Adjoint Fractional Differential Expressions and Operators

In this article we present the notions of adjoint differential expressions for fractional-order differential expressions, adjoint boundary conditions for fractional differential equations, and adjoint fractional-order operators. These notions are based on new formulas obtained for various types of fractional derivatives. The introduced notions can be used in many fields of modelling and control of real dynamical systems and processes. INTRODUCTION Adjoint differential operators play an important role in many fields of mathematics and its applications in various fields, including mathematical physics and control theory. In this article we present a consistent extension of the notion of adjoint operators for the case of fractional-order differential operators. For simplicity, we reduce this presentation to Riemann-Liouville and Caputo derivatives and give only basic formulas that are necessary for obtaining similar results in case of other types of fractional derivatives. More general results will be published elsewhere. 1 SOME BASIC DEFINITIONS Left-sided Riemann-Liouville derivatives are defined as follows [1]: aD t f (t) = 1 Γ(n−α) dn dtn t Z a f (τ)dτ (t− τ)α+1−n , (n−1 < α < n) (1) Another definition of the left-sided fractional derivative was introduced by [2]: C aD α t f (t) = 1 Γ(n−α) t Z a f n(τ)dτ (t− τ)α+1−n , (n−1 < α < n), (2) Relationship between the left-sided Riemann-Liouville and the left-sided Caputo fractional derivatives has been discussed by [1]. Inspite of the remark made by [1], the definition of the rightsided Caputo derivative did not appear in the literature until very recent works by Agrawal [3, 4] The right-sided Caputo derivative is defined by analogy with the right-sided Riemann–Liouville derivative: C t D α b f (t) = (−1)n Γ(n−α) b Z t f n(τ)dτ (τ− t)α+1−n , (n−1 < α < n), (3) Copyright c © 2007 by ASME Copyright c © 2007 by ASME The Riesz potential [1,5] (sometimes it is written with a factor of [2cos(πα/2)]−1 in the right hand side) aRb f (t) = 1 Γ(α) b Z a f (τ)|τ− t|α−1dτ (4) is the sum of the left-sided and the right-sided Riemann– Liouville fractional integrals: aRb f (t) = 1 Γ(α) t Z a f (τ)(t− τ)α−1dτ+ + 1 Γ(α) b Z t f (τ)(τ− t)α−1dτ. (5) Based on this, we can consider the fractional Riesz derivative: aRb f (t) = 1 Γ(n−α) dn dtn b Z a f (τ)dτ |t− τ|α+1−n , (n−1< α < n), (6) and we can also introduce the fractional Riesz-Caputo derivative: C aR α b f (t) = 1 Γ(n−α) b Z a f (n)(τ)dτ |t− τ|α+1−n , (n−1 < α < n), (7) BASIC INTEGRATIONS BY PARTS In the subsequent sections we will need the two basic formulas for integration by parts. The first one is the formula known for fractional integrals [6]: b Z a f (t) ( aD t g(t) ) dt = b Z a g(t) ( tD b f (t) ) dt, (α > 0) (8) Due to an analogy with the classical formula for integration by parts, the formula (8) is called fractional integration by parts. Besides this, we will need also the classical formula for repeated integration by parts, which, under the assumption that that the functions w(t), y(t), and z(t) are k times continuously differentiable, is b Z a wzykdt = [ wzy(k−1)− (wz)′y(k−2) + . . . 2 . . .+(−1)k−1(wz)(k−1)y ]t=b t=a +(−1)k b Z a y(wz)(k)dt (9) CASE OF THE LEFT-SIDED RIEMANN-LIOUVILLE DERIVATIVES Putting w(t) = 1, y(t) = aD t g(t), z(t) = f (t), and taking into account definition (1), the equation (9) takes the form: