By Kenneth S. Miller

ISBN-10: 0471588849

ISBN-13: 9780471588849

Commences with the ancient improvement of fractional calculus, its mathematical theory—particularly the Riemann-Liouville model. quite a few examples and theoretical functions of the speculation are offered. gains 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 essential equations.

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**Extra resources for An Introduction to the Fractional Calculus and Fractional Differential Equations**

**Sample text**

2* +- U p sin 2u) = h(x) for any N > 2, 1 < p ::; 2*, ), > 0, and CHAPTER 1 CLASSICAL VARIATIONAL METHOD 36 §5. The Monotonicity Method. Uniqueness of Solutions In the study of various differential equations, the monotonicity method proves very helpful. 3 where the weak lower semicontinuousness of T was proved. However, because of the relative simplicity of problems investigated in this book, the existence results obtained by this method are as a rule weaker than those obtained by other methods.

2). Furthermore, any function u E E~(BR) can be extended onto IRN by u(x) = { u(x), 0, By this extension we get U E Ep and analogously U E E~ (B Ro) for all Ro ;::: R (Gilbarg and Trudinger [1]). We shall identify u and U and consider u E E~(BR) as an element of Ep. Remark. The definitions of this section can be generalized to the case of smooth bounded domains n E IRN. 2 THE THEOREM ON ApPROXIMATION BY BOUNDED DOMAINS The method of approximation by bounded domains is based on the following theorem.

Radial Solutions We saw that the spaces £p,q are not compactly embedded into Lr(IR N ) with any p, q, and r for which we defined these spaces, which does not allow us to use the corresponding existence theorems. However, if we consider some subspaces of the above-mentioned spaces, the situation may improve. Such a case arises if functions a, b depend only on Ixl. In this situation we shall find solutions to appropriate equations in the class of radial functions. 1 RADIAL LEMMAS As we study radial solutions, we need to know some facts about radial functions, collected in the following radial lemmas, which are due to Coleman, Glazer, and Martin [1] and Strauss [1].

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

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