مشخصات پژوهش

Fractional calculus is used to model complex systems that exhibit memory effects, but finding analytical solutions to fractional boundary value problems (FBVPs) is difficult because of nonlocal operators such as Caputo derivatives. Conformable fractional calculus offers a solution by introducing localized derivatives, which allows for more manageable solutions to conformable fractional boundary value problems (CFBVPs). However, CFBVPs lack specific numerical methods despite their potential, which restricts their use in areas like anomalous diffusion and control theory. It is essential to develop effective numerical techniques to fully utilize their capabilities, filling a gap in existing research and enhancing real-world modeling. In this work we investigate the fractional boundary value problem 𝑇𝜁𝜘(𝜎) = 𝜎 𝜉𝜇(𝜎, 𝜘(𝜎)), 0 ≤ 𝜎 ≤ 𝐴, (1.1) 𝜘′(0) = 0, 𝜛𝜘(0)+ 𝜍𝜘(𝐴) = 𝛾, (1.2) where 𝑇𝜁 is the conformable fractional derivative of order 1 < 𝜁 < 2,, 𝜉 > (2 − 𝜁) ∈ 𝑅 and 𝜇: [0,1] × 𝑅 + → 𝑅 + is a continuous function. Firstly, we compute the Green function of the above problem by transforming it into an integral equation. Then, using the properties of the Green function and applying some fixed-point theorems, like the Banach fixed point theorem and some α-ψ- contractions, we will prove the existence and uniqueness of the solution for the problem. Finally, a numerical method to solve the problem will be examined using Bernstein polynomials.