Electron. J. Differential Equations, Vol. 2017 (2017), No. 210, pp. 1-18.

Even-order self-adjoint boundary value problems for proportional derivatives

Douglas R. Anderson

In this study, even order self-adjoint differential equations incorporating recently introduced proportional derivatives, and their associated self-adjoint boundary conditions, are discussed. Using quasi derivatives, a Lagrange bracket and bilinear functional are used to obtain a Lagrange identity and Green's formula; this also leads to the classification of self-adjoint boundary conditions. Next we connect the self-adjoint differential equations with the theory of Hamiltonian systems and (n,n)-disconjugacy. Specific formulas of Green's functions for two and four iterated proportional derivatives are also derived.

Submitted July 7, 2017. Published September 11, 2017.
Math Subject Classifications: 26A24, 34A05, 49J15, 49K15.
Key Words: Proportional derivatives; PD controller; Green's function; self-adjoint boundary value problem.

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Douglas R. Anderson
Department of Mathematics
Concordia College
Moorhead, MN 56562, USA
email: andersod@cord.edu

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