Note on the Conformable Boundary Value Problems: Sturm’s Theorems and Green’s Function

Abstract

Recently, the conformable derivative and its properties have been introduced. In this paper, we propose and prove some new results on conformable Boundary Value Problems. First, we introduce a conformable version of classical Sturm´s separation, and comparison theorems. For a conformable Sturm-Liouville problem, Green's function is constructed, and its properties are also studied. In addition, we propose the applicability of the Green´s Function in solving conformable inhomogeneous linear differential equations with homogeneous boundary conditions, whose associated homogeneous boundary value problem has only trivial solution. Finally, we prove the generalized Hyers-Ulam stability of the conformable inhomogeneous boundary value problem.

New approximate analytical solutions for the nonlinear fractional Schrödinger equation with second‐order spatio‐temporal dispersion via double Laplace transform method

“…From this newly defined derivative, many essential elements of the mathematical analysis of functions of a real variable have been successfully developed, among which we can mention: mean value theorem, Rolle's theorem, chain rule, conformable integration by parts formulas, conformable power series expansion, CDs, and integrals of complex‐valued functions of a real variable, conformable boundary value problems: Sturm's theorems and Green's function, complex conformable integral (ComI), and conformable single and double Laplace transform (DLTr) definitions 9,11,16‐21 . The conformable partial derivative of the order γ ∈ (0, 1] of the real‐valued functions of several variables and conformable gradient vector are defined, and a conformable Clairaut's theorem for partial derivatives is proven in Atangana et al 22 .…”

“…For a comparative review and analysis of all definitions of FDs and fractional operators, we refer to Teodoro et al 15 From this newly defined derivative, many essential elements of the mathematical analysis of functions of a real variable have been successfully developed, among which we can mention: mean value theorem, Rolle's theorem, chain rule, conformable integration by parts formulas, conformable power series expansion, CDs, and integrals of complex-valued functions of a real variable, conformable boundary value problems: Sturm's theorems and Green's function, complex conformable integral (ComI), and conformable single and double Laplace transform (DLTr) definitions. 9,11,[16][17][18][19][20][21] The conformable partial derivative of the order γ ∈ (0, 1] of the real-valued functions of several variables and conformable gradient vector are defined, and a conformable Clairaut's theorem for partial derivatives is proven in Atangana et al 22 In Gözütok, 23 the conformable Jacobian matrix is introduced; chain rule for multivariable CD is defined; and the relation between conformable Jacobian matrix and conformable partial derivatives is investigated. In Martínez et al, 24 two new results on homogeneous functions involving their conformable partial derivatives are introduced, specifically the homogeneity of the conformable partial derivatives of a homogeneous function and conformable Euler's theorem.…”