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 crucial operators, for the 1st time studied within the literature. taken with the aim to resolve numerically equations whose linked capabilities might be additionally non-differentiable within the usual feel. that's between others extending the classical Newton approach concept which calls for traditional differentiability of function.
Chapters are self-contained and will be learn independently and several other complex classes could be taught out of this publication. an intensive checklist of references is given in line with bankruptcy. The book’s effects are anticipated to discover functions in lots of components of utilized arithmetic, stochastics, laptop 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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Additional info for Intelligent Numerical Methods: Applications to Fractional Calculus
19) and 0 < λ < 21 . Equivalently, we have |J f (x) − J f (y)| ≤ 2λ Jbα f (x2 ) |x − y| , all x, y ∈ a, b∗ . 5 Applications to Fractional Calculus 33 We observe that |F (y) − F (x) − A (x) (y − x)| ≤ |F (y) − F (x)| + |A (x)| |y − x| ≤ λ |y − x| + |A (x)| |y − x| = (λ + |A (x)|) |y − x| =: (ψ1 ) , ∀ x, y ∈ a, b∗ . 21) We have that Jbα f (x) ≤ (b − a)α (α + 1) f ∞ < ∞, ∀ x ∈ a, b∗ . 22) Hence |A (x)| = Jbα f (x) 2 Jbα f (x2 ) ≤ (b − a)α f ∞ < ∞, 2 (α + 1) Jbα f (x2 ) ∀ x ∈ a, b∗ . 23) Therefore we get (ψ1 ) ≤ λ + (b − a)a f ∞ 2 (α + 1) Jbα f (x2 ) Call 0 < γ1 := λ + |y − x| , ∀ x, y ∈ a, b∗ .
366(1), 164–174 (2010) 2. 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.
1) Let U (x0 , r ) stand for the ball defined by U (x0 , r ) := x ∈ X : x − x0 ≤ r for some r ∈ K . 1) using the preceding notation. 6 Let F : D ⊂ X , A (·) : D → L (X, Y ) and x0 ∈ D be as defined previously. Suppose: (H1 ) There exists an operator M ∈ B (I − A (x)) for each x ∈ D. (H2 ) There exists an operator N ∈ L + (E, E) satisfying for each x, y ∈ D F (y) − F (x) − A (x) (y − x) ≤ N y−x . (H3 ) There exists a solution r ∈ K of R0 (t) := (M + N ) t + F (x0 ) ≤ t. (H4 ) U (x0 , r ) ⊆ D. (H5 ) (M + N )k r → 0 as k → ∞.