The Analysis of Fractional Differential Equations
An Application-Oriented Exposition Using Differential Operators of Caputo Type
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Article
Open AccessA Hilbert Space Approach to Fractional Differential Equations
We study fractional differential equations of Riemann–Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on
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Book
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Chapter
Introduction
This chapter serves as a short introduction to the topic of fractional calculus. We briefly mention the history of the field, elaborate on traditional and modern applications and recall some basic ideas from c...
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Chapter
Caputo’s Approach
The Riemann–Liouville fractional derivatives play a very important role in pure mathematics. However they have some properties that are rather undesirable in applications in physics, engineering, and other are...
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Chapter
Existence and Uniqueness Results for Riemann-Liouville Fractional Differential Equations
This short chapter is devoted to a study of the most important existence and uniqueness results for fractional differential equations with Riemann-Liouville operators. These results are provided for a later co...
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Chapter
Mittag-Leffler Functions
This chapter is devoted to a brief summary of the most important properties of Mittag-Leffler functions. These functions play a fundamental role in many questions related to fractional differential equations, ...
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Chapter
Multi-Term Caputo Fractional Differential Equations
In the previous chapter we have taken a very detailed look at initial value problems for differential equations with one differential operator of Caputo’s type. This chapter forms the basis of an analytical th...
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Chapter
Single-Term Caputo Fractional Differential Equations: Advanced Results for Special Cases
In this chapter we present additional results about the behaviour of the solutions of certain special classes of fractional differential equations. This includes the question of the long-term behaviour of solu...
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Chapter
Riemann-Liouville Differential and Integral Operators
This chapter begins with the definition of fractional integral operators in the sense of Riemann and Liouville and with an investigation of their fundamental properties. Based on these integral operators we th...
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Chapter
Single-Term Caputo Fractional Differential Equations: Basic Theory and Fundamental Results
This chapter is devoted to a detailed study of the basic analytic properties of single-term fractional differential equations involving Caputo differential operators. In particular, we shall deal with the exis...
