# Adam-Bashforth Iterative Scheme for Initial Value Problems

ABSTRACT

In this project work, we studied the Adams-Bashforth scheme for solving initial value problems. We gave an indebt explanation on the Adam-Bashforth scheme, its consistency, stability, and convergence, the two and three step methods were also derived. Numerical solutions were obtained using four (4) examples.

Cover page                        i

Title page                           ii

Certification                       iii

Dedication                          iv

Acknowledgement             v

Abstract                                vi

CHAPTER ONE: INTRODUCTION

1.1.  Background of Study                  1

1.2.  Objectives Of The Study           10

1.3. Scope of the Study                     11

1.4. Significance of the Study          11

CHAPTER TWO: BASIC CONCEPTS AND DEFINITIONS

2.1. Differential Equation                    13

2.2. Ordinary Differential Equation     14

2.3. Initial Value Problems                   15

2.4. Iteration                                             16

2.5. Numerical Method                          16

2.6. Mesh Points                                    17

2.7. Local Truncation Error or Discretization Error                                 18

2.8. Single and Multi-Step Method       18

2.8.1. Single Step Methods                   18

2.8.2. Multi-Step Methods                      26

2.9. Order Of A Method                           28

2.10. Predictor-Corrector Methods        28

2.11.  Interval of Validity                           30

2.12. Solution                                             30

2.13. Existence and Uniqueness of An Initial Value Problem                                 30

2.14. Mean Value Theorem                      31

CHAPTER THREE: ADAM-BASHFORTH LINEAR MULTI-STEP ITERATIVE SCHEME FOR INITIAL VALUE PROBLEMS

3.1. The Adam-Bashforth Iterative Scheme                                                  32

3.2. Derivation, Order and Truncation Error of the Adam Bashforth Iterative.         33

3.3. Consistency, Stability and Convergence of Linear Multi-Step Methods                                                   41

CHAPTER FOUR: NUMERICAL EXPERIMENTS OF THE ADAM-BASHFORTH METHOD                             49

CHAPTER FIVE:  SUMMARY AND CONCLUSION

5.1. Summary                                           61

5.2. Conclusion                                         61

BIBLIOGRAPHY

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APA

MLA 8th

Diedemise, Precious "Adam-Bashforth Iterative Scheme for Initial Value Problems" Afribary. Afribary, 05 Nov. 2018, https://afribary.com/works/adam-bashforth-iterative-scheme-for-initial-value-problems. Accessed 17 Jul. 2024.

MLA7

Diedemise, Precious . "Adam-Bashforth Iterative Scheme for Initial Value Problems". Afribary, Afribary, 05 Nov. 2018. Web. 17 Jul. 2024. < https://afribary.com/works/adam-bashforth-iterative-scheme-for-initial-value-problems >.

Chicago

Diedemise, Precious . "Adam-Bashforth Iterative Scheme for Initial Value Problems" Afribary (2018). Accessed July 17, 2024. https://afribary.com/works/adam-bashforth-iterative-scheme-for-initial-value-problems

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