# 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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