Solution of a Concentration Problem Using Numerical Methods

American University of Sharjah

Faculty of Engineering

Department of Chemical Engineering

Advance Numerical Methods (NGN-509)

Solution of a concentration problem using

Numerical Methods

Submitted by:

Salma Elgaili Abdel-Karim Ahmed

@g00050051

January - 2013

Solution of a concentration problem using Numerical Methods NGN 509

2

Abstract

In this project the dimensionless concentration profile of benzene in the reaction of benzene

dehydrogenation into cyclohexane in absence of inter- and intra-phase gradients is calculated

numerically using the first order explicit Euler method, the second order modified midpoint and

modified Euler methods, the Fourth-order Runge-Kutta method and the fourth-order Adams-

Bashforth-Moulton method, both by hand and using MATLAB.

Finally all numerical results were compared to analytical results graphically and by hand

calculations and errors were found.

The stability criterion for each method was discussed in the conclusion.


3

Table of contents Abstract……………………………………………...………………………………..............…..2

Table of contents…………..……………………………………………………………...……....3

List of tables……………………………………...…………………………………………….....4

Section1 General Background and Problem Modeling

Introduction…………………………………………………………………………………….....6

Problem Modeling ……………………………………………………………………………..…7

Section2 Numerical Solution

Solving using Euler method…………………………………………………………………......12

Solving using the modified midpoint method……………………………………………….......15

Solving using the modified Euler method………………………………………………….........22

Solving using the fourth-order Runge-Kutta method method……………………………….......28

Solving using the fourth-order Adams-Bashforth-Moulton method……..………………….......35

Section3 MATLAB Results

Solving using MATLAB………………………………………………………………….…......44

Section4 Discussion………………………………………………………………….…......48

Section5 Conclusion………………………………………………………………….…......49

References…………….………………………………………………………………….…......51

Appendix A…………….………………………………………………………………….…......52


4

List of Tables

Table 1.1: Analytical results of the problem using explicit Euler method ….……………….......8

Table 2.1: Numerical results of the problem using explicit Euler method ….……………….....12

Table 2.2: Numerical results of the problem using the modified midpoint method ….……......16

Table 2.3: Numerical results of the problem using the modified Euler method ….……...........22

Table 2.4: Numerical results of the problem using the fourth-order Runge-Kutta method........29

Table 2.5: Numerical results of the problem using the fourth-order Adams-Bashforth-Moulton

method…………………………………………………………..…………………….36


5


6

1.1. Introduction

The Hydrogenation process of benzene to give cyclohexane is an important case in industry and

considered as a large-scale process that represents the first stage of a caprolactam1 production

process according to the polish technology named CYCLOPOL.

The reaction is governed by equation

C6H6 + 3H2 → C6H6; ΔH298 = -206.2 [kJ/mol] (1.1)

And the complete process happens as shown in figure (1.1)

The Hydrogenation reaction is a well-known reaction that has been appreciably examined in the

laboratory as a model reaction for catalysis studies. The catalyst itself is based on most Group

VIII metals, mainly nickel, palladium or platinum.

Extensive care has been given to the kinetics of the benzene hydrogenation on nickel catalysts,

given that, in industry, standard nickel catalysts are used for the gas-phase hydrogenation and

Raney-nickel for the liquid phase hydrogenation.

One of the most important efforts was by Kehoe and Butt [1] who had studied the reaction on a

supported Ni/kieselguhr catalyst.

1 a nylon precursor

Figure 1.1 Hydrogenation of benzene


7

They have found that in the presence of a large excess of Hydrogen, the reaction is pseudo-first-

order at temperatures below 200°C with the rate given by:

[

]

Where

Rg= gas constant, 1.987cal/(mole0K)

-Q – Ea= 2700cal/mole

PH2 = hydrogen partial pressure(torr)

k0= 4.22 mole/(gcat·s·torr)

K0 = 2.63 x10-6

cm3/(mole

0K)

T = absolute temperature (K)

CB = concentration of benzene (mole/cm3)

Price and Butt [2] studied this reaction in an isothermal tubular, plug flow reactor where a typical

run, in which (PH2 = 685 torr, ρH = density of the reactor bed = 1,2 gcat/ cm3, ϴ = constant time =

0.266 s, T = 1500C),

is used.

1.2. Problem modeling

Let

CB = feed concentration of benzene (mole/cm3)

z = axial reactor coordinate (cm)

L = reactor length

y = dimensionless concentration of benzene (CB/ CB0)

x = dimensionless axial coordinate (z/L).

The one-dimensional steady-state material balance for the reactor that expresses the fact that the

change in the axial convection of benzene is equal to the amount converted by reaction is

(

)

With CB = CB0 at x = 0

Since ϴ is constant,

[

]

mole/(g of catalyst.s) (1.2)

(1.3)


8

Gathering the constants and defining as

[

]

Using the data provided, it’s found = 21.6. Substituting back in (1.4)

With (y = 1 at x = 0), and the analytical solution

y = exp (-21.6 x) (1.6)

The analytical results were as follows

Table0.1 Analytical results of the problem

x Analytical results

(yn) fn

0.00 1.00000E+00 -2.16000E+01

0.01 8.05735E-01 -1.74039E+01

0.02 6.49209E-01 -1.40229E+01

0.03 5.23091E-01 -1.12988E+01

0.04 4.21473E-01 -9.10381E+00

0.05 3.39596E-01 -7.33526E+00

0.06 2.73624E-01 -5.91028E+00

0.07 2.20469E-01 -4.76212E+00

0.08 1.77639E-01 -3.83701E+00

0.09 1.43130E-01 -3.09161E+00

0.10 1.15325E-01 -2.49102E+00

0.11 9.29215E-02 -2.00710E+00

0.12 7.48701E-02 -1.61720E+00

0.13 6.03255E-02 -1.30303E+00

0.14 4.86064E-02 -1.04990E+00

0.15 3.91639E-02 -8.45940E-01

0.16 3.15557E-02 -6.81604E-01

0.17 2.54256E-02 -5.49192E-01

0.18 2.04863E-02 -4.42504E-01

0.19 1.65065E-02 -3.56541E-01

0.20 1.32999E-02 -2.87277E-01

0.21 1.07162E-02 -2.31470E-01

0.22 8.63441E-03 -1.86503E-01

0.23 6.95705E-03 -1.50272E-01

0.24 5.60554E-03 -1.21080E-01

0.25 4.51658E-03 -9.75581E-02

0.26 3.63917E-03 -7.86060E-02

(1.5)


9


(yn) fn

0.27 2.93221E-03 -6.33357E-02

0.28 2.36258E-03 -5.10318E-02

0.29 1.90362E-03 -4.11181E-02

0.30 1.53381E-03 -3.31303E-02

0.31 1.23585E-03 -2.66943E-02

0.32 9.95764E-04 -2.15085E-02

0.33 8.02322E-04 -1.73302E-02

0.34 6.46460E-04 -1.39635E-02

0.35 5.20875E-04 -1.12509E-02

0.36 4.19688E-04 -9.06525E-03

0.37 3.38157E-04 -7.30419E-03

0.38 2.72465E-04 -5.88525E-03

0.39 2.19535E-04 -4.74195E-03

0.40 1.76887E-04 -3.82076E-03

0.41 1.42524E-04 -3.07852E-03

0.42 1.14837E-04 -2.48047E-03

0.43 9.25279E-05 -1.99860E-03

0.44 7.45530E-05 -1.61035E-03

0.45 6.00700E-05 -1.29751E-03

0.46 4.84005E-05 -1.04545E-03

0.47 3.89980E-05 -8.42357E-04

0.48 3.14221E-05 -6.78717E-04

0.49 2.53179E-05 -5.46866E-04

0.50 2.03995E-05 -4.40629E-04

0.51 1.64366E-05 -3.55031E-04

0.52 1.32435E-05 -2.86061E-04

0.53 1.06708E-05 -2.30489E-04

0.54 8.59784E-06 -1.85713E-04

0.55 6.92758E-06 -1.49636E-04

0.56 5.58180E-06 -1.20567E-04

0.57 4.49745E-06 -9.71449E-05

0.58 3.62375E-06 -7.82731E-05

0.59 2.91979E-06 -6.30674E-05

0.60 2.35258E-06 -5.08156E-05

0.61 1.89555E-06 -4.09439E-05

0.62 1.52731E-06 -3.29900E-05

0.63 1.23061E-06 -2.65812E-05

0.64 9.91546E-07 -2.14174E-05

0.65 7.98924E-07 -1.72568E-05

0.66 6.43721E-07 -1.39044E-05

0.67 5.18669E-07 -1.12032E-05


10


(yn) fn

0.68 4.17910E-07 -9.02685E-06

0.69 3.36725E-07 -7.27325E-06

0.70 2.71311E-07 -5.86032E-06

0.71 2.18605E-07 -4.72187E-06

0.72 1.76138E-07 -3.80457E-06

0.73 1.41920E-07 -3.06548E-06

0.74 1.14350E-07 -2.46996E-06

0.75 9.21360E-08 -1.99014E-06

0.76 7.42372E-08 -1.60352E-06

0.77 5.98156E-08 -1.29202E-06

0.78 4.81955E-08 -1.04102E-06

0.79 3.88328E-08 -8.38789E-07

0.80 3.12890E-08 -6.75842E-07

0.81 2.52106E-08 -5.44550E-07

0.82 2.03131E-08 -4.38763E-07

0.83 1.63670E-08 -3.53527E-07

0.84 1.31875E-08 -2.84849E-07

0.85 1.06256E-08 -2.29513E-07

0.86 8.56142E-09 -1.84927E-07

0.87 6.89824E-09 -1.49002E-07

0.88 5.55815E-09 -1.20056E-07

0.89 4.47840E-09 -9.67334E-08

0.90 3.60840E-09 -7.79415E-08

0.91 2.90742E-09 -6.28003E-08

0.92 2.34261E-09 -5.06004E-08

0.93 1.88752E-09 -4.07705E-08

0.94 1.52084E-09 -3.28502E-08

0.95 1.22540E-09 -2.64686E-08

0.96 9.87347E-10 -2.13267E-08

0.97 7.95540E-10 -1.71837E-08

0.98 6.40995E-10 -1.38455E-08

0.99 5.16472E-10 -1.11558E-08

1.00 4.16140E-10 -8.98862E-09


11


12

2.1. Solving using Euler method

The explicit Euler finite difference equation (FDE) is given by

yi+1 = yi + h fn, i= 0, 1, 2, N-1 (2.1)

where (h) is the step used, (fn) is the derivative function and (N= 1/h).

Thus, and from (1.5), the desired equation forming the solution is

yi+1 = yi – 21.6 h yi (2.2)

Let h = 0.01 (N=100), for the first step,

y1 = 1 - 21.6 * 0.01 * 1 = 0.784

These results and the results of subsequent steps are summarized in Table 2.1.

Table2.1 Numerical results of the problem using explicit Euler method

x Numerical results

(yn) Error

0 7.84000E-01 -2.16000E-01

0.01 6.14656E-01 -1.91079E-01

0.02 4.81890E-01 -1.67319E-01

0.03 3.77802E-01 -1.45289E-01

0.04 2.96197E-01 -1.25276E-01

0.05 2.32218E-01 -1.07377E-01

0.06 1.82059E-01 -9.15650E-02

0.07 1.42734E-01 -7.77342E-02

0.08 1.11904E-01 -6.57356E-02

0.09 8.77325E-02 -5.53978E-02

0.1 6.87823E-02 -4.65428E-02

0.11 5.39253E-02 -3.89962E-02

0.12 4.22775E-02 -3.25927E-02

0.13 3.31455E-02 -2.71800E-02

0.14 2.59861E-02 -2.26203E-02

0.15 2.03731E-02 -1.87908E-02

0.16 1.59725E-02 -1.55832E-02

0.17 1.25224E-02 -1.29031E-02

0.18 9.81760E-03 -1.06687E-02

0.19 7.69700E-03 -8.80952E-03

0.2 6.03444E-03 -7.26544E-03

0.21 4.73100E-03 -5.98518E-03


13

x Numerical results

(yn) Error

0.22 3.70911E-03 -4.92530E-03

0.23 2.90794E-03 -4.04911E-03

0.24 2.27983E-03 -3.32571E-03

0.25 1.78738E-03 -2.72920E-03

0.26 1.40131E-03 -2.23786E-03

0.27 1.09863E-03 -1.83358E-03

0.28 8.61323E-04 -1.50126E-03

0.29 6.75277E-04 -1.22834E-03

0.3 5.29417E-04 -1.00439E-03

0.31 4.15063E-04 -8.20782E-04

0.32 3.25409E-04 -6.70355E-04

0.33 2.55121E-04 -5.47201E-04

0.34 2.00015E-04 -4.46445E-04

0.35 1.56812E-04 -3.64064E-04

0.36 1.22940E-04 -2.96747E-04

0.37 9.63852E-05 -2.41772E-04

0.38 7.55660E-05 -1.96899E-04

0.39 5.92437E-05 -1.60291E-04

0.4 4.64471E-05 -1.30440E-04

0.41 3.64145E-05 -1.06109E-04

0.42 2.85490E-05 -8.62876E-05

0.43 2.23824E-05 -7.01455E-05

0.44 1.75478E-05 -5.70052E-05

0.45 1.37575E-05 -4.63125E-05

0.46 1.07859E-05 -3.76147E-05

0.47 8.45612E-06 -3.05419E-05

0.48 6.62960E-06 -2.47925E-05

0.49 5.19760E-06 -2.01203E-05

0.5 4.07492E-06 -1.63246E-05

0.51 3.19474E-06 -1.32419E-05

0.52 2.50467E-06 -1.07389E-05

0.53 1.96366E-06 -8.70713E-06

0.54 1.53951E-06 -7.05832E-06

0.55 1.20698E-06 -5.72060E-06

0.56 9.46271E-07 -4.63552E-06


14

x Numerical results

(yn) Error

0.57 7.41877E-07 -3.75557E-06

0.58 5.81631E-07 -3.04212E-06

0.59 4.55999E-07 -2.46379E-06

0.6 3.57503E-07 -1.99507E-06

0.61 2.80282E-07 -1.61527E-06

0.62 2.19741E-07 -1.30757E-06

0.63 1.72277E-07 -1.05833E-06

0.64 1.35065E-07 -8.56481E-07

0.65 1.05891E-07 -6.93033E-07

0.66 8.30188E-08 -5.60703E-07

0.67 6.50867E-08 -4.53582E-07

0.68 5.10280E-08 -3.66882E-07

0.69 4.00059E-08 -2.96719E-07

0.7 3.13647E-08 -2.39946E-07

0.71 2.45899E-08 -1.94015E-07

0.72 1.92785E-08 -1.56859E-07

0.73 1.51143E-08 -1.26806E-07

0.74 1.18496E-08 -1.02501E-07

0.75 9.29011E-09 -8.28459E-08

0.76 7.28344E-09 -6.69538E-08

0.77 5.71022E-09 -5.41053E-08

0.78 4.47681E-09 -4.37187E-08

0.79 3.50982E-09 -3.53230E-08

0.8 2.75170E-09 -2.85373E-08

0.81 2.15733E-09 -2.30533E-08

0.82 1.69135E-09 -1.86217E-08

0.83 1.32602E-09 -1.50410E-08

0.84 1.03960E-09 -1.21479E-08

0.85 8.15045E-10 -9.81055E-09

0.86 6.38995E-10 -7.92242E-09

0.87 5.00972E-10 -6.39726E-09

0.88 3.92762E-10 -5.16539E-09

0.89 3.07925E-10 -4.17047E-09

0.9 2.41414E-10 -3.36699E-09

0.91 1.89268E-10 -2.71815E-09


15

x Numerical results

(yn) Error

0.92 1.48386E-10 -2.19422E-09

0.93 1.16335E-10 -1.77119E-09

0.94 9.12065E-11 -1.42964E-09

0.95 7.15059E-11 -1.15389E-09

0.96 5.60606E-11 -9.31286E-10

0.97 4.39515E-11 -7.51588E-10

0.98 3.44580E-11 -6.06537E-10

0.99 2.70151E-11 -4.89457E-10

1 2.11798E-11 -3.94960E-10

2.2. Solving using the modified midpoint method

The modified midpoint method is given by

⁄

⁄

where the superscript P denotes that ⁄ and ⁄

are predictor values where ⁄ is

evaluated using ⁄ , and the superscript C denotes that

is the corrected second-order

result.

Thus, the desired equations forming the solution s are obtained as

⁄

⁄ ⁄

⁄


⁄

⁄ ⁄

⁄


(2.3)


16

Table2.2 Numerical results of the problem using modified midpoint method

xn yn fn

Error yn+1/2 fn+1/2

xn+1 yn+1

0

1.00000E+00 -2.16000E+01 0.00000E+00

8.92000E-01 -1.92672E+01

0.01

8.07328E-01 -1.74383E+01 1.59270E-03

7.20137E-01 -1.55550E+01

0.02

6.51778E-01 -1.40784E+01 2.56912E-03

5.81386E-01 -1.25579E+01

0.03

5.26199E-01 -1.13659E+01 3.10812E-03

4.69370E-01 -1.01384E+01

0.04

4.24815E-01 -9.17601E+00 3.34240E-03

3.78935E-01 -8.18500E+00

0.05

3.42965E-01 -7.40805E+00 3.36969E-03

3.05925E-01 -6.60798E+00

0.06

2.76885E-01 -5.98073E+00 3.26132E-03

2.46982E-01 -5.33481E+00

0.07

2.23537E-01 -4.82841E+00 3.06875E-03

1.99395E-01 -4.30694E+00

0.08

1.80468E-01 -3.89811E+00 2.82863E-03

1.60977E-01 -3.47711E+00

0.09

1.45697E-01 -3.14705E+00 2.56656E-03

1.29962E-01 -2.80717E+00

0.1

1.17625E-01 -2.54070E+00 2.30002E-03

1.04922E-01 -2.26631E+00

0.11

9.49621E-02 -2.05118E+00 2.04055E-03

8.47062E-02 -1.82965E+00

0.12

7.66655E-02 -1.65598E+00 1.79539E-03

6.83857E-02 -1.47713E+00

0.13

6.18942E-02 -1.33692E+00 1.56871E-03

5.52097E-02 -1.19253E+00

0.14

4.99689E-02 -1.07933E+00 1.36255E-03

4.45723E-02 -9.62762E-01

0.15

4.03413E-02 -8.71373E-01 1.17744E-03

3.59845E-02 -7.77264E-01


17

xn yn fn

Error yn+1/2 fn+1/2

xn+1 yn+1

0.16

3.25687E-02 -7.03484E-01 1.01295E-03

2.90513E-02 -6.27507E-01

0.17

2.62936E-02 -5.67942E-01 8.68044E-04

2.34539E-02 -5.06604E-01

0.18

2.12276E-02 -4.58516E-01 7.41292E-04

1.89350E-02 -4.08996E-01

0.19

1.71376E-02 -3.70172E-01 6.31094E-04

1.52867E-02 -3.30194E-01

0.2

1.38357E-02 -2.98851E-01 5.35790E-04

1.23414E-02 -2.66575E-01

0.21

1.11699E-02 -2.41270E-01 4.53741E-04

9.96357E-03 -2.15213E-01

0.22

9.01779E-03 -1.94784E-01 3.83385E-04

8.04387E-03 -1.73748E-01

0.23

7.28032E-03 -1.57255E-01 3.23270E-04

6.49404E-03 -1.40271E-01

0.24

5.87760E-03 -1.26956E-01 2.72065E-04

5.24282E-03 -1.13245E-01

0.25

4.74515E-03 -1.02495E-01 2.28574E-04

4.23268E-03 -9.14258E-02

0.26

3.83090E-03 -8.27474E-02 1.91728E-04

3.41716E-03 -7.38106E-02

0.27

3.09279E-03 -6.68043E-02 1.60583E-04

2.75877E-03 -5.95894E-02

0.28

2.49690E-03 -5.39329E-02 1.34313E-04

2.22723E-03 -4.81082E-02

0.29

2.01581E-03 -4.35416E-02 1.12198E-04

1.79811E-03 -3.88391E-02

0.3

1.62742E-03 -3.51523E-02 9.36123E-05

1.45166E-03 -3.13559E-02

0.31

1.31386E-03 -2.83795E-02 7.80188E-05

1.17197E-03 -2.53145E-02

0.32

1.06072E-03 -2.29115E-02 6.49551E-05

9.46162E-04 -2.04371E-02

0.33 8.56348E-04 -1.84971E-02 5.40260E-05


18

xn yn fn

Error yn+1/2 fn+1/2

xn+1 yn+1

7.63863E-04 -1.64994E-02

0.34

6.91354E-04 -1.49332E-02 4.48945E-05

6.16688E-04 -1.33205E-02

0.35

5.58149E-04 -1.20560E-02 3.72742E-05

4.97869E-04 -1.07540E-02

0.36

4.50610E-04 -9.73317E-03 3.09221E-05

4.01944E-04 -8.68199E-03

0.37

3.63790E-04 -7.85786E-03 2.56327E-05

3.24501E-04 -7.00921E-03

0.38

2.93698E-04 -6.34387E-03 2.12326E-05

2.61978E-04 -5.65873E-03

0.39

2.37110E-04 -5.12158E-03 1.75756E-05

2.11502E-04 -4.56845E-03

0.4

1.91426E-04 -4.13480E-03 1.45390E-05

1.70752E-04 -3.68824E-03

0.41

1.54543E-04 -3.33814E-03 1.20194E-05

1.37853E-04 -2.97762E-03

0.42

1.24767E-04 -2.69497E-03 9.93062E-06

1.11292E-04 -2.40392E-03

0.43

1.00728E-04 -2.17573E-03 8.20017E-06

8.98495E-05 -1.94075E-03

0.44

8.13206E-05 -1.75653E-03 6.76760E-06

7.25380E-05 -1.56682E-03

0.45

6.56524E-05 -1.41809E-03 5.58241E-06

5.85620E-05 -1.26494E-03

0.46

5.30030E-05 -1.14487E-03 4.60251E-06

4.72787E-05 -1.02122E-03

0.47

4.27908E-05 -9.24282E-04 3.79282E-06

3.81694E-05 -8.24459E-04

0.48

3.45462E-05 -7.46199E-04 3.12416E-06

3.08152E-05 -6.65609E-04

0.49

2.78901E-05 -6.02427E-04 2.57227E-06

2.48780E-05 -5.37365E-04

0.5

2.25165E-05 -4.86356E-04 2.11699E-06

2.00847E-05 -4.33830E-04


19

xn yn fn

Error yn+1/2 fn+1/2

xn+1 yn+1

0.51

1.81782E-05 -3.92649E-04 1.74160E-06

1.62150E-05 -3.50243E-04

0.52

1.46758E-05 -3.16997E-04 1.43222E-06

1.30908E-05 -2.82761E-04

0.53

1.18482E-05 -2.55920E-04 1.17736E-06

1.05686E-05 -2.28281E-04

0.54

9.56535E-06 -2.06612E-04 9.67513E-07

8.53229E-06 -1.84297E-04

0.55

7.72237E-06 -1.66803E-04 7.94794E-07

6.88836E-06 -1.48789E-04

0.56

6.23449E-06 -1.34665E-04 6.52693E-07

5.56116E-06 -1.20121E-04

0.57

5.03328E-06 -1.08719E-04 5.35827E-07

4.48968E-06 -9.69772E-05

0.58

4.06351E-06 -8.77717E-05 4.39752E-07

3.62465E-06 -7.82924E-05

0.59

3.28058E-06 -7.08606E-05 3.60795E-07

2.92628E-06 -6.32076E-05

0.6

2.64851E-06 -5.72077E-05 2.95930E-07

2.36247E-06 -5.10293E-05

0.61

2.13821E-06 -4.61854E-05 2.42660E-07

1.90729E-06 -4.11974E-05

0.62

1.72624E-06 -3.72868E-05 1.98925E-07

1.53981E-06 -3.32598E-05

0.63

1.39364E-06 -3.01026E-05 1.63030E-07

1.24313E-06 -2.68516E-05

0.64

1.12513E-06 -2.43027E-05 1.33579E-07

1.00361E-06 -2.16780E-05

0.65

9.08345E-07 -1.96203E-05 1.09421E-07

8.10244E-07 -1.75013E-05

0.66

7.33333E-07 -1.58400E-05 8.96113E-08

6.54133E-07 -1.41293E-05

0.67

5.92040E-07 -1.27881E-05 7.33710E-08

5.28100E-07 -1.14070E-05

0.68 4.77970E-07 -1.03242E-05 6.00605E-08


20

xn yn fn

Error yn+1/2 fn+1/2

xn+1 yn+1

4.26350E-07 -9.20915E-06

0.69

3.85879E-07 -8.33498E-06 4.91542E-08

3.44204E-07 -7.43481E-06

0.7

3.11531E-07 -6.72907E-06 4.02198E-08

2.77886E-07 -6.00233E-06

0.71

2.51508E-07 -5.43256E-06 3.29027E-08

2.24345E-07 -4.84585E-06

0.72

2.03049E-07 -4.38586E-06 2.69114E-08

1.81120E-07 -3.91219E-06

0.73

1.63927E-07 -3.54083E-06 2.20069E-08

1.46223E-07 -3.15842E-06

0.74

1.32343E-07 -2.85861E-06 1.79928E-08

1.18050E-07 -2.54988E-06

0.75

1.06844E-07 -2.30784E-06 1.47082E-08

9.53051E-08 -2.05859E-06

0.76

8.62583E-08 -1.86318E-06 1.20211E-08

7.69424E-08 -1.66196E-06

0.77

6.96388E-08 -1.50420E-06 9.82322E-09

6.21178E-08 -1.34174E-06

0.78

5.62213E-08 -1.21438E-06 8.02583E-09

5.01494E-08 -1.08323E-06

0.79

4.53891E-08 -9.80404E-07 6.55624E-09

4.04870E-08 -8.74520E-07

0.8

3.66439E-08 -7.91507E-07 5.35488E-09

3.26863E-08 -7.06025E-07

0.81

2.95836E-08 -6.39006E-07 4.37298E-09

2.63886E-08 -5.69993E-07

0.82

2.38837E-08 -5.15887E-07 3.57058E-09

2.13042E-08 -4.60172E-07

0.83

1.92820E-08 -4.16490E-07 2.91498E-09

1.71995E-08 -3.71509E-07

0.84

1.55669E-08 -3.36244E-07 2.37942E-09

1.38856E-08 -2.99930E-07

0.85

1.25676E-08 -2.71459E-07 1.94197E-09

1.12103E-08 -2.42142E-07


21

xn yn fn

Error yn+1/2 fn+1/2

xn+1 yn+1

0.86

1.01462E-08 -2.19157E-07 1.58473E-09

9.05037E-09 -1.95488E-07

0.87

8.19127E-09 -1.76931E-07 1.29303E-09

7.30661E-09 -1.57823E-07

0.88

6.61304E-09 -1.42842E-07 1.05489E-09

5.89883E-09 -1.27415E-07

0.89

5.33889E-09 -1.15320E-07 8.60495E-10

4.76229E-09 -1.02866E-07

0.9

4.31024E-09 -9.31012E-08 7.01834E-10

3.84473E-09 -8.30462E-08

0.91

3.47978E-09 -7.51632E-08 5.72357E-10

3.10396E-09 -6.70456E-08

0.92

2.80932E-09 -6.06813E-08 4.66711E-10

2.50591E-09 -5.41278E-08

0.93

2.26804E-09 -4.89897E-08 3.80520E-10

2.02309E-09 -4.36988E-08

0.94

1.83106E-09 -3.95508E-08 3.10211E-10

1.63330E-09 -3.52793E-08

0.95

1.47826E-09 -3.19305E-08 2.52864E-10

1.31861E-09 -2.84820E-08

0.96

1.19344E-09 -2.57784E-08 2.06096E-10

1.06455E-09 -2.29943E-08

0.97

9.63499E-10 -2.08116E-08 1.67959E-10

8.59441E-10 -1.85639E-08

0.98

7.77860E-10 -1.68018E-08 1.36865E-10

6.93851E-10 -1.49872E-08

0.99

6.27988E-10 -1.35645E-08 1.11516E-10

5.60165E-10 -1.20996E-08

1

5.06992E-10 -1.09510E-08 9.08527E-11


22

2.3. Solving using the modified Euler method

The modified Euler method is given by

where the superscript P denotes that ⁄ and ⁄

are predictor values where ⁄ is

evaluated using ⁄ , and the superscript C denotes that

is the corrected second-order

result.

Thus, the desired equations forming the solution s are obtained as



Table2.3 Numerical results of the problem using modified Euler method

xn yn fn

Error yn+1P fn+1

xn+1 yn+1C

0 1.00000E+00 -2.16000E+01 0.00000E+00

7.84000E-01 -1.69344E+01

0.01 8.07328E-01 -1.74383E+01 1.59270E-03

6.32945E-01 -1.36716E+01

0.02 6.51778E-01 -1.40784E+01 2.56912E-03

5.10994E-01 -1.10375E+01

0.03 5.26199E-01 -1.13659E+01 3.10812E-03

4.12540E-01 -8.91086E+00

0.04 4.24815E-01 -9.17601E+00 3.34240E-03

3.33055E-01 -7.19399E+00

(2.4)


23

xn yn fn

Error yn+1P fn+1

xn+1 yn+1C

0.05 3.42965E-01 -7.40805E+00 3.36969E-03

2.68885E-01 -5.80791E+00

0.06 2.76885E-01 -5.98073E+00 3.26132E-03

2.17078E-01 -4.68889E+00

0.07 2.23537E-01 -4.82841E+00 3.06875E-03

1.75253E-01 -3.78547E+00

0.08 1.80468E-01 -3.89811E+00 2.82863E-03

1.41487E-01 -3.05612E+00

0.09 1.45697E-01 -3.14705E+00 2.56656E-03

1.14226E-01 -2.46729E+00

0.1 1.17625E-01 -2.54070E+00 2.30002E-03

9.22181E-02 -1.99191E+00

0.11 9.49621E-02 -2.05118E+00 2.04055E-03

7.44503E-02 -1.60813E+00

0.12 7.66655E-02 -1.65598E+00 1.79539E-03

6.01058E-02 -1.29828E+00

0.13 6.18942E-02 -1.33692E+00 1.56871E-03

4.85251E-02 -1.04814E+00

0.14 4.99689E-02 -1.07933E+00 1.36255E-03

3.91757E-02 -8.46194E-01

0.15 4.03413E-02 -8.71373E-01 1.17744E-03

3.16276E-02 -6.83156E-01

0.16 3.25687E-02 -7.03484E-01 1.01295E-03

2.55339E-02 -5.51531E-01

0.17 2.62936E-02 -5.67942E-01 8.68044E-04

2.06142E-02 -4.45267E-01

0.18 2.12276E-02 -4.58516E-01 7.41292E-04

1.66424E-02 -3.59476E-01

0.19 1.71376E-02 -3.70172E-01 6.31094E-04

1.34359E-02 -2.90215E-01

0.2 1.38357E-02 -2.98851E-01 5.35790E-04

1.08472E-02 -2.34299E-01

0.21 1.11699E-02 -2.41270E-01 4.53741E-04


24

xn yn fn

Error yn+1P fn+1

xn+1 yn+1C

8.75722E-03 -1.89156E-01

0.22 9.01779E-03 -1.94784E-01 3.83385E-04

7.06995E-03 -1.52711E-01

0.23 7.28032E-03 -1.57255E-01 3.23270E-04

5.70777E-03 -1.23288E-01

0.24 5.87760E-03 -1.26956E-01 2.72065E-04

4.60804E-03 -9.95337E-02

0.25 4.74515E-03 -1.02495E-01 2.28574E-04

3.72020E-03 -8.03563E-02

0.26 3.83090E-03 -8.27474E-02 1.91728E-04

3.00342E-03 -6.48739E-02

0.27 3.09279E-03 -6.68043E-02 1.60583E-04

2.42475E-03 -5.23745E-02

0.28 2.49690E-03 -5.39329E-02 1.34313E-04

1.95757E-03 -4.22834E-02

0.29 2.01581E-03 -4.35416E-02 1.12198E-04

1.58040E-03 -3.41366E-02

0.3 1.62742E-03 -3.51523E-02 9.36123E-05

1.27590E-03 -2.75594E-02

0.31 1.31386E-03 -2.83795E-02 7.80188E-05

1.03007E-03 -2.22495E-02

0.32 1.06072E-03 -2.29115E-02 6.49551E-05

8.31604E-04 -1.79626E-02

0.33 8.56348E-04 -1.84971E-02 5.40260E-05

6.71377E-04 -1.45017E-02

0.34 6.91354E-04 -1.49332E-02 4.48945E-05

5.42022E-04 -1.17077E-02

0.35 5.58149E-04 -1.20560E-02 3.72742E-05

4.37589E-04 -9.45193E-03

0.36 4.50610E-04 -9.73317E-03 3.09221E-05

3.53278E-04 -7.63081E-03

0.37 3.63790E-04 -7.85786E-03 2.56327E-05

2.85211E-04 -6.16056E-03


25

xn yn fn

Error yn+1P fn+1

xn+1 yn+1C

0.38 2.93698E-04 -6.34387E-03 2.12326E-05

2.30259E-04 -4.97359E-03

0.39 2.37110E-04 -5.12158E-03 1.75756E-05

1.85895E-04 -4.01532E-03

0.4 1.91426E-04 -4.13480E-03 1.45390E-05

1.50078E-04 -3.24168E-03

0.41 1.54543E-04 -3.33814E-03 1.20194E-05

1.21162E-04 -2.61710E-03

0.42 1.24767E-04 -2.69497E-03 9.93062E-06

9.78175E-05 -2.11286E-03

0.43 1.00728E-04 -2.17573E-03 8.20017E-06

7.89708E-05 -1.70577E-03

0.44 8.13206E-05 -1.75653E-03 6.76760E-06

6.37554E-05 -1.37712E-03

0.45 6.56524E-05 -1.41809E-03 5.58241E-06

5.14715E-05 -1.11178E-03

0.46 5.30030E-05 -1.14487E-03 4.60251E-06

4.15544E-05 -8.97575E-04

0.47 4.27908E-05 -9.24282E-04 3.79282E-06

3.35480E-05 -7.24637E-04

0.48 3.45462E-05 -7.46199E-04 3.12416E-06

2.70842E-05 -5.85020E-04

0.49 2.78901E-05 -6.02427E-04 2.57227E-06

2.18659E-05 -4.72303E-04

0.5 2.25165E-05 -4.86356E-04 2.11699E-06

1.76529E-05 -3.81303E-04

0.51 1.81782E-05 -3.92649E-04 1.74160E-06

1.42517E-05 -3.07837E-04

0.52 1.46758E-05 -3.16997E-04 1.43222E-06

1.15058E-05 -2.48525E-04

0.53 1.18482E-05 -2.55920E-04 1.17736E-06

9.28896E-06 -2.00641E-04

0.54 9.56535E-06 -2.06612E-04 9.67513E-07


26

xn yn fn

Error yn+1P fn+1

xn+1 yn+1C

7.49923E-06 -1.61983E-04

0.55 7.72237E-06 -1.66803E-04 7.94794E-07

6.05434E-06 -1.30774E-04

0.56 6.23449E-06 -1.34665E-04 6.52693E-07

4.88784E-06 -1.05577E-04

0.57 5.03328E-06 -1.08719E-04 5.35827E-07

3.94609E-06 -8.52355E-05

0.58 4.06351E-06 -8.77717E-05 4.39752E-07

3.18579E-06 -6.88130E-05

0.59 3.28058E-06 -7.08606E-05 3.60795E-07

2.57198E-06 -5.55547E-05

0.6 2.64851E-06 -5.72077E-05 2.95930E-07

2.07643E-06 -4.48509E-05

0.61 2.13821E-06 -4.61854E-05 2.42660E-07

1.67636E-06 -3.62094E-05

0.62 1.72624E-06 -3.72868E-05 1.98925E-07

1.35337E-06 -2.92328E-05

0.63 1.39364E-06 -3.01026E-05 1.63030E-07

1.09261E-06 -2.36005E-05

0.64 1.12513E-06 -2.43027E-05 1.33579E-07

8.82098E-07 -1.90533E-05

0.65 9.08345E-07 -1.96203E-05 1.09421E-07

7.12143E-07 -1.53823E-05

0.66 7.33333E-07 -1.58400E-05 8.96113E-08

5.74933E-07 -1.24185E-05

0.67 5.92040E-07 -1.27881E-05 7.33710E-08

4.64159E-07 -1.00258E-05

0.68 4.77970E-07 -1.03242E-05 6.00605E-08

3.74729E-07 -8.09414E-06

0.69 3.85879E-07 -8.33498E-06 4.91542E-08

3.02529E-07 -6.53463E-06

0.7 3.11531E-07 -6.72907E-06 4.02198E-08

2.44240E-07 -5.27559E-06


27

xn yn fn

Error yn+1P fn+1

xn+1 yn+1C

0.71 2.51508E-07 -5.43256E-06 3.29027E-08

1.97182E-07 -4.25913E-06

0.72 2.03049E-07 -4.38586E-06 2.69114E-08

1.59191E-07 -3.43851E-06

0.73 1.63927E-07 -3.54083E-06 2.20069E-08

1.28519E-07 -2.77601E-06

0.74 1.32343E-07 -2.85861E-06 1.79928E-08

1.03757E-07 -2.24115E-06

0.75 1.06844E-07 -2.30784E-06 1.47082E-08

8.37659E-08 -1.80934E-06

0.76 8.62583E-08 -1.86318E-06 1.20211E-08

6.76265E-08 -1.46073E-06

0.77 6.96388E-08 -1.50420E-06 9.82322E-09

5.45968E-08 -1.17929E-06

0.78 5.62213E-08 -1.21438E-06 8.02583E-09

4.40775E-08 -9.52075E-07

0.79 4.53891E-08 -9.80404E-07 6.55624E-09

3.55850E-08 -7.68637E-07

0.8 3.66439E-08 -7.91507E-07 5.35488E-09

2.87288E-08 -6.20542E-07

0.81 2.95836E-08 -6.39006E-07 4.37298E-09

2.31936E-08 -5.00981E-07

0.82 2.38837E-08 -5.15887E-07 3.57058E-09

1.87248E-08 -4.04456E-07

0.83 1.92820E-08 -4.16490E-07 2.91498E-09

1.51171E-08 -3.26528E-07

0.84 1.55669E-08 -3.36244E-07 2.37942E-09

1.22044E-08 -2.63616E-07

0.85 1.25676E-08 -2.71459E-07 1.94197E-09

9.85297E-09 -2.12824E-07

0.86 1.01462E-08 -2.19157E-07 1.58473E-09

7.95458E-09 -1.71819E-07

0.87 8.19127E-09 -1.76931E-07 1.29303E-09


28

xn yn fn

Error yn+1P fn+1

xn+1 yn+1C

6.42196E-09 -1.38714E-07

0.88 6.61304E-09 -1.42842E-07 1.05489E-09

5.18463E-09 -1.11988E-07

0.89 5.33889E-09 -1.15320E-07 8.60495E-10

4.18569E-09 -9.04110E-08

0.9 4.31024E-09 -9.31012E-08 7.01834E-10

3.37923E-09 -7.29913E-08

0.91 3.47978E-09 -7.51632E-08 5.72357E-10

2.72814E-09 -5.89279E-08

0.92 2.80932E-09 -6.06813E-08 4.66711E-10

2.20251E-09 -4.75742E-08

0.93 2.26804E-09 -4.89897E-08 3.80520E-10

1.77815E-09 -3.84080E-08

0.94 1.83106E-09 -3.95508E-08 3.10211E-10

1.43555E-09 -3.10078E-08

0.95 1.47826E-09 -3.19305E-08 2.52864E-10

1.15896E-09 -2.50335E-08

0.96 1.19344E-09 -2.57784E-08 2.06096E-10

9.35659E-10 -2.02102E-08

0.97 9.63499E-10 -2.08116E-08 1.67959E-10

7.55384E-10 -1.63163E-08

0.98 7.77860E-10 -1.68018E-08 1.36865E-10

6.09842E-10 -1.31726E-08

0.99 6.27988E-10 -1.35645E-08 1.11516E-10

4.92343E-10 -1.06346E-08

1 5.06992E-10 -1.09510E-08 9.08527E-11

3.97482E-10 -8.58561E-09

2.4. Solving using the fourth-order Runge-Kutta method

The fourth-order Runge-Kutta method is given by

yn+1 = yn + (1/6) (∆y1 + 2 ∆y2 + 2 ∆y3 + ∆y4)

∆y1 = h f(xn, yn) ∆y2 = h f(xn+ h/2, yn+∆y1/2 )

(2.5)


29

∆y3 = h f(xn+ h/2, yn+∆y2/2 ) ∆y4 = h f(xn+ h, yn+∆y3 )


∆y1 = 0.01*(-21.6*1) = -0.21600

∆y2 = 0.01* [-21.6*(1-0.21600/2)] = -0.192672

∆y3 = 0.01* [-21.6*(1-0.87856/2)] = -0.19519

∆y4 = 0.01* [-21.6*(1-0.19082)] = -0.17384

y1 = 1 + (1/6) (-0.216 + 2*-0.192672+ 2*-0.19519-0.17384) = 0.805739


Table2.4 Numerical results of the problem using the fourth-order Runge-Kutta method

xn yn ∆y1 ∆y2 Error

∆y3 ∆y4

xn+1 yn+1

0 1.00000E+00 -2.16000E-01 -1.92672E-01 0

-1.95191E-01 -1.73839E-01

0.01 8.05739E-01 -1.74040E-01 -1.55243E-01 3.78139E-06

-1.57273E-01 -1.40069E-01

0.02 6.49215E-01 -1.40231E-01 -1.25086E-01 6.09361E-06

-1.26721E-01 -1.12859E-01

0.03 5.23098E-01 -1.12989E-01 -1.00786E-01 7.36478E-06

-1.02104E-01 -9.09347E-02

0.04 4.21481E-01 -9.10398E-02 -8.12075E-02 7.9121E-06

-8.22694E-02 -7.32696E-02

0.05 3.39603E-01 -7.33544E-02 -6.54321E-02 7.96884E-06

-6.62877E-02 -5.90362E-02

0.06 2.73632E-01 -5.91045E-02 -5.27212E-02 7.70495E-06

-5.34106E-02 -4.75678E-02

0.07 2.20476E-01 -4.76228E-02 -4.24795E-02 7.24286E-06

-4.30350E-02 -3.83272E-02

0.08 1.77646E-01 -3.83715E-02 -3.42274E-02 6.66953E-06

-3.46750E-02 -3.08817E-02

0.09 1.43136E-01 -3.09174E-02 -2.75784E-02 6.04563E-06

-2.79390E-02 -2.48826E-02

0.1 1.15331E-01 -2.49114E-02 -2.22210E-02 5.41243E-06


30


∆y3 ∆y4

xn+1 yn+1

-2.25115E-02 -2.00489E-02

0.11 9.29263E-02 -2.00721E-02 -1.79043E-02 4.7971E-06

-1.81384E-02 -1.61542E-02

0.12 7.48744E-02 -1.61729E-02 -1.44262E-02 4.21658E-06

-1.46148E-02 -1.30161E-02

0.13 6.03292E-02 -1.30311E-02 -1.16237E-02 3.68058E-06

-1.17757E-02 -1.04875E-02

0.14 4.86096E-02 -1.04997E-02 -9.36571E-03 3.1937E-06

-9.48818E-03 -8.45023E-03

0.15 3.91667E-02 -8.46000E-03 -7.54632E-03 2.75709E-06

-7.64499E-03 -6.80868E-03

0.16 3.15581E-02 -6.81655E-03 -6.08036E-03 2.36959E-06

-6.15987E-03 -5.48602E-03

0.17 2.54276E-02 -5.49236E-03 -4.89919E-03 2.02859E-06

-4.96325E-03 -4.42030E-03

0.18 2.04880E-02 -4.42541E-03 -3.94747E-03 1.73066E-06

-3.99908E-03 -3.56161E-03

0.19 1.65080E-02 -3.56573E-03 -3.18063E-03 1.47193E-06

-3.22222E-03 -2.86973E-03

0.2 1.33011E-02 -2.87304E-03 -2.56276E-03 1.24841E-06

-2.59627E-03 -2.31225E-03

0.21 1.07172E-02 -2.31492E-03 -2.06491E-03 1.05618E-06

-2.09191E-03 -1.86307E-03

0.22 8.63530E-03 -1.86522E-03 -1.66378E-03 8.9153E-07

-1.68554E-03 -1.50115E-03

0.23 6.95780E-03 -1.50288E-03 -1.34057E-03 7.5099E-07

-1.35810E-03 -1.20953E-03

0.24 5.60617E-03 -1.21093E-03 -1.08015E-03 6.3141E-07

-1.09428E-03 -9.74569E-04

0.25 4.51711E-03 -9.75696E-04 -8.70321E-04 5.29948E-07

-8.81701E-04 -7.85248E-04

0.26 3.63961E-03 -7.86156E-04 -7.01251E-04 4.44079E-07

-7.10421E-04 -6.32705E-04


31


∆y3 ∆y4

xn+1 yn+1

0.27 2.93258E-03 -6.33437E-04 -5.65026E-04 3.71573E-07

-5.72414E-04 -5.09795E-04

0.28 2.36289E-03 -5.10385E-04 -4.55263E-04 3.10479E-07

-4.61216E-04 -4.10762E-04

0.29 1.90388E-03 -4.11237E-04 -3.66823E-04 2.59099E-07

-3.71620E-04 -3.30967E-04

0.3 1.53403E-03 -3.31350E-04 -2.95564E-04 2.15964E-07

-2.99429E-04 -2.66673E-04

0.31 1.23603E-03 -2.66981E-04 -2.38147E-04 1.79811E-07

-2.41262E-04 -2.14869E-04

0.32 9.95914E-04 -2.15117E-04 -1.91885E-04 1.49554E-07

-1.94394E-04 -1.73128E-04

0.33 8.02447E-04 -1.73328E-04 -1.54609E-04 1.24267E-07

-1.56631E-04 -1.39496E-04

0.34 6.46563E-04 -1.39658E-04 -1.24575E-04 1.0316E-07

-1.26203E-04 -1.12398E-04

0.35 5.20961E-04 -1.12528E-04 -1.00375E-04 8.55649E-08

-1.01687E-04 -9.05631E-05

0.36 4.19758E-04 -9.06678E-05 -8.08757E-05 7.09126E-08

-8.19333E-05 -7.29702E-05

0.37 3.38216E-04 -7.30546E-05 -6.51647E-05 5.87241E-08

-6.60168E-05 -5.87950E-05

0.38 2.72514E-04 -5.88630E-05 -5.25058E-05 4.8595E-08

-5.31923E-05 -4.73734E-05

0.39 2.19575E-04 -4.74282E-05 -4.23059E-05 4.01852E-08

-4.28591E-05 -3.81706E-05

0.4 1.76920E-04 -3.82147E-05 -3.40876E-05 3.32089E-08

-3.45333E-05 -3.07556E-05

0.41 1.42551E-04 -3.07911E-05 -2.74657E-05 2.74266E-08

-2.78248E-05 -2.47810E-05

0.42 1.14859E-04 -2.48096E-05 -2.21302E-05 2.26376E-08

-2.24195E-05 -1.99670E-05

0.43 9.25466E-05 -1.99901E-05 -1.78311E-05 1.86743E-08


32


∆y3 ∆y4

xn+1 yn+1

-1.80643E-05 -1.60882E-05

0.44 7.45684E-05 -1.61068E-05 -1.43672E-05 1.53965E-08

-1.45551E-05 -1.29629E-05

0.45 6.00827E-05 -1.29779E-05 -1.15763E-05 1.26874E-08

-1.17276E-05 -1.04447E-05

0.46 4.84110E-05 -1.04568E-05 -9.32744E-06 1.04499E-08

-9.44941E-06 -8.41570E-06

0.47 3.90066E-05 -8.42543E-06 -7.51548E-06 8.60293E-09

-7.61376E-06 -6.78086E-06

0.48 3.14292E-05 -6.78870E-06 -6.05552E-06 7.07918E-09

-6.13470E-06 -5.46360E-06

0.49 2.53237E-05 -5.46992E-06 -4.87917E-06 5.82279E-09

-4.94297E-06 -4.40224E-06

0.5 2.04043E-05 -4.40733E-06 -3.93134E-06 4.78739E-09

-3.98274E-06 -3.54705E-06

0.51 1.64405E-05 -3.55116E-06 -3.16763E-06 3.93452E-09

-3.20905E-06 -2.85800E-06

0.52 1.32468E-05 -2.86130E-06 -2.55228E-06 3.23235E-09

-2.58566E-06 -2.30280E-06

0.53 1.06734E-05 -2.30547E-06 -2.05647E-06 2.65451E-09

-2.08337E-06 -1.85546E-06

0.54 8.60002E-06 -1.85760E-06 -1.65698E-06 2.17919E-09

-1.67865E-06 -1.49502E-06

0.55 6.92937E-06 -1.49674E-06 -1.33510E-06 1.78837E-09

-1.35255E-06 -1.20459E-06

0.56 5.58326E-06 -1.20598E-06 -1.07574E-06 1.46716E-09

-1.08981E-06 -9.70587E-07

0.57 4.49865E-06 -9.71709E-07 -8.66765E-07 1.20325E-09

-8.78099E-07 -7.82040E-07

0.58 3.62474E-06 -7.82944E-07 -6.98386E-07 9.86516E-10

-7.07518E-07 -6.30120E-07

0.59 2.92060E-06 -6.30849E-07 -5.62717E-07 8.08577E-10

-5.70075E-07 -5.07712E-07


33


∆y3 ∆y4

xn+1 yn+1

0.6 2.35324E-06 -5.08299E-07 -4.53403E-07 6.62543E-10

-4.59332E-07 -4.09084E-07

0.61 1.89610E-06 -4.09557E-07 -3.65325E-07 5.42733E-10

-3.70102E-07 -3.29615E-07

0.62 1.52776E-06 -3.29996E-07 -2.94356E-07 4.44469E-10

-2.98205E-07 -2.65583E-07

0.63 1.23097E-06 -2.65891E-07 -2.37174E-07 3.63901E-10

-2.40276E-07 -2.13991E-07

0.64 9.91844E-07 -2.14238E-07 -1.91101E-07 2.97863E-10

-1.93600E-07 -1.72421E-07

0.65 7.99168E-07 -1.72620E-07 -1.53977E-07 2.43749E-10

-1.55991E-07 -1.38926E-07

0.66 6.43921E-07 -1.39087E-07 -1.24065E-07 1.99419E-10

-1.25688E-07 -1.11938E-07

0.67 5.18832E-07 -1.12068E-07 -9.99644E-08 1.63114E-10

-1.01272E-07 -9.01931E-08

0.68 4.18043E-07 -9.02973E-08 -8.05452E-08 1.33389E-10

-8.15985E-08 -7.26721E-08

0.69 3.36834E-07 -7.27561E-08 -6.48984E-08 1.09057E-10

-6.57471E-08 -5.85547E-08

0.7 2.71400E-07 -5.86224E-08 -5.22912E-08 8.91446E-11

-5.29750E-08 -4.71798E-08

0.71 2.18678E-07 -4.72344E-08 -4.21331E-08 7.28532E-11

-4.26840E-08 -3.80146E-08

0.72 1.76197E-07 -3.80586E-08 -3.39483E-08 5.95273E-11

-3.43922E-08 -3.06299E-08

0.73 1.41969E-07 -3.06653E-08 -2.73534E-08 4.86295E-11

-2.77111E-08 -2.46797E-08

0.74 1.14390E-07 -2.47082E-08 -2.20397E-08 3.97194E-11

-2.23279E-08 -1.98854E-08

0.75 9.21684E-08 -1.99084E-08 -1.77583E-08 3.24358E-11

-1.79905E-08 -1.60224E-08

0.76 7.42637E-08 -1.60410E-08 -1.43085E-08 2.64832E-11


34


∆y3 ∆y4

xn+1 yn+1

-1.44956E-08 -1.29099E-08

0.77 5.98372E-08 -1.29248E-08 -1.15289E-08 2.16193E-11

-1.16797E-08 -1.04020E-08

0.78 4.82132E-08 -1.04140E-08 -9.28932E-09 1.76457E-11

-9.41079E-09 -8.38131E-09

0.79 3.88472E-08 -8.39100E-09 -7.48477E-09 1.44001E-11

-7.58264E-09 -6.75315E-09

0.8 3.13007E-08 -6.76096E-09 -6.03077E-09 1.17495E-11

-6.10963E-09 -5.44128E-09

0.81 2.52202E-08 -5.44757E-09 -4.85923E-09 9.58538E-12

-4.92277E-09 -4.38425E-09

0.82 2.03209E-08 -4.38932E-09 -3.91527E-09 7.81865E-12

-3.96647E-09 -3.53256E-09

0.83 1.63734E-08 -3.53664E-09 -3.15469E-09 6.3766E-12

-3.19594E-09 -2.84632E-09

0.84 1.31927E-08 -2.84961E-09 -2.54185E-09 5.19977E-12

-2.57509E-09 -2.29339E-09

0.85 1.06298E-08 -2.29604E-09 -2.04807E-09 4.23952E-12

-2.07485E-09 -1.84788E-09

0.86 8.56487E-09 -1.85001E-09 -1.65021E-09 3.45613E-12

-1.67179E-09 -1.48891E-09

0.87 6.90105E-09 -1.49063E-09 -1.32964E-09 2.81711E-12

-1.34703E-09 -1.19967E-09

0.88 5.56045E-09 -1.20106E-09 -1.07134E-09 2.29594E-12

-1.08535E-09 -9.66621E-10

0.89 4.48027E-09 -9.67739E-10 -8.63223E-10 1.87095E-12

-8.74510E-10 -7.78844E-10

0.9 3.60993E-09 -7.79745E-10 -6.95532E-10 1.52443E-12

-7.04627E-10 -6.27545E-10

0.91 2.90866E-09 -6.28271E-10 -5.60418E-10 1.24194E-12

-5.67746E-10 -5.05638E-10

0.92 2.34362E-09 -5.06222E-10 -4.51550E-10 1.01167E-12

-4.57455E-10 -4.07412E-10


35


∆y3 ∆y4

xn+1 yn+1

0.93 1.88835E-09 -4.07883E-10 -3.63832E-10 8.24002E-13

-3.68589E-10 -3.28268E-10

0.94 1.52152E-09 -3.28647E-10 -2.93153E-10 6.71068E-13

-2.96987E-10 -2.64498E-10

0.95 1.22594E-09 -2.64804E-10 -2.36205E-10 5.46457E-13

-2.39294E-10 -2.13117E-10

0.96 9.87792E-10 -2.13363E-10 -1.90320E-10 4.44935E-13

-1.92808E-10 -1.71716E-10

0.97 7.95902E-10 -1.71915E-10 -1.53348E-10 3.62235E-13

-1.55353E-10 -1.38359E-10

0.98 6.41290E-10 -1.38519E-10 -1.23559E-10 2.94875E-13

-1.25174E-10 -1.11481E-10

0.99 5.16712E-10 -1.11610E-10 -9.95559E-11 2.40016E-13

-1.00858E-10 -8.98245E-11

1 4.16335E-10 -8.99284E-11 -8.02161E-11 1.95344E-13

-8.12650E-11 -7.23751E-11

2.5. Solving using the fourth-order Adams-Bashforth-Moulton method

The fourth-order Adams-Bashforth-Moulton method is given by

yP

n+1 = yn + (h/24) (55 fn – 59 fn-1 +37 fn-2 – 9 fn-3)

yC

n+1 = yn + (h/24) (9 fP

n+1 +19 fn -5 fn-1 + fn-2)

Rather than using starting values obtained by the fourth-order Runge-Kutta as done in the

general case, the first three exact values at x= 0.01, 0.02, 0.03 are used with their corresponding

(fn). These values are shown in Table 2.5.

For x4= 0.04:

yP

4= 5.23091E-01 + (0.01/24) (55*-1.12988E-01 – 59*-1.40229E-01 +37* -1.74039E-01

-9*-2.16000E-01) = 5.22076E-1

fP

4= -21.6* 5.22076E-1= -1.12768E-1

yC

4= 5.23091E-01 + (0.01/24) (9*-1.12768E-1+19*-1.12988E-01 -5*-1.40229E-01 +

-1.74039E-01) = 5.21993E-01

(2.6)


36


Table2.5 Numerical results of the problem using the fourth-order Adams-Bashforth-Moulton method

xn yn fn

yn+1

Error

yP

n+1 fP

n+1

xn+1 yC

n+1

0 1.00000E+00 -2.16000E+01

Exact Solution

0.01 8.05735E-01 -1.74039E+01

0.02 6.49209E-01 -1.40229E+01

0.03 5.23091E-01 -1.12988E+01

4.21581E-01 -9.10615E+00

0.04 4.21457E-01 -9.10347E+00 4.21473E-01 -1.57236E-05

3.39675E-01 -7.33698E+00

0.05 3.39570E-01 -7.33472E+00 3.39596E-01 -2.50607E-05

2.73673E-01 -5.91135E+00

0.06 2.73594E-01 -5.90963E+00 2.73624E-01 -2.99894E-05

2.20502E-01 -4.76285E+00

0.07 2.20436E-01 -4.76143E+00 2.20469E-01 -3.21918E-05

1.77660E-01 -3.83745E+00

0.08 1.77607E-01 -3.83631E+00 1.77639E-01 -3.24042E-05

1.43142E-01 -3.09186E+00

0.09 1.43099E-01 -3.09094E+00 1.43130E-01 -3.13158E-05

1.15330E-01 -2.49113E+00

0.1 1.15296E-01 -2.49039E+00 1.15325E-01 -2.94285E-05

9.29221E-02 -2.00712E+00

0.11 9.28944E-02 -2.00652E+00 9.29215E-02 -2.70925E-05


37

xn yn fn

yn+1

Error

yP

n+1 fP

n+1

xn+1 yC

n+1

7.48679E-02 -1.61715E+00

0.12 7.48456E-02 -1.61666E+00 7.48701E-02 -2.45533E-05

6.03215E-02 -1.30294E+00

0.13 6.03035E-02 -1.30256E+00 6.03255E-02 -2.19782E-05

4.86014E-02 -1.04979E+00

0.14 4.85869E-02 -1.04948E+00 4.86064E-02 -1.94769E-05

3.91584E-02 -8.45822E-01

0.15 3.91468E-02 -8.45570E-01 3.91639E-02 -1.71180E-05

3.15502E-02 -6.81484E-01

0.16 3.15408E-02 -6.81281E-01 3.15557E-02 -1.49405E-05

2.54202E-02 -5.49076E-01

0.17 2.54126E-02 -5.48912E-01 2.54256E-02 -1.29630E-05

2.04812E-02 -4.42394E-01

0.18 2.04751E-02 -4.42262E-01 2.04863E-02 -1.11899E-05

1.65018E-02 -3.56439E-01

0.19 1.64969E-02 -3.56333E-01 1.65065E-02 -9.61651E-06

1.32956E-02 -2.87185E-01

0.2 1.32917E-02 -2.87100E-01 1.32999E-02 -8.23210E-06

1.07124E-02 -2.31387E-01

0.21 1.07092E-02 -2.31318E-01 1.07162E-02 -7.02266E-06

8.63101E-03 -1.86430E-01

0.22 8.62844E-03 -1.86374E-01 8.63441E-03 -5.97243E-06


38

xn yn fn

yn+1

Error

yP

n+1 fP

n+1

xn+1 yC

n+1

6.95406E-03 -1.50208E-01

0.23 6.95198E-03 -1.50163E-01 6.95705E-03 -5.06521E-06

5.60292E-03 -1.21023E-01

0.24 5.60125E-03 -1.20987E-01 5.60554E-03 -4.28508E-06

4.51431E-03 -9.75091E-02

0.25 4.51296E-03 -9.74800E-02 4.51658E-03 -3.61689E-06

3.63721E-03 -7.85636E-02

0.26 3.63612E-03 -7.85402E-02 3.63917E-03 -3.04659E-06

2.93052E-03 -6.32992E-02

0.27 2.92965E-03 -6.32803E-02 2.93221E-03 -2.56137E-06

2.36114E-03 -5.10005E-02

0.28 2.36043E-03 -5.09853E-02 2.36258E-03 -2.14969E-06

1.90238E-03 -4.10914E-02

0.29 1.90181E-03 -4.10792E-02 1.90362E-03 -1.80130E-06

1.53276E-03 -3.31076E-02

0.3 1.53230E-03 -3.30978E-02 1.53381E-03 -1.50714E-06

1.23495E-03 -2.66750E-02

0.31 1.23459E-03 -2.66671E-02 1.23585E-03 -1.25929E-06

9.95010E-04 -2.14922E-02

0.32 9.94713E-04 -2.14858E-02 9.95764E-04 -1.05085E-06

8.01685E-04 -1.73164E-02

0.33 8.01447E-04 -1.73112E-02 8.02322E-04 -8.75879E-07


39

xn yn fn

yn+1

Error

yP

n+1 fP

n+1

xn+1 yC

n+1

6.45923E-04 -1.39519E-02

0.34 6.45730E-04 -1.39478E-02 6.46460E-04 -7.29228E-07

5.20424E-04 -1.12412E-02

0.35 5.20269E-04 -1.12378E-02 5.20875E-04 -6.06500E-07

4.19309E-04 -9.05707E-03

0.36 4.19184E-04 -9.05437E-03 4.19688E-04 -5.03934E-07

3.37839E-04 -7.29733E-03

0.37 3.37739E-04 -7.29516E-03 3.38157E-04 -4.18330E-07

2.72199E-04 -5.87950E-03

0.38 2.72118E-04 -5.87775E-03 2.72465E-04 -3.46967E-07

2.19313E-04 -4.73715E-03

0.39 2.19247E-04 -4.73574E-03 2.19535E-04 -2.87543E-07

1.76701E-04 -3.81675E-03

0.4 1.76649E-04 -3.81561E-03 1.76887E-04 -2.38112E-07

1.42369E-04 -3.07518E-03

0.41 1.42327E-04 -3.07426E-03 1.42524E-04 -1.97036E-07

1.14708E-04 -2.47769E-03

0.42 1.14674E-04 -2.47695E-03 1.14837E-04 -1.62932E-07

9.24208E-05 -1.99629E-03

0.43 9.23933E-05 -1.99570E-03 9.25279E-05 -1.34643E-07

7.44640E-05 -1.60842E-03

0.44 7.44418E-05 -1.60794E-03 7.45530E-05 -1.11196E-07


40

xn yn fn

yn+1

Error

yP

n+1 fP

n+1

xn+1 yC

n+1

5.99961E-05 -1.29592E-03

0.45 5.99782E-05 -1.29553E-03 6.00700E-05 -9.17772E-08

4.83392E-05 -1.04413E-03

0.46 4.83248E-05 -1.04382E-03 4.84005E-05 -7.57069E-08

3.89472E-05 -8.41259E-04

0.47 3.89356E-05 -8.41009E-04 3.89980E-05 -6.24168E-08

3.13800E-05 -6.77808E-04

0.48 3.13706E-05 -6.77606E-04 3.14221E-05 -5.14331E-08

2.52830E-05 -5.46114E-04

0.49 2.52755E-05 -5.45951E-04 2.53179E-05 -4.23614E-08

2.03707E-05 -4.40007E-04

0.5 2.03646E-05 -4.39876E-04 2.03995E-05 -3.48732E-08

1.64128E-05 -3.54516E-04

0.51 1.64079E-05 -3.54411E-04 1.64366E-05 -2.86957E-08

1.32239E-05 -2.85636E-04

0.52 1.32199E-05 -2.85551E-04 1.32435E-05 -2.36023E-08

1.06546E-05 -2.30139E-04

0.53 1.06514E-05 -2.30070E-04 1.06708E-05 -1.94049E-08

8.58445E-06 -1.85424E-04

0.54 8.58189E-06 -1.85369E-04 8.59784E-06 -1.59475E-08

6.91654E-06 -1.49397E-04

0.55 6.91448E-06 -1.49353E-04 6.92758E-06 -1.31011E-08


41

xn yn fn

yn+1

Error

yP

n+1 fP

n+1

xn+1 yC

n+1

5.57270E-06 -1.20370E-04

0.56 5.57104E-06 -1.20334E-04 5.58180E-06 -1.07588E-08

4.48996E-06 -9.69830E-05

0.57 4.48862E-06 -9.69541E-05 4.49745E-06 -8.83211E-09

3.61758E-06 -7.81398E-05

0.58 3.61651E-06 -7.81165E-05 3.62375E-06 -7.24796E-09

2.91471E-06 -6.29577E-05

0.59 2.91384E-06 -6.29390E-05 2.91979E-06 -5.94599E-09

2.34840E-06 -5.07254E-05

0.6 2.34770E-06 -5.07103E-05 2.35258E-06 -4.87634E-09

1.89212E-06 -4.08698E-05

0.61 1.89156E-06 -4.08576E-05 1.89555E-06 -3.99788E-09

1.52449E-06 -3.29290E-05

0.62 1.52404E-06 -3.29192E-05 1.52731E-06 -3.27670E-09

1.22829E-06 -2.65311E-05

0.63 1.22793E-06 -2.65232E-05 1.23061E-06 -2.68484E-09

9.89642E-07 -2.13763E-05

0.64 9.89347E-07 -2.13699E-05 9.91546E-07 -2.19928E-09

7.97361E-07 -1.72230E-05

0.65 7.97123E-07 -1.72179E-05 7.98924E-07 -1.80105E-09

6.42438E-07 -1.38767E-05

0.66 6.42247E-07 -1.38725E-05 6.43721E-07 -1.47454E-09


42

xn yn fn

yn+1

Error

yP

n+1 fP

n+1

xn+1 yC

n+1

5.17616E-07 -1.11805E-05

0.67 5.17462E-07 -1.11772E-05 5.18669E-07 -1.20692E-09

4.17047E-07 -9.00820E-06

0.68 4.16922E-07 -9.00552E-06 4.17910E-07 -9.87635E-10

3.36017E-07 -7.25796E-06

0.69 3.35917E-07 -7.25580E-06 3.36725E-07 -8.07998E-10

2.70731E-07 -5.84779E-06

0.7 2.70650E-07 -5.84604E-06 2.71311E-07 -6.60883E-10

2.18129E-07 -4.71160E-06

0.71 2.18064E-07 -4.71019E-06 2.18605E-07 -5.40433E-10

1.75748E-07 -3.79616E-06

0.72 1.75696E-07 -3.79503E-06 1.76138E-07 -4.41840E-10

1.41601E-07 -3.05859E-06

0.73 1.41559E-07 -3.05768E-06 1.41920E-07 -3.61158E-10

1.14089E-07 -2.46432E-06

0.74 1.14055E-07 -2.46359E-06 1.14350E-07 -2.95149E-10

9.19222E-08 -1.98552E-06

0.75 9.18949E-08 -1.98493E-06 9.21360E-08 -2.41157E-10

7.40623E-08 -1.59975E-06

0.76 7.40402E-08 -1.59927E-06 7.42372E-08 -1.97003E-10

5.96724E-08 -1.28892E-06

0.77 5.96547E-08 -1.28854E-06 5.98156E-08 -1.60903E-10


43

xn yn fn

yn+1

Error

yP

n+1 fP

n+1

xn+1 yC

n+1

4.80784E-08 -1.03849E-06

0.78 4.80641E-08 -1.03818E-06 4.81955E-08 -1.31395E-10

3.87371E-08 -8.36721E-07

0.79 3.87255E-08 -8.36472E-07 3.88328E-08 -1.07279E-10

3.12107E-08 -6.74151E-07

0.8 3.12014E-08 -6.73950E-07 3.12890E-08 -8.75739E-11

2.51466E-08 -5.43168E-07

0.81 2.51392E-08 -5.43006E-07 2.52106E-08 -7.14764E-11

2.02608E-08 -4.37633E-07

0.82 2.02548E-08 -4.37503E-07 2.03131E-08 -5.83282E-11

1.63243E-08 -3.52604E-07

0.83 1.63194E-08 -3.52499E-07 1.63670E-08 -4.75910E-11

1.31525E-08 -2.84095E-07

0.84 1.31486E-08 -2.84010E-07 1.31875E-08 -3.88243E-11

1.05971E-08 -2.28897E-07

0.85 1.05939E-08 -2.28829E-07 1.06256E-08 -3.16677E-11

8.53814E-09 -1.84424E-07

0.86 8.53559E-09 -1.84369E-07 8.56142E-09 -2.58264E-11

6.87923E-09 -1.48591E-07

0.87 6.87718E-09 -1.48547E-07 6.89824E-09 -2.10595E-11

5.54263E-09 -1.19721E-07

0.88 5.54098E-09 -1.19685E-07 5.55815E-09 -1.71701E-11


44

xn yn fn

yn+1

Error

yP

n+1 fP

n+1

xn+1 yC

n+1

4.46573E-09 -9.64598E-08

0.89 4.46440E-09 -9.64311E-08 4.47840E-09 -1.39970E-11

3.59807E-09 -7.77183E-08

0.9 3.59700E-09 -7.76951E-08 3.60840E-09 -1.14088E-11

2.89899E-09 -6.26181E-08

0.91 2.89812E-09 -6.25994E-08 2.90742E-09 -9.29796E-12

2.33573E-09 -5.04518E-08

0.92 2.33503E-09 -5.04367E-08 2.34261E-09 -7.57667E-12

1.88191E-09 -4.06493E-08

0.93 1.88135E-09 -4.06372E-08 1.88752E-09 -6.17327E-12

1.51627E-09 -3.27514E-08

0.94 1.51582E-09 -3.27416E-08 1.52084E-09 -5.02919E-12

1.22167E-09 -2.63880E-08

0.95 1.22130E-09 -2.63801E-08 1.22540E-09 -4.09664E-12

9.84303E-10 -2.12610E-08

0.96 9.84010E-10 -2.12546E-08 9.87347E-10 -3.33662E-12

7.93059E-10 -1.71301E-08

0.97 7.92823E-10 -1.71250E-08 7.95540E-10 -2.71729E-12

6.38972E-10 -1.38018E-08

0.98 6.38782E-10 -1.37977E-08 6.40995E-10 -2.21266E-12

5.14824E-10 -1.11202E-08

0.99 5.14670E-10 -1.11169E-08 5.16472E-10 -1.80155E-12


45

xn yn fn

yn+1

Error

yP

n+1 fP

n+1

xn+1 yC

n+1

4.14797E-10 -8.95961E-09

1 4.14673E-10 -8.95694E-09 4.16140E-10 -1.4667E-12


46


47

3.1. Solving using MATLAB

In this section, graphs comparing the MATLAB results obtained using previous numerical

methods and the analytical solution are made. The MATLAB codes used to obtain the solution of

the ODE are shown in Appendix A.

Figure 3.1 Solution using Euler approximation


48

Figure 3.2 Solution using Modified midpoint and Euler approximation

Figure 3.3 Solution using fourth-order Runge-Kutta approximation


49

Figure 3.4 Solution using fourth-order Adams-Bashforth-Moulton approximation


50


51

4.1. Discussion

The problem was defined and modeled in the first section and an analytical solution was

presented.

The numerical solution of the problem was in Section 2, in which five approximation methods

were used and the results were compared to the exact solution by obtaining the error in each step.

Section 3 shows the figures compares the numerical solution to the analytical one graphically

using MATLAB, were the MATLAB codes were shown in Appendix A.

Approximating with Euler method gave the results in Table 2.1, while Figure 3.1 graphs the

solution.

The modified midpoint was applied introducing the predictor and corrected values and the results

were in Table 2.2 and Figure 3.2, successively.

The modified Euler was also applied using the predictor and corrected values as well and it gave

the results in Table 2.3 and Figure 3.2.

The forth-order Runge-Kutta method which evaluates ∆y =(yn+1 -yn) as the weighted sum of

several ∆yi was used and the procedure and the results were summarized in Table 2.4 and Figure

3.3.

The last method was the fourth-order Adams-Bashforth-Moulton which is concidered as a multi-

points method that use starting value from either the fourth-order Runge-Kutta or the analytical

values.

The results were presented in Table 2.5 and Figure 3.4.


52

5.1. Conclusion

Most of the problems in the engineering field can be explained in a way that gives an analytical

solution, but most of the times it’s complicated and might take long time to solve.

Numerical methods are used as fast ways that offer approximate solutions which are simple to

obtain and have accuracy that is good enough and close to the exact solution.

For the aim of this project, four single-point methods and one multi-points method were used.

The first, the fastest and the simplest was the 1st order Explicit Euler Method.

This method relatively has low accuracy but this was modified using a small step size. The

importance of small step size is extreme in this method as the solution might diverge if the step

was large as the method is conditionally stable.

The limit of stability for this method is [-1 ]. The right-hand inequality is

always satisfied for (21h 1), while the left-hand side is satisfied only if h 0.095 (or 21h 2).

The second numerical method used was the modified midpoint method which is considered as a

second order single-point method.

The errors presented in Table 2.2 for the second-order modified midpoint method are

approximately1 5 times smaller than the errors presented in Table 2.1 for the first-order explicit

Euler method. This illustrates the advantage of the second-order method.

The third method used was the modified Euler method which is considered as an alternate

approach for solving the implicit midpoint FDE.

Both, the modified midpoint and the modified Euler methods have a limit of stability of (21h 2)

or (h 0.095) which is the same as the first-order Euler method.

The fourth-order Runge-Kutta method is one ne of the most popular method of the Runge-Kutta

family. It gave the most accurate results with extremely small error as described in Table 2.4 and

Figure 3.3.

The stability condition for this method is

G = 1 + 21 h + 0.5 ( -21 h)2 – (1/6)*(-21h)

3 + (1/24) (-21h)

4

This implies that |G| 1 if (21h 2.785), where G = yn+1/yn.

Last method applied was the fourth-order Adams-Bashforth-Moulton method as an example of

the multi-points methods.

Comparing the results in Table 2.5 obtained using this method to the results in Table 2.4

obtained using the fourth-order Runge-Kutta method shows how different they are regarding the

(5.1)


53

errors. For example, at x=1, Table 2.4 gives an error of 1.95344E-13 which is approximately 8

times smaller than the corresponding error in Table 2.5. However, the Runge-Kutta method

requires four derivative function evaluations per step compared to two for the Adams-Bashforth-

Moulton method.

The stability condition for this method is

(

)

For stability, the whole for roots of G should be 1. Solving this equation gives the results

shown in Figure 5.1 [7]

In this project, the method is stable as (21h= 0.21) gives G 1.

(5.2)

Figure 5.1 Solution of equation (5.2)


54


55

References

[1] Kehoe, J. P. G. and J. B. Butt,(1972), "Interactions of Inter- and. Intraphase. Gradients in a

Diffusion Limited Catalytic Reaction," A.I.Ch.E. J., 18, 347

[2] Price. T. H, Gradients in a Diffusion Liniiteci Catalytic Reaction,” A.I.Ch.E. J., 18, 347, and

J. B. Butt, “Catalyst Poisoning and Fixed Bed Reactor Dynamics-TI.” Chern. Eng, Sci., 32, 393,

(1977).

[3] http://chemelab.ucsd.edu/CAPE/tutorial/aspentutorial01.pdf, cited on January 2013

[4] S.B. Halligudi*, H.C. Bajaj, K.N. Bhatt and M. Krishnaratnam, Hydrogenation of Benzene to

Cyclohexane Catalyzed by Rhodium(I) Complex Supported on Montmorillonite Clay, Central

Salt and Marine Chemicals Research Institute, Bhavnagar 364 002, India, September 1992

[5] John K. Marangozis , Basil G. Mantzouranls , Anastasios N. Sophos, Intrinsic Kinetics of

Hydrogenation of Benzene on Nickel Catalysts Supported on Kieselguhr, March 1979

[6] K. M. M. Al-Abrahemee, Solution of Some Application of System of Ordinary Initial Value

Problems Using Osculatory Interpolation Technique, Department of Mathematics, College of

Education ,University of Al- Qadysea, December 2011

[7] Joe D. Hoffman, Numerical Methods for Engineers and Scientists, Second Edition, 2001

http://chemelab.ucsd.edu/CAPE/tutorial/aspentutorial01.pdf

http://pubs.acs.org/action/doSearch?action=search&author=Marangozis%2C+John+K.&qsSearchArea=author

http://pubs.acs.org/action/doSearch?action=search&author=Mantzouranls%2C+Basil+G.&qsSearchArea=author

http://pubs.acs.org/action/doSearch?action=search&author=Sophos%2C+Anastasios+N.&qsSearchArea=author


56

APPENDIX A

A.1. Euler Approximation code

% The problem to be solved is:

%y'(x)=-21.6*y

%Note: this problem has a known exact solution % y(x)=exp(-21.6*x)

h=0.01; %h is the time step. x=0:h:1; %initialize time variable.

clear ystar; %wipe out old variable.

ystar(1)=1.0; %initial condition (same for approximation).

for i=1:length(x)-1, %Set up "for" loop. k1=-21.6*ystar(i); %Calculate derivative; ystar(i+1)=ystar(i)+h*k1; %Estimate new value of y; end

%exact solution y=exp(-21.6*x);

%Plot numerical and exact solution. plot(x,ystar,'b--',x,y,'r-'); legend('Numerical','Exact'); title('Euler Approximation, h=0.01'); xlabel('x'); ylabel('y*(x), y(x)');

%Print results for i=1:length(x) disp(sprintf('x=%5.3f, y(x)=%6.4f, y*(x)=%6.4f',x(i),y(i),ystar(i))); end

A.2. Modified Euler Approximation code


%y'(x)=-21.6*y




57

clear ystar; %wipe out old variable.

ystar(1)=1.0; %initial condition (same for approximation).

for i=1:length(x)-1, %Set up "for" loop. k1=-21.6*ystar(i); %Calculate derivative; ystar(i+1)=ystar(i)+ h *

subs(k1,{x(i),ystar},{(x(i)+1/2*h),(ystar(i)+1/2*h*subs(k1,{x,y},{x(i),ystar(

i)}))}); %Estimate new value of y;

end


%Plot numerical and exact solution. plot(x,ystar,'b--',x,y,'r-'); legend('Numerical','Exact'); title('Modified Euler Approximation, h=0.01'); xlabel('x'); ylabel('y*(x), y(x)');


A.3. Fourth-order Runge-Kutta Approximation code


%y'(x)=-21.6*y



clear ystar; %wipe out old variable. ystar = zeros(1,length(x)); ystar(1)=1.0; %initial condition (same for approximation).

for i=1:length(x)-1, %Set up "for" loop. F_xy = @(t,r) -21.6*r; %Calculate derivative;

k_1 = F_xy(x(i),ystar(i)); k_2 = F_xy(x(i)+0.5*h,ystar(i)+0.5*h*k_1); k_3 = F_xy((x(i)+0.5*h),(ystar(i)+0.5*h*k_2)); k_4 = F_xy((x(i)+h),(ystar(i)+k_3*h));

ystar(i+1) = ystar(i) + (1/6)*(k_1+2*k_2+2*k_3+k_4)*h; %Estimate new

value of y;


58

end


%Plot numerical and exact solution. plot(x,ystar,'s',x,y,'r-'); legend('Numerical','Exact'); title('Forth-Order Runge-Kutta Approximation, h=0.01'); xlabel('x'); ylabel('y*(x), y(x)');


A.4. Fourth-order Adams-Bashforth-Moulton Approximation code


%y'(x)=-21.6*y



clear ystar; %wipe out old variable. ystar = zeros(1,length(x)); ystar(1)=1.0; %initial condition (same for approximation).

for i=1:3, % generate starting estimates using Runge-Kutta F_xy = @(t,r) -21.6*r; %Calculate derivative;

k_1 = F_xy(x(i),ystar(i)); k_2 = F_xy(x(i)+0.5*h,ystar(i)+0.5*h*k_1); k_3 = F_xy((x(i)+0.5*h),(ystar(i)+0.5*h*k_2)); k_4 = F_xy((x(i)+h),(ystar(i)+k_3*h));

ystar(i+1) = ystar(i) + (1/6)*(k_1+2*k_2+2*k_3+k_4)*h; %Estimate new

value of y;

end

% iterate for i = 4:length(x)-1 % Adams-Bashforth -- *predict* ystar(i+1) = ystar(i) + h/24*(55*F_xy(x(i), ystar(i)) - 59*F_xy(x(i-

1),ystar(i-1))+ 37*F_xy(x(i-2),ystar(i-2)) - 9*F_xy(x(i-3),ystar(i-3)));


59

% Adams-Moulton -- *correct* ystar(i+1) = ystar(i) + h/24*(9*F_xy(x(i+1),ystar(i+1)) +

19*F_xy(x(i),ystar(i))- 5*F_xy(x(i-1),ystar(i-1)) + F_xy(x(i-2),ystar(i-2))); end


%Plot numerical and exact solution. plot(x,ystar,'s',x,y,'r-'); legend('Numerical','Exact'); title('Forth-Order Adams-Bashforth-Moulton Approximation, h=0.01'); xlabel('x'); ylabel('y*(x), y(x)');


Solution of a Concentration Problem Using Numerical Methods

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