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

Abstract

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.

Keywords

• Proportional derivatives
• PD controller
• Green's function
• self-adjoint boundary value problem