Abstract
The Sturm–Liouville differential equation is an important tool for physics, applied mathematics, and other fields of engineering and science and has wide applications in quantum mechanics, classical mechanics, and wave phenomena. In this paper, we investigate the coupled hybrid version of the Sturm–Liouville differential equation. Indeed, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with multi-point boundary coupled hybrid condition. Furthermore, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with an integral boundary coupled hybrid condition. We give an application and some examples to illustrate our results.
Highlights
Introduction and PreliminariesVarious papers have been published on fractional differential equations (FDEs)
During the history of mathematics, an important framework of problems called Sturm–Liouville differential equations has been in the spotlight of the mathematicians of applied mathematics and engineering; scientists of physics, quantum mechanics, and classical mechanics; and certain phenomena; for some examples see in [18,19] and the list of references of these papers
We take into account the existence and uniqueness of solution for the following fractional coupled hybrid Sturm–Liouville differential equation:
Summary
Various papers have been published on fractional differential equations (FDEs) (see, e.g., in [1,2,3,4,5,6]). In 2019, El-Sayed et al [23] investigated the following fractional Sturm–Liouville differential equation: Dcα ( p(t)u0 (t)) + q(t)u(t) = h(t) f (u(t)), t ∈ I with multi-point boundary hybrid condition u (0) = 0,. Where α ∈ (0, 1], Dcα denotes the Caputo fractional derivative and p ∈ C ( I, R), q(t), and h(t) are absolutely continuous functions on I = [0, T ], T < ∞ with p(t) 6= 0 for all t ∈ I, f : R → R is defined and differentiable on the interval I, 0 ≤ a1 < a2 < .
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