Numerical Solutions of Riccati Equation | PDF

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Bachelor of Science (Hons.) Mathematics
Center of Mathematics Studies Faculty of Computer and Mathematical
Sciences
TECHNICAL REPORT
NUMERICAL SOLUTIONS OF RICCATI EQUATIONS USING
ADAM-BASHFORTH AND ADAM-MOULTON METHODS
MOHAMAD NAZRI BIN MOHAMAD KHATA2014885526
NUR HABIBAH BINTI RADZALI2014246052
MOHAMAD ALIFF AFIFUDDIN BIN HILMY2014675202
K162/15
UNIVERSITI TEKNOLOGI MARA
TECHNICAL REPORT
NUMERICAL SOLUTIONS OF RICCATIEQUATIONS USING ADAM-BASHFORTH ANDADAM-MOULTON METHODS
MOHAMAD NAZRI BIN MOHAMAD KHATA
2014885526CS2495C
NUR HABIBAH BINTI RADZALI
2014246052CS2495C
MOHAMAD ALIFF AFIFUDDIN BIN HILMY
2014675202CS2495C
Report submitted in partial fulfillment of the requirementfor the degree ofBachelor of Science (Hons.) MathematicsCenter of Mathematics StudiesFaculty of Computer and Mathematical Sciences
JULY 2017
ACKNOWLEDGEMENTS
IN THE NAME OF ALLAH, THE MOST GRACIOUS, THE MOST MERCIFUL
Firstly, we are grateful to Allah S.W.T for giving us the strength to complete this project
successfully.We would like to express our gratitude to all people involved in our final year
project.Throughout the process, we are in contact with many lecturers and friends.They have
contributed in their own way towards ours understanding and thoughts about the whole research
project.
We also would like to express our deepest gratitude and appreciation to the our beloved
supervisor, Miss Farahanie binti Fauzi, for her valuable time, advices and the guidance that we
need to complete our project from the beginning up to the end of the writing.To all lecturers
and friends who have willingly sacrificed their time to help us,we want to record or sincere
thanks.
In addition,we are also indebted to allour undergraduate friends who have helped us
by giving brilliant ideas.Our sincere appreciation also extends to those who have provided
assistance at various occasions. Their views and tips are useful indeed.
Lastbutnotleast,we would like to thanks to our beloved parents for their continuous
encouragement and supports for our team.Without their support it would be hard for us to
finish this study.
ii
TABLE OF CONTENTS
ACKNOWLEDGEMENTSii
TABLE OF CONTENTSiii
LIST OF FIGURESv
LIST OF TABLESvi
ABSTRACTvii
1INTRODUCTION1
1.1Problem Statement3
1.2Research Objective3
1.3Significant Of Project4
1.4Scope Of Project4
2LITERATURE REVIEW5
3METHODOLOGY7
3.1Introduction of Adam-Bashforth and Adam-Moulton Methods7
3.2Introduction to Riccati Equation9
3.3Accuracy of Analysis11
3.4Analysis of apply both Adam-Bashforth and Moulton methods11
4IMPLEMENTATION12
4.1Derivation of Adam-Bashforth and Adam-Moulton Methods12
4.2Solving Riccati Equation20
4.3Analysis of Results Obtained23
5RESULTS AND DISCUSSION25
iii
5.1Result25
5.2Discussion26
6CONCLUSIONS AND RECOMMENDATIONS27
REFERENCES28
iv
LIST OF FIGURES
Figure 4.1Coding of Adam Bashforth21
Figure 4.2Coding of Adam Moulton22
v
LIST OF TABLES
Table 4.1Result Value23
Table 4.2Error of Adam-Bashforth23
Table 4.3Error of Adam-Moulton24
Table 4.4Comparison of Error Analysis24
Table 5.1Magnitude Error25
vi
ABSTRACT
A differential equation can be solved analytically or numerically.In many complicated
cases, it is enough to just approximate the solution if the differential equation cannot be solved
analytically.Euler’s method,the improved Euler’s method and Runge-Kutta methods are ex-
amples of commonly used numerical techniques in approximately solved differential equations.
These methods are also called as single-step methods or starting methods because they use the
value from one starting step to approximate the solution of the next step.While, multistep or
continuing methods such as Adam-Bashforth and Adam-Moulton methods use the values from
several computed steps to approximate the value of the next step.So, in terms of minimizing
the calculating time in solving differential , multistep method is recommended by previous re-
searchers.In this project, a Riccati differential equation is solved using the two multistep meth-
ods in order to analyze the accuracy of both methods.Both methods give small errors when
they are compared to the exact solution but it is identified that Adam-Bashforth method is more
accurate than Adam-Moulton method.
vii
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