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Initial and boundary value problems for fractional order differential equations

  • Sabir Umarov
Part of the Developments in Mathematics book series (DEVM, volume 41)

Abstract

In this chapter we will discuss boundary value problems for fractional order differential and pseudo-differential equations. For methodological clarity we first consider in detail the Cauchy problem for pseudo-differential equations of time-fractional order β, \(m - 1 <\beta <m,\) (\(m \in \mathbb{N}\))
$$\displaystyle\begin{array}{rcl} D_{{\ast}}^{\beta }u(t,x) = A(D)u(t,x) + h(t,x),\quad t> 0,\ x \in \mathbb{R}^{n},& &{}\end{array}$$
(5.1)
$$\displaystyle\begin{array}{rcl} \frac{\partial ^{k}u(0,x)} {\partial t^{k}} =\varphi _{k}(x),\quad x \in \mathbb{R}^{n},\ k = 0,\ldots,m - 1,& &{}\end{array}$$
(5.2)
where h(t, x) and \(\varphi _{k},\ k = 0,\ldots,m - 1,\) are given functions in certain spaces described later, \(D = (D_{1},\ldots,D_{n})\), \(D_{j} = -i \frac{\partial } {\partial x_{j}},\ j = 1,\ldots,n\), A(D) is a ΨDOSS with a symbol A(ξ) ∈ XS p (G) defined in an open domain \(G \subset \mathbb{R}^{n}\), and \(D_{{\ast}}^{\beta }\) is the fractional derivative of order β > 0 in the sense of Caputo-Djrbashian (see Section 3.5)

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Sabir Umarov
    • 1
  1. 1.Department of MathematicsUniversity of New HavenWest HavenUSA

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