Mathematical analysis

Chemistry

Ecology

Dirichlet boundary condition

Eigenvalues and eigenvectors

Sturm–Liouville theory

Interval (graph theory)

Ordinary differential equation

Biochemistry

Combinatorics

Physics

Dirichlet distribution

Operator (biology)

Pure mathematics

Mathematics

Differential operator

Biology

Gene

Quantum mechanics

Boundary value problem

Differential equation

Transcription factor

Repressor

Type (biology)

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas

We introduce and present the general solution of three two-term fractional differential equations of mixed Caputo/Riemann–Liouville type. We then solve a Dirichlet type Sturm–Liouville eigenvalue problem for a fractional differential equation derived from a special composition of a Caputo and a Riemann–Liouville operator on a finite interval where the boundary conditions are induced by evaluating Riemann–Liouville integrals at those end-points. For each $$1/2<\alpha <1$$ 1/2<α<1 it is shown that there is a finite number of real eigenvalues, an infinite number of non-real eigenvalues, that the number of such real eigenvalues grows without bound as $$\alpha \rightarrow 1^-$$ α→1-, and that the fractional operator converges to an ordinary two term Sturm–Liouville operator as $$\alpha \rightarrow 1^-$$ α→1- with Dirichlet boundary conditions. Finally, two-sided estimates as to their location are provided as is their asymptotic behavior as a function of $$\alpha $$ α.

Fractional Sturm–Liouville eigenvalue problems, I