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Numerical Solutions of Riccati Equation | PDF

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Added on  2021-10-01

Numerical Solutions of Riccati Equation | PDF

   Added on 2021-10-01

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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 KHATA 2014885526
NUR HABIBAH BINTI RADZALI 2014246052
MOHAMAD ALIFF AFIFUDDIN BIN HILMY 2014675202
K162/15
Numerical Solutions of Riccati Equation | PDF_1
UNIVERSITI TEKNOLOGI MARA
TECHNICAL REPORT
NUMERICAL SOLUTIONS OF RICCATIEQUATIONS USING ADAM-BASHFORTH ANDADAM-MOULTON METHODS
MOHAMAD NAZRI BIN MOHAMAD KHATA
2014885526 CS2495C
NUR HABIBAH BINTI RADZALI
2014246052 CS2495C
MOHAMAD ALIFF AFIFUDDIN BIN HILMY
2014675202 CS2495C
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
Numerical Solutions of Riccati Equation | PDF_2
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 all our 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.
Last but not least, 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
Numerical Solutions of Riccati Equation | PDF_3
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
TABLE OF CONTENTS iii
LIST OF FIGURES v
LIST OF TABLES vi
ABSTRACT vii
1 INTRODUCTION 1
1.1 Problem Statement 3
1.2 Research Objective 3
1.3 Significant Of Project 4
1.4 Scope Of Project 4
2 LITERATURE REVIEW 5
3 METHODOLOGY 7
3.1 Introduction of Adam-Bashforth and Adam-Moulton Methods 7
3.2 Introduction to Riccati Equation 9
3.3 Accuracy of Analysis 11
3.4 Analysis of apply both Adam-Bashforth and Moulton methods 11
4 IMPLEMENTATION 12
4.1 Derivation of Adam-Bashforth and Adam-Moulton Methods 12
4.2 Solving Riccati Equation 20
4.3 Analysis of Results Obtained 23
5 RESULTS AND DISCUSSION 25
iii
Numerical Solutions of Riccati Equation | PDF_4
5.1 Result 25
5.2 Discussion 26
6 CONCLUSIONS AND RECOMMENDATIONS 27
REFERENCES 28
iv
Numerical Solutions of Riccati Equation | PDF_5
LIST OF FIGURES
Figure 4.1 Coding of Adam Bashforth 21
Figure 4.2 Coding of Adam Moulton 22
v
Numerical Solutions of Riccati Equation | PDF_6
LIST OF TABLES
Table 4.1 Result Value 23
Table 4.2 Error of Adam-Bashforth 23
Table 4.3 Error of Adam-Moulton 24
Table 4.4 Comparison of Error Analysis 24
Table 5.1 Magnitude Error 25
vi
Numerical Solutions of Riccati Equation | PDF_7
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
Numerical Solutions of Riccati Equation | PDF_8

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