomial.3. The following table gives the values of density of saturated water for various temperature of saturated stream.

Temp C : 100 150 200 250 300 Density : 958 917 865 799 712

Find by interpolation, the density when the temperature is 130 C and 275 C

4. Find a polynomial of degree two which takes the following values using Newton’s backward formula.

x : 0 1 2 3 4 5 6 7 y : 1 2 4 7 11 16 22 29 . Also find y(8).

5. From the following table, find the value of e using Gauss’s forward formula. x : 1.00 1.05 1.10 1.15 1.20 1.25 1.30 e : 2.7183 2.8577 3.0042 3.1582 3.3201 3.4903 3.6693

6. Apply Gauss’s forward and backward formulas to obtain sin 45 , given in the following table. x : 20 30 40 50 60 70 y = sin x : 0.3420 0.5020 0.6428 0.7660 0.8660 0.9397

7. Given that sin(0.1) = 0.0998, sin(0.2) = 0.1986, sin(0.3) = 0.2995, sin (0.4) = 0.3894 and sin(0.5) = 0.4794. Find sin (0.35) and sin (0.37) using Bessel’s formula.

8. The following table gives the values of e for certain equidistant values of x. Find the value of e when x = 0.644 using Bessel’s and Everett’ formula.

x = 0.61 0.62 0.63 0.64 0.65 0.66 0.67 e = 1.84043 1.85892 1.87761 1.89648 1.91554 1.93479 1.95423

9. Given that = 2, = 2.2361, = 2.4495, = 2.6458, = 2.8284 and = 3, find by using Gauss’s forward formula and by using Gauss’s backward formula.10. Use stirling’s formula to find u(32) from the following :

u(20) = 14.035,u(25) = 13.674,u(30) = 13.257,u(35) = 12.734,u(40) = 12.084, u(45) = 11.309

11. Use Everett’s formula to find y(26) and y(28) from the following table. x : 15 20 25 30 35 40 y : 12.849 16.351 19.524 22.396 24.999 27.356

12. Compute the value of Bessel function (x) for x = 1.72 from the values in the following table using Newton’s forward formula and Newton’s backward formula.

x: 1.7 1.8 1.9 2.0 y: 0.3979849 0.3399864 0.2818186 0.2238908

13. Set up Newton’s forward difference formula for the data which is given below. x : 0.5 0.6 0.7 0.8 cosh x : 1.127626 1.185465 1.255169 1.337435 . Also find cosh (0.56)

:Interpolation with unequal Space Subinterval:

1. Using Lagrange’s interpolation formula to fit a polynomial to the data : x : 0 1 3 4 y : -12 0 6 12 Also find the value of y when x = 2

2. Given the table of values x : 150 152 154 156 y = :. 12.247 12.329 12.410 12.490

Evaluate and using Lagrange’s interpolation formula.3. Calculate the Lagrange’s polynomial of the error function

f(x) = erf(x) = (2/ ) for the following data

x : 0.25 0.5 1 f(x) : 0.27633 0.52050 0.84270. Also find f(0.75)

4. Find the polynomial of the lowest possible degree which takes the values 3, 12, 15, -29 when x has the values 3, 2, 1, -1 respectively. Also find f(2.5)5.Using Newton’s DD interpolation formula, find f(x) as a polynomial in power of (x+3)

x : -4 -1 0 2 5 f (x) : 1245 33 5 9 1335. Hence find f(1).

6. If y(1) = 4, y(3) = 12, y(4) = 19 and y(x) = 7 then find x by Newton’s formula7. Prove that the third DD of the function f(x) = 1/x with arguments p, q, r, s is ( - 1 / pqrs ).8. Given sin 45 = 0.7071, sin 50 = 0.7660, sin 55 = 0.8192 and sin 60 = 0.8660. Find sin 52 using Newton’s interpolation formula.9. Given x : 2 2.5 4

f(x) : 0.5 0.4 0.25 obtain f(x) at (x – 3) by Newton’s formula.10. Use Newton’s divided difference formula, to find f(x) from the following data

x : 0 2 3 4 6 7 f(x) : 0 8 0 -72 0 1008

11. Given the table of values x : 150 152 154 156 y : 12.247 12.329 12.410 12.490Evaluate f(151) using Newton’s divided interpolation formula.

Tutorial C : Numerical Integration.

1. Estimate using five strips by Trapezoidal rule.

2. Calculate by Simpson’s rule approximate value of by taking seven equidistant

ordinates. Compare it with the exact value and value obtained by using the Trapezoidal rule.

3. Using Simpson’s rule, find the volume of the solid of revolution formed by rotating about x- axis. The area between the x-axis, the lines x = 0 and x = 1 and a curve through the points (0,1), (0.25,0.9896), (0.50,0.9589), (0.75,0.9089) and (1,0.8415).

4. The velocity V of a particle at distance S from a point on its path is given by following table S (ft) : 0 10 20 30 40 50 60

V(ft/s) : 47 58 64 65 61 52 38 Estimate the time taken to travel 60 ft. using Simpson’s 1/3 & also using Simpson’s 3/8 rule.

5. Find the value of h so that the integral obtain by Simpson’s 1/3 rule is correct upto 4D

6. Evaluate using strip width 0.5 by Simpson’s 3/8 rule.

7. Compute the error in the evaluation of by (a)Trapezoidal rule (b) Simpson’s 1/3 rule

(c) Simpson’s 3/8 rule.

Tutorial D : Solution of System of linear equation.

1. Solve the following system of equations using Gauss elimination method (i) 2x + y + z = 10 (ii) 2x - 6y + 8z = 24

3x + 2y + 3z = 18 5x + 4y - 3z = 2 x + 4y + 9z = 16 3x + y + 2z = 16

2. Solve the following system of equations using partial pivoting method (i) 2x + 2y + z = 6 (ii) x + 2y + 3z = 18

4x + 2y + 3z = 4 2x + y - 4z = -30 x - y + z = 0 -5x + 8 y + 17z = 96

3. Solve the following system of equations using Gauss-Jordan method 10x + y + z = 12 x + 10y - z = 10 x - 2y + 10z = 9

4. Solve the following system of equations using Gauss Jacobi method x + y + 54z = 110 6x + 15y - z = 72 27x + 6y + 4z = 85

5. Solve the following system of equations using Gauss Seidel method (i) x + y + 54z = 110 (ii) 83x + 11y - 4z = 95 (iii) x + 3y + 10z = 24 27x + 6y - z = 85 7x + 52y + 13z = 104 28x + 4y – z = 32 6x + 15y + 2z = 72 3x + 8y + 29z = 71 2x + 17y + 4z = 35

Government Engineering College, ModasaB.E Semester IV : Tutorial of Complex Variable

Tutorial 1 : Complex numbers.

1. Determine the principal value of the argument : (i) 1- I (ii) 3 ± 4i (iii) -5+5i 2. Find all roots of the following(i) (ii) (iii) (iv) 3. Find the roots common to the equations , 4. Solve the equations

(i) (ii) (iii)

(iv) (v) (vi)

(vii) (vii) 5. Find out and give reason whether f(z) is continuous at z = 0 if f(z) = 0 and for z ≠ 0 the

function is equal to (i) (ii) (iii)

6. Find the value of derivatives (i) (ii) at i (iii)

7. Find argument and modulus of following complex numbers using De-Moivre’s Theorem

(i) (ii) (iii)

(iv) (v)

8. Seperate real and imaginary parts of (i) (ii) (iii)

9. Prove that .

Tutorial 2 : Complex function-Analytic function.

1. State and prove Necessary and Sufficient condition for the function to be analytic.2. Check the analyticity (Show the details of your work)

(i) (ii) (iii)

(iv) (v) (vi)

(vii) (viii) (ix) w = sin z

(x) w = log z 3. Show that if f (z) is analytic inDomain D and in D,then f(z) = const in D. 4. Show that the function f(z) =u + iv, where

satisfies the Cauchy- Riemann equation at z = 0 .Is the function analytic at z = 0? Justify your answer. 5. Examine the nature of the function

in the region including the origin.

6. Examine the nature of the function

prove that as z 0 along

any radius vector but not as z 0 in any manner and also that f(z) is not analytic at z = 0. 7. Derive C-R Equation in polar form. 8. Find p such that the function f (z) expressed in polar coordinates as is analytic.

9. If n is real show that is analytic except possible when r = 0 and its derivative is .

10. Every analytic function f(z) define two families of curves which form orthogonal system. 11. Determine whether the following functions are harmonic. If yes, find the corresponding analytic function .

(i) (ii) (iii)

(iv) (v) u = sin x cosh y (vi) (vii) (viii)

12. Determine a and b such that the given function are harmonic and find conjugate harmonic function. (i) (ii) (iii) (iv) u = cos ax cosh 2y13. Show that the function which is harmonic remains harmonic under the transformation .

14. If potential function is , find flux function and the complex potential function.

15. In a two dimensional fluid flow, the steam function is ψ = , find the velocity

potential φ.

16. Deduce the following with the polar form of Cauchy-Riemann Equations

(a) (b)

17. Find analytic function f(z) =u (r,θ) + i v (r,θ) such that

18. If , find f (z).

19. Determine the analytic function (Milne Thomson Method), real part is

(i) (ii) (iii) cos x cosh y

(iv) (v) (vi) 20. Determine the analytic function (Milne Thomson Method), imaginary part is

(i) (ii) sinh x cos y (iii)

(iv) (v) (vi) tan y/x

21. If and f (z) = u + iv is an analytic function

of z = x + iy, find f(z) in terms of z.22. Show that the function is harmonic. Find the conjugate function v and express u + iv as an analytic function of z.

23. If f (z) is a regular function of z, prove that

Tutorial 3 : Conformal Mapping

1. For the conformal transformation , show that (a) The coefficient of magnification at z = 2 + i is .

(b) The angle of rotation at z = 2 + i is tan 0.5. (c) The coefficient of magnification at z = 1 + i is . (d) The angle of rotation at z = 1 + i is /4.

2. Find the image of under the mapping .

3. Show that the function transforms the straight line x = c in the z-plane into a circle

in the w-plane.

4. Show that under the transformation the image of the hyperbola is the

lemniscates .5. Find the bilinear transformation that maps the points z = 1, i,-1 into the points w = i,,0,-1 .Hence find the image of │z│< 16. Find the bilinear transformation that maps the points i, -i, 1 of the z –plane 0,1,∞ of the w-plane respectively.7. Find the bilinear transformation that maps the points -1, 0, 1 of the z –plane 1,-1, ∞ of

the w-plane respectively. 8. Find the bilinear transformation that maps the points i,-i, 0 of the z –plane 0, ∞,-1 of the w- plane respectively.

9. Show that maps the real axis of the z-plane into the circle │w│= 1 and the half

plane y > 0 into the interior of the unit circle │w│< 1 in the w-plane.

10.Find the image of under the mapping .

11.Find all the linear fractional transformation whose only fixed points are –i and i.

12. For the conformal transformation , show that the circle│z - 1│=1 transforms into the cardioid R = 2(1+cosφ) where w =R in the w- plane.

Tutorial 4 : Complex Integration

:Open Curve:

1. Find the value of the integral

a) Along the straight line z = 0 to z = 1+ i ; b) Along real axis from z = 0 to z = 1 and then along a line parallel to the imaginary axis from z = 1 to z = 1+ i .

2. Evaluate along the line joining the points (1,-1) and (2, 3).

3. Evaluate the shortest path from 1+ i to 3 + 2i.

: Closed Curve using Cauchy Integral Theorem:

1. Evaluate , where c is the circle │z - i│= 5.

2. Evaluate the integral , where c is the unit circle │z│= 1.

3. Evaluate , where C the boundary of the square with vertices 0, i, 1 + i, 1,

clockwise.4. State and prove Cauchy’s Integral Theorem.

5. Find the value of the integral , where c is the circle(a) │z + i│= (b) │z - i│= 1

6. Find the value of the integral , where c is the circle │z + 1│= 1.

7. Find the value of the integral , where c is the circle │z - 2│= 4.

8. State and prove Cauchy’s Integral Formula.

9. Evaluate a) , C the circle │z│=

b) , C the circle │z + 1│= 1

c) , C the square with vertices ±1and ±i.

d) , C the circle │z │= 1/2

e) , over the circular path │z│= 2.

f) , C the ellipse .

g) ,C the circle │z │= 1/2

10. Evaluate where c is the circle, (i) (ii) (iii)

11. Use Cauchy’s Integral Formula to evaluate where c is circle .

12.Evaluate where c is the circle , .

13.Integrate the following function around the unit circle

i) ii) iii)

Tutorial 4 : Expansion of function : Laurent series

1. Expand in the region a) b)

2. Find the first four terms of the of the Taylor’s series expansion of the complex variable

function about z = 2.Find the region of convergence.

3. Find the tree terms of Taylor’s series expansion of about z = - i. Find the

region of convergence.

4. Expand in Laurent’s series valid for

(i) (ii) (iii) (iv)

5. Expand the function in Laurent’s series about the point z = o.

Tutorial 5 : Cauchy Residue Theory

1. Determine poles and residue at simple pole of the function

2. Determine the poles and residue of the function (i) at its double pole

(ii) at z = I (iii)

3. State residue theorem .Evaluate the following using residue theorem

(i) where c is the circle

(ii) where c is the circle

(iii) where c is the circle .

4. Using contour integration, evaluate the real integral

(i) (ii) (iii) .