Applied Numerical Methods with MATLAB for Engineers and Scientists 4th Edition

-40 -20 0 20 40 60 0 4 8 12 $1,200 $1,300 $1,400 $1,500 $1,600 $1,700 M M J A S Bi-monthly Monthly 0.0% 1.0% 2.0% 0 0.5 1 1.5 2 2.5 relative error 0 20 40 60 0 4 8 12 -40 -20 0 20 40 60 0 5 10 15 4.4 4.5 4.6 0 0.2 0.4 0.6 0.8 1 -0.5 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10 -0.5 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 V y : 0 400 800 1200 1600 020000 40000 60000 80000 100000 v-analytical v-numerical

0 20 40 60 80 0 2 4 6 8 10 1.6 2 2.4 V r 1.2 0.20.20.7 0.30.5 1.2 0.5 0.3 0.1 1.2 0.20.20.7 0.30.5 1.2 0.5 0.3 0.1 50 60 70 80 0 5 10 15 20 24 28 32 36 40 0:00 1:00 2:00 3:00 4:00 5:00 Constant Ta Cooling Ta 0 20 40 60 0 2 4 6 8 10 0 100 200 300 v x 0.0 0.2 0.4 0.6 0.8 0 0.0625 0.125 0.1875 0.25 v -40 0 40 80 120 0 5 10 15 v x -100 0 100 200 300 400 0 5 10 15 x 0 0.001 0.002 0.003 0.004 0.005 0.006 0 1 2 3 4 y-Euler y-analytical -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -40 -30 -20 -10 0 10 20 00.02 0.04 0.06 0.08 0.1 i q 0 200 400 600 800 1000 0 50 100 150 200 0

4 1 2 3 F1 F5 F3 F2F4 V2V3 H2

-2 -1 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 Reactor Gas Liquid 0 20 40 60 80 100 120 0 1 2 3 4 5 6 7 Gas Liquid

-2 -1 0 1 2 -1 -0.5 0 0.5 1 1.5 2 0 1 2 3 4 5 6 0 2 4 6 -4 0 4 8 12 0 1 2 3 4 5 1st eq 2nd eq

0 4 0 4 0 4 The solution along with its second derivative can be substituted into the simultaneous ODEs. After simplification, the result is where LC 2. MATLAB can be employed to determine values for the eigenvalues and eigenvectors : : :

For four interior points (h 3/5), the resulting system of equations is Setting the determinant equal to zero and expanding it gives 4 )2 + 4 )2 + 4 )2 + which can be solved for the first four eigenvalues

0 1 2 3 4 5 6 0.8 1.2 1.6 2.0 2.4 Frequency 0 1 2 3 4 5 6 0.8 1.2 1.6 2.0 2.4 Frequency 3131.431.832.232.63333.433.8 3131.431.832.232.63333.433.8 31 31.4 31.8 32.2 32.6 33 33.4 33.8 0 1 2 3 4 5 6 7 8 28.2 28.6 29 29.4 29.8 30.2 30.6 0 5 10 15 20 0 2 4 6 8 10 12 14 0 4 8 12 0 5 10 15 20 y y versus x x versus y y x 0 4 8 12 0 5 10 15 20 y y versus x x versus y y x y = 1.9842x - 1.2941 R2 = 0.9481 2 3 4 5 6 7 8 2 2.5 3 3.5 4 4.5 y = 29.688x R2 = 0.9997 0 5000 10000 15000 0100 200 300 400

0100 200 300 400 0 100 200 300 x y Fit (untransformed) 10-1 100101102103 10-5 100 105 log(x) log(y) Fit (transformed) 0.0001 0.001 0.01 0.1 1 10 10 100 1000 y = 38148x-3.0134 R2 = 0.9757 0 0.5 1 1.5 2 2.5 050 100 150 200 250 300 350 y = 1.05897x + 0.81793 R2 = 0.90583 0 20 40 60 80 020 40 60 80 y = 1.07514x R2 = 0.90556 0 20 40 60 80 020 40 60 80 1.8 22.2 2.4 2.6 2.8 33.2 3.4 3.6 x 10-3 0 1 2 3 4 5 -3 -2 -1 0 1 2 0 1 2 3 4

0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0 100 200 (a) Distribution of drag cd (kg/m) 60 65 70 75 0 100 200 (b) Distribution of mass m (kg) 32 32.5 33 33.5 34 0 100 200 (c) Distribution of velocity v (m/s) 1.5 1.55 1.6 1.65 0 200 400 600 800 1000 1200 Distribution of Flow Q (m3/s) 1500 1600 1700 1800 1900 2000 2100 2200 2300 0.5 1 1.5 2 2.5 3 3.5 4 rhos (kg/m3) terminal velocity (10 -3 m/s) Plot of terminal velocity

-500 0 500 1000 1500 2000 020 40 60 80 100 0 500 1000 1500 2000 2500 020 40 60 80 100 -2 0 2 4 6 8 0 5 10 15 0 4 8 12 16 0 5 10 15 20 25 30 4 8 12 16 4 8 12 16 6 8 10 12 14 16 5 10 15 0 2 4 6 8 10 1 2 3 4 5 6 0100 200 300 400 500 600 700 0 50 100 150 200 250

10-2 10-1 100101102 10-12 10-10 10-8 10-6 10-4 y = 0.4945x + 20.6 R2 = 0.8385 0 10 20 30 40 50 010 20 30 40 50 60 y = 9.9529x0.3851 R2 = 0.9553 0 10 20 30 40 50 010 20 30 40 50 60 0 10 20 30 40 50 010 20 30 40 50 60 y = 50.092 x x + 9.891369 R2= 0.98919 y = -0.0161x2 + 1.3779x + 11.767 R2 = 0.98 0 10 20 30 40 50 010 20 30 40 50 60 0.0E+00 5.0E-04 1.0E-03 1.5E-03 2.0E-03 010 20

050 100 150 200 250 300 350 400 2 4 6 8 10 12 14 16 18 20 22 Plot of temperature (C) versus time (days) t (days) T (C) 6 7 8 9 10 0 6 12 18 24 0 5 10 15 20 25 6.5 7 7.5 8 8.5 9 9.5 Time vs. pH Time (hours) pH 0 100 200 300 400 060 120 180 240 300 360 0100 200 300 400 100 150 200 250 300 350 400 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 24 -0.7 0 0.7 2 4 00.1 0.2 0.3

y = -2.015000E-06x + 9.809420E+00 R2 = 9.999671E-01 9.5 9.6 9.7 9.8 9.9 020000 40000 60000 80000 100000 120000

0 2 4 6 8 10 0 2 4 6 0 1 2 3 050 100 0 0.2 0.4 0.6 0.8 1 1.2 0500 1000 y = 0.0182x4 - 0.1365x3 + 0.1818x2 + 0.263x + 0.1237 R2 = 1 0.4 0.45 0.5 0.55 0.6 1.5 1.7 1.9 2.1 2.3 2.5 2.7 -1 -0.5 0 0.5 1 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 0 0.5 1not-a-knot 0 2 4 6 8 0 0.5 1clamped 0 2 4 6 8 0 0.5 1pchip 0 2 4

0 2 4 6 8 10 4.5 5 5.5 6 6.5 7 7.5 8 8.5

020 40 60 80 100 120 140 160 180 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 01000 2000 3000 4000 5000 6000 7000 0 5 10 15 0 1 2 3 4 5 6 7 x 106 0 2 4 6 8 x 1024

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -20 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -10 -8 -6 -4 -2 0 2 4 6 8 10 -5 -4 -3 -2 -1 0 1 2 3 4 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 z frequency 0 5 10 15 20 25 30 35 0.994 0.995 0.996 0.997 0.998 0.999 1 1.001 010 20 30 40 50 60 70 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 60 65 70 75 80 85 90

050 100 150 200 250 300 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 4 Heat required versus change in temperature Change in temperature (C) Heat required (cal) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.5 1 1.5 2 2.5 x 104 0 5 10 15 20 25 0 1 2 3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -20 0 20 40 60 80 100 120 140

01000 2000 3000 4000 5000 6000 7000 3 4 5 6 7 8 9 10 11 12 13

0 0.4 0.8 1.2 1.6 010 20 30 40 50 60 70 80 0 0.01 0.02 0.03 0.04 0.05 VDerivative

y = -0.1618x R2 = 0.9758 -10 -8 -6 -4 -2 0 010 20 30 40 50 60 0 1 2 3 0 0.2 0.4 0.6 0.8 t(d) SOD (g/m2/d) 0 1 2 3 0 0.2 0.4 0.6 0.8 t(d) SOD (g/m2/d) 0 1 2 3 0 2 4 6 8 10 SOD (g/m2/d) o(mg/L) 0 1 2 3 0 2 4 6 8 10 SOD (g/m2/d) o(mg/L) -1.E-03 -8.E-04 -6.E-04 -4.E-04 -2.E-04 0.E+00 0 1 2 3 -150000 -100000 -50000 0 50000 100000 0 1 2 3 -300000 -200000 -100000 0 100000 0 1 2 3 V=dM/dx V=EI*d2Q/dx2 Vtrue -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0 1 2 3 y, m dy/dx -2500000 -2000000 -1500000 -1000000 -500000 0 500000 0 1 2 3 M (Eq. 1) M (Eq. 2) 0 400,000 800,000 1,200,000 1,600,000 0 1 2

0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 3 0 0.5 1 1.5 2 Euler (h=0.5) Euler (h=0.25) Midpoint Analytical RK4 0 2 4 6 0 0.2 0.4 0.6 0.8 1 yanal yEuler yHeun yRalston yRK4 0 1 2 3 4 5 0 1 2 3 Heun Ralston Midpoint Heun (iter) Exact y = 0.01777x - 26.80753 R = 0.99763 7.8 8 8.2 8.4 8.6 8.8 1940 1960 1980 2000 0 2000 4000 6000 8000 10000 1950 1970 1990 2010 2030 2050 pdata panal p-rk4 0 1 2 3 4 00.5 y z -4 -3 -2 -1 0 1 2 3 4 0 2 4

0 400 800 1200 1960 1990 2020 moose wolves Wolves 0 100 200 300 0 500 1000 1500 Moose 0 400 800 1200 1960 1990 2020 Wolves Moose 0 100 200 300 0500 1000 1500 0 400 800 1200 1960 1990 2020 Wolves Moose 0 100 200 300 0 500 1000 1500 0 5 10 15 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 t (s) x (m) Displacements for mass-spring system underdamped critically damped overdamped 0 1 2 3 02000 4000 6000 8000 10000 0 0.5 1 1.5 2 2.5 3 3.5 4 20 25 30 35 40 t (hr) T (C) 0 2 4 6 8 10 0 2 4 6 8 10 12 time (min) Concentration Reactor concentrations versus time

-1 -0.5 0 0.5 1 1.5 0 1 2 3 4 -2 0 2 4 0 0.1 0.2 x1 x2 -10 -5 0 5 10 15 0 1 2 3 4 -30 -20 -10 0 10 20 30 0 1 2 3 4 5 RK4 Analytical ODE45 ODE23s ODE23tb 0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 3 4 5 Implicit numerical Analytical Implicit numerical Analytical 0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 3 4 5

h= 0.1 h= 0.2 analytical Explicit Euler -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 h= 0.1 h= 0.2 analytical Implicit Euler 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 35 -5 0 5 10 time (s) displacement (m) (a) Displacement of first mass 0 2 4 6 8 10 0 1 2 (b) Power spectrum Frequency (Hz) 020 40 60 80 100 120 140 -2 0 2 time (s) displacement (m) (a) Displacement of first

0 500 1000 1500 2000 0200 400 600 800

0 80 160 240 0 2 4 6 8 10 0 80 160 240 0 2 4 6 8 10 0 80 160 240 0 2 4 6 8 10 0 80 160 240 0 2 4 6 8 10 0 4 8 12 0 5 10 15 20 0 4 8 12 0 5 10 15 20 0 100 200 300 400 0 2 4 6 8 10 0 100 200 300 400 500 0 2 4 6 8 10 0 100 200 300 0 2 4 6 8 10 0 0.05 0.1 0 1 2 3 4

0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 cb ca cc sum 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 cb ca cc sum 0 40 80 120 0 0.004 0.008 0.012 0 0.5 1 1.5 2 2.5 3 -3 -2.5 -2 -1.5 -1 -0.5 0 x 10-4 xi xi1xi+1 xi1/2 xi+1/2

0 400 800 1200 0 5 10 15 20 T0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 5 6 7 Ta 5 6 7 Ta T T13 x/2 T14 0 100 200 300 400 500 0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 1 2 3 4 5 6 7 8 9 10 -600 -500 -400 -300 -200 -100 0 0 1 2 3 4 5 6 0 50 100 150 Volume versus depth for horizontal cylindrical tank -10 0 10 20 30 40 50 0 500 1000 1500 2000 2500 0 2 4 6 8 10 12 14 16 18 -8 -6 -4 -2 0 2 4 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6

-20 -15 -10 -5 05 -30 -20 -10 0 10 -20 -10 0 10 0 2 4 6 -5 0 5 10 15 -5 0 5 10 15 -1 -0.8 -0.6 -0.4 -0.2 0 -1 -0.5 0 0 0.5 1 010 20 30 40 50 60 0 5 10 Velocity vs. time time (s) velocity (m/s) 010 20 30 40 50 60 0 200 400 600 Elevation vs. time time (s) elevation (m) -100 0 100 0 10 20 30 40 50 60 70 80 90 (a) Temperature(C) altitude (km) 0 1 2 x 105 0 10 20 30 40 50 60 70 80 90 (b) Pressure(atm) 0 1 2 0 10 20 30 40 50 60 70 80 90 (c) Density(kg/m3) 050 100 150 200 250 300 350 -100 -50 0 50 t (days) T (C) South Pole Death

0 1 2 3 4 -1 0 1 2 3 f(x)=x-1-0.5*sin(x) 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1st Order Taylor Series Error 0 1 2 3 4 0 0.05 0.1 0.15 0.2 2nd/3rd Order Taylor Series Error 0 1 2 3 4 0 0.005 0.01 0.015 4th Order Taylor Series Error First Derivative Approximations Compared to Theoretical -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

0 0.5 1 1.5 2 2.5 3 3.5 4 -4 -2 0 2 4x 10-3 x(m) dy/dx Slope versus distance 0 0.5 1 1.5 2 2.5 3 3.5 4 -4 -3 -2 -1 0x 10-3 x(m) y(m) Deflection versus distance 0 0.5 1 1.5 2 2.5 3 3.5 4 -5 -4 -3 -2 -1 0 1 2 3 4 H (m) f(H) Plot of f(H) versus H -12 -10 -8 -6 -4 -2 0 2 -3 -2 -1 0 1 2 3 -500 0 500 1000 0 4 8 12 0<x<3 3<x<6 6<x<10 10<x<12

-40 -30 -20 -10 0 10 0 0.5 1 1.5 2 2.5 0 5 10 15 20 25 30 35 40 45 50 0 1 2 3 4 5 6 7 8 9 -0.2 0 0.2 0.4 0.6 0.8 0 5 10 15 20 -10000 -5000 0 5000 0 2 4 6 8 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -3500 -3000 -2500 -2000 -1500 -1000 -500 0 500 1000 1500 010 20 30 40 50 60 70 80 90 100 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x 105 -4.E+06 -2.E+06 0.E+00 2.E+06 4.E+06 6.E+06 8.E+06 0.0E+00 1.0E+10 2.0E+10 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 0200

| -60 -40 -20 20 40 60 -2 0 2 4 6 8 Newton Raphson Modified secant -12 -8 -4 0 4 0 2 4 6 -1 -0.5 0 0.5 1 2.6 2.8 3 3.2 3.4 -1 -0.5 0 0.5 1 2.6 2.8 3 3.2 3.4 -10 -5 0 5 10 -5 5 10 15 20 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

010 20 30 40 50 60 70 -150 -100 -50 0 50 020 40 60 80 100 0 5 10 15 20 25 30 angle=37.959 angle=51.5317 -3 -2 -1 0 1 2 3 4 8 12 16 -3 -2 -1 0 1 2 3 4 8 12 16 iteration

0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5

-160 -120 -80 -40 0 40 -2 -1 0 1 2 -400 -200 0 200 400 -2 -1 012 -6 0 6 12 18 24 -2 -1.5 -1 -0.5 0 0.5 1 2 4 6 8 10 12 14 16 18 20 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 1.21598 5.3623 0 20 20 40 40 60 60 80 80 100 100 120 140 x1 x2 (a) Contour plot -2 0 2 4 6 -2 -1 0 1 2 3 4 5 6 -5 0 5 10 -2 0 2 4 6 -50 0 50 100 150 200 x1 (b) Mesh plot x2 f(x1,x2) 0.2 0.4 0.6 0.8 1 1.2 1.4 x1 x2 (a) Contour plot -4 -3 -2 -1 0 1 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 -4 -2 0 2 -2 0 2 4 0 0.5 1 1.5 x1 (b) Mesh plot x2 f(x1,x2) (a) The impact of different angles on the deflections

-0.6 -0.4 -0.2 0 0100 200 300 400 500 600 -120 -80 -40 0 40 0 2 4 6 8 10 0 10 20 30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 0 500 1000 1500 2000 2500 3000 3500 4000 Velocity Drag d A s 1 b1 b2 d A s 1 b1 b1 b2 b2 h d x y 0 0.5 1 1.5 2 2.5 3 3.5 4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0x 10-3 x (m) y (m) Deflection versus distance 1e+03 2e+03 3e+03 4e+03 5e+03 6e+03 7e+03 8e+03 9e+03 -2000 -1500 -1000 -500 0 500 1000 1500 2000 0 500 1000 1500 2000 2500 3000 3500 4000 4500

-2 -1 01234 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 2D rotation through angle of 45.00 degrees

0 20 40 60 80 100 120 140 0200 400 600 800 1000 10 12 14 16 18 20 22 510 15 20 14.3 14.35 14.4 14.45 14.5 14.55 14.6 14.65 14.7 9.75 10 10.25 0 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 90

0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 x c 0 0.5 1 1.5 2 2.5 3 -3 -2 -1 0x 10-4 x y 0 2 4 6 8 10 12 280 300 320 340 360 380 400 420 440 460 x T