Mathematical Modeling for the Solution of Equations and Systems of Equations with Applications. Volume I

Ioannis K. Argyros
Cameron University, Department of Mathematical Sciences, Lawton, OK, USA

Santhosh George
Department of Mathematical and Computational Sciences, NIT Karnakata, India

Narayan Thapa
Cameron University, Department of Mathematical Sciences, Lawton, OK, USA

Series: Mathematics Research Developments
BISAC: MAT017000




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This book is dedicated to the approximation of solutions of nonlinear equations using iterative methods. The study about convergence matter of iterative methods is usually based on two categories: semi-local and local convergence analysis. The semi-local convergence category is, based on the information around an initial point, to provide criteria ensuring the convergence of the method; while the local one is, based on the information around a solution, to find estimates of the radii of the convergence balls. The book is divided into two volumes. The chapters in each volume are self-contained so they can be read independently. Each chapter contains semi-local and local convergence results for single, multi-step and multi-point old and new contemporary iterative methods involving Banach, Hilbert or Euclidean valued operators. These methods are used to generate a sequence defined on the aforementioned spaces that converges with a solution of a nonlinear equation, an inverse problem or an ill-posed problem. It is worth mentioning that most problems in computational and related disciplines can be brought in the form of an equation using mathematical modelling. The solutions of equations can be found in analytical form only in special cases. Hence, it is very important to study the convergence of iterative methods.

The book is a valuable tool for researchers, practitioners, graduate students, and can also be used as a textbook for seminars in all computational and related disciplines.


Chapter 1. Newton-Secant Methods (pp. 1-12)

Chapter 2. Newton-Gauss Method (pp. 13-24)

Chapter 3. Newton Traub Method (pp. 25-36)

Chapter 4. The MMN-HSS Method (pp. 37-50)

Chapter 5. The Kantorovich Theorem and Finite Element Methods (pp. 51-58)

Chapter 6. Newton's Method for Solving Optimal Shape Design Problems (pp. 59-70)

Chapter 7. Meuller's Method (pp. 71-78)

Chapter 8. Meuller's Method for Non-Differentiable Functions (pp. 79-88)

Chapter 9. The Shadowing Lemma for Operators with Chaotic Behaviour (pp. 89-94)

Chapter 10. Secant-Like Methods (pp. 95-104)

Chapter 11. Modified Secant Method (pp. 105-110)

Chapter 12. Newton's Method for Generalized Equations (pp. 111-128)

Chapter 13. Variants of Jarratt's Method (pp. 129-138)

Chapter 14. Super-Halley Methods (pp. 139-150)

Chapter 15. Semidefinite Programs (pp. 151-158)

Chapter 16. Moore's Theorem (pp. 159-166)

Chapter 17. Miranda Results for Intermediate Value Theorem (pp. 167-172)

Chapter 18. Directional Newton Methods (pp. 173-182)

Chapter 19. Newton's Method for a Class of Nonsmooth Operators (pp. 183-192)

Chapter 20. Newton-HSS Methods (pp. 193-206)

Chapter 21. Cauchy-Type Methods (pp. 207-216)

Chapter 22. Second Derivative Free Cauchy-Type Methods (pp. 217-224)

Chapter 23. Mesh Independence Principle of Newton's Method (pp. 225-236)

Chapter 24. High Convergence Order Method (pp. 237-248)

Chapter 25. Two Step Method with Memory (pp. 249-262)

Chapter 26. Broyden's Method (pp. 263-270)

Chapter 27. Existence and Uniqueness of Weak Solution (pp. 271-278)

Chapter 28. Existence of Optimal Parameters (pp. 279-282)

Chapter 29. Weak G^Ateaux Derivative of the Solution Map of Initial Boundary Value Problem (pp. 283-288)

Chapter 30. Spectral Method for Optimal Parameters (pp. 289-294)

Chapter 31. Variational Inequality Problems (pp. 295-300)

Chapter 32. Newton's Method on Generalized Banach Spaces (pp. 301-316)

Authors Contact Information (pp. 317-318)

Index (pp. 319)

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