Download e-book for iPad: An Introduction to the Fractional Calculus and Fractional by Kenneth S. Miller

Calculus

By Kenneth S. Miller

ISBN-10: 0471588849

ISBN-13: 9780471588849

Commences with the historic improvement of fractional calculus, its mathematical theory—particularly the Riemann-Liouville model. a variety of examples and theoretical purposes of the speculation are awarded. good points subject matters linked to fractional differential equations. Discusses Weyl fractional calculus and a few of its makes use of. contains chosen actual difficulties which bring about fractional differential or indispensable equations.

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Extra info for An Introduction to the Fractional Calculus and Fractional Differential Equations

Example text

A). ~(A) > O. n(A) is concave in ( -1 / a, (0). ~(O) .. =I i=l ai' Noting lim A ... n(A) = 00, we know that, if I:=l ai < a, the curve of ! (A) = 1 + a' A (illustrated in Fig. 1a) on some A > O. (A) = 1 + a' A is just a tangent of ! (A) at point A = 0 (illustrated in Fig. (A) = 1 + a . A. (A) = 1 + a . (A) = 1 + a' A on some A E (-1/ a, 0) (illustrated in Fig. 1c). Now, the proof is complete. • If there is some Xi such that 1'( {x;}) = IL(X), then Eq. , I' is a A-fuzzy measure for any A E (-1/IL(X), (0)] only when 1L({Xj}) = 0 for all j oF i, otherwise, it has no solution in (-1/IL(X), (0).

12, we have ~ A(y), we can suppose x e A(y) without any loss of generality. In this case, if there exists Z E A(x) n A(y), we have A(x) E A(y) and y => E = A(y). So, when A(x) Chapter 2 26 the result that, from x e A(y) and Z E A(y), there exists E E Cfi such that x e E, but Z E E. Thus, if we take F E Cfi satisfying x E F and set G = F - E, then x E G E Cfi, but Z e G. This contradicts the fact that Z E A(x). Therefore, we have A(x) (') A(y) = 0. 2. If ~ is an algebra, then d[~] is a partition of X.

4). 4. The concepts of S-precompact and S-compact were introduced by Wang [1990b]. 5. The concept of u-algebra can be generalized to fuzzy u-field, which is a class of fuzzy sets. This issue is discussed by Qioa [1990]. 1. 2. 3. Let X = (-00,00). ; I}; (b) {xlx5}. Prove the following equalities: (a) (E - G) () (F - G) = (E () F) - G; (b) (E - F) - G = E - (F u G); (c) E - (F - G) = (E - F) u (E () G); (d) (E - F) () (G - H) = (E () G) - (F u H).

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An Introduction to the Fractional Calculus and Fractional Differential Equations by Kenneth S. Miller


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