Question

? ≤ 0.9 (subsonic) and over the range 1.2 ≤ ? ≤ 3.2 (supersonic). 

(a) Using Newton-Raphson method. (b) Using Bi-section method. (c) Using Simple method. (d) Using false-position method. Stop the solution when the relative error < 1% and show all your steps

Answer 1

The red bold font is the answer in all the tables.

A 1[1 + 0.2 M?]3 A**M 1.728 Now, A=4 4= Cito, 2 m2 3 1.728 m 6 (Itoizma)? 6.912m=0 if(M)= (1+0.2m?)"_6.912M (a) Newton Raphson Method: f(m)= (1+0.2m?) ?-6.912M f'(M)= 3 (1+ 0 12 M2) (0 +0,2 (2m))-6.912 = 1.2m (1+0, 2m²)²-6.912 3 nitial guessa mpol $ cm= f (0.1) = 0.31481 f'(M)=f(0) =-6.79176

. 0,31481 =0.14635 M2=mi- f(mi) f'(m) 1-6.79 176) error = 1 me-mi +100=31.6770 following Doing same steps, we get table, we stop when error is less than to For Sub Sonic Flow (M between0.1 and 0.9) Iteration No. Xi f(xi) f'(xi) Error 1 0.10000 0.31481 -6.79176 - 2 0.14635 0.00132 -6.73563 31.67% 3 0.14655 0.00000 -6.73539 0.13% Solution : m=0014 655 For supersonic flow, solution will converge e to to m= only. 0014655 For Super Sonic Flow (M between 1.2 and 3.2) Iteration No. Xi f(xi) f'(xi) Error 1 1.20000 -6.15768 -5.05728 - 2 -0.01759 1.12175 -6.93311 6923.0879% 3 0.14421 0.01576 -6.73823 112.1958% 4 0.14655 0.00001 -6.73539 1.5959% 5 0.14655 0.00000 -6.73539 0.0005%

(6) Bi-section method For Subsonic flow I guess is ol to 0.9 b=0.9 a=0.) - Initial guess JC;=a=oo) C=atb 0,170g - = 0.5 f(a) = 0+3)48) f(b) =-4.65182 +6)=-2,29838 Now, - if fla) f(c) >0 then a=c else a=a is f(b) f(c)20 then b=c else b=b here f(a) f(o zo so, a=0) f(b) f(c) zo Sos b= 0.5

So in nex iteration a=ool, b=0.5, c= atb_o.3 following table is obtained For Sub Sonic Flow (M between0.1 and 0.9) Iteration No. a b c=(a+b)/2 f(a) f(b) f(c) Error 1 0.10000 0.90000 0.50000 0.31481 -4.65182 -2.29838 - 2 0.10000 0.50000 0.30000 0.31481 -2.29838 -1.01862 66.67% 3 0.10000 0.30000 0.20000 0.31481 -1.01862 -0.35821 50.00% 4 0.10000 0.20000 0.15000 0.31481 -0.35821 -0.02324 33.33% 5 0.10000 0.15000 0.12500 0.31481 -0.02324 0.14540 20.00% 6 0.12500 0.15000 0.13750 0.14540 -0.02324 0.06099 9.09% 7 0.13750 0.15000 0.14375 0.06099 -0.02324 0.01885 4.35% 8 0.14375 0.15000 0.14688 0.01885 -0.02324 -0.00220 2.13% 9 0.14375 0.14688 0.14531 0.01885 -0.00220 0.00832 1.08% 10 0.14531 0.14688 0.14609 0.00832 -0.00220 0.00306 0.53% For subsonic flow For Super Sonic Flow (M between 1.2 and 3.2) Iteration No. a b c=(a+b)/2 f(a) f(b) f(c) Error 1 1.20000 3.20000 2.20000 -6.15768 6.19845 -7.58429 - 2 2.20000 3.20000 2.70000 -7.58429 6.19845 -3.81174 18.52% 3 2.70000 3.20000 2.95000 -3.81174 6.19845 0.19169 8.47% 4 2.70000 2.95000 2.82500-3.81174 0.19169 -2.02887 4.42% 5 2.82500 2.95000 2.88750 -2.02887 0.19169 -0.97699 2.16% 6 2.88750 2.95000 2.91875 -0.97699 0.19169 -0.40773 1.07% 7 2.91875 2.95000 2.93438 -0.40773 0.19169 -0.11185 0.53%

(d) False position It is same as bisection method but eqreation of 'c is changed. c- Ofcb-bfca) f(b)-fca) Table obtained For Sub Sonic Flow (M between0.1 and 0.9), Iteration No. a b c=(a*f(b)-b*f(a)/(f(b)-f(a) f(a) f(b) f(c) Error 1 0.10000 0.90000 0.15071 0.31481 -4.65182 -0.02801 - 20.10000 0.15071 0.14657 0.31481 -0.02801 -0.00012 2.83% 3 0.10000 0.14657 0.14655 0.31481 -0.00012 0.00000 0.01% For Super Sonic Flow (M between 1.2 and 3.2) Iteration No. a b c=(a*f(b)-b*f(a)/(f(b)-f(a)) f(a) f(b) f(c) Error 1 1.20000 3.20000 2.19670 -6.15768_6.19845 -7.59514 - 2 2.19670 3.20000 2.74915 -7.59514 6.19845 -3.15933 20.10% 3 2.74915 3.20000 2.90136 -3.15933 6.19845 -0.72815 5.25% 4 2.90136 3.20000 2.93276 -0.72815 6.19845 -0.14288 1.07% 5 2.93276 3.20000 2.93878 -0.14288 6.19845 -0.02713 0.20%

(c) SImple Method:

For Sub Sonic Flow (M between0.1 and 0.9) Xi 0.10000 0.11000 0.12000 0.13000 0.14000 0.15000 0.16000 f(xi) 0.31481 0.24696 0.17922 0.11161 0.04413 -0.02324 -0.09048 Ans= 0.1470.15 = olub

For Super Sonic Flow (M between 1.2 and 3.2) Xi f(xi) 1.20000 -6.15768 1.30000 -6.59025 1.40000 -6.97957 1.50000 -7.31938 1.60000 -7.60255 1.70000 -7.82105 1.80000 -7.96579 1.90000 -8.02658 2.00000 -7.99200 2.10000 -7.84930 2.20000 -7.58429 2.30000 -7.18122 2.40000 -6.62266 2.50000 -5.88938 2.60000 -4.96016 2.70000 -3.81174 2.80000 -2.41861 2.90000 -0.75284 3.00000 1.21600 3.10000 3.52108 3.20000 6.19845 3.30000 9.28720 3.40000 12.82967 Ans=2-9+3= 2.95 2

Comparison of Answers:

Method Name	Subsonic Flow	Super Sonic Flow
Bisection	0.14609	2.93438
False-Position	0.14655	2.93878
Newton Raphson	0.14655	-
Simple Method	0.14500	2.95000