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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