Intelligent Numerical Methods: Applications to Fractional by George A. Anastassiou, Ioannis K. Argyros

By George A. Anastassiou, Ioannis K. Argyros

In this monograph the authors current Newton-type, Newton-like and different numerical equipment, which contain fractional derivatives and fractional quintessential operators, for the 1st time studied within the literature. focused on the aim to resolve numerically equations whose linked capabilities could be additionally non-differentiable within the usual experience. that's between others extending the classical Newton process conception which calls for ordinary differentiability of function.

Chapters are self-contained and will be learn independently and a number of other complicated classes could be taught out of this e-book. an in depth record of references is given in keeping with bankruptcy. The book’s effects are anticipated to discover purposes in lots of components of utilized arithmetic, stochastics, desktop technology and engineering. As such this monograph is acceptable for researchers, graduate scholars, and seminars of the above topics, additionally to be in all technology and engineering libraries.

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A. Magreñán, Different anomalies in a Jarratt family of iterative root finding methods. Appl. Math. Comput. 233, 29–38 (2014) 14. A. Magreñán, A new tool to study real dynamics: the convergence plane. Appl. Math. Comput. 248, 215–224 (2014) 15. W. Meyer, Newton’s method in generalized Banach spaces. Numer. Func. Anal. Optimiz. 9, 3, 4, 244–259 (1987) 16. A. Potra, V. Ptak, Nondiscrete Induction and Iterative Processes (Pitman, London, 1984) 17. D. Proinov, New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems.

Rn = Pn∞ (0) , Pn (t) = Mt + Nrn−1 . (C3 ) An aposteriori bound is given by the sequence {sn } defined by sn := Rn∞ (0) , Rn (t) = (M + N ) t + N an−1 , bn := /xn − x0 / ≤ r − rn ≤ r, where an−1 := /xn − xn−1 / for each n = 1, 2, . . 44 3 Convergence of Iterative Methods and Generalized … Proof Let us define for each n ∈ N the statement: (In ) xn ∈ X and rn ∈ K are well defined and satisfy rn + an−1 ≤ rn−1 . We use induction to show (In ). 4 and (H3 ), (H5 ) there exists q ≤ r such that: Mq + /F (x0 ) / = q and M k q ≤ M k r → 0 as k → ∞.

G. Anastassiou, Fractional Differentiation Inequalities (Springer, New York, 2009) 3. A. Anastassiou, Fractional representation formulae and right fractional inequalities. Math. Comput. Model. 54(11–12), 3098–3115 (2011) 4. G. Anastassiou, Intelligent Mathematics: Computational Analysis (Springer, Heidelberg, 2011) 5. G. Anastassiou, I. Argyros, Semilocal convergence of Newton-like methods under general conditions with applications in fractional calculus (submitted) (2015) 6. K. Argyros, Newton-like methods in partially ordered linear spaces.

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