Study of some fractional boundary value problems

Abstract

تتناول هذه المذكرة دراسة وجود ووحدانية الحلول لمسائل حدودل۰ من رتبة كسرية حيث تم التعامل مع نوعين من المشتقات اܳـܝཏ ๤ل۰ ᆇᆅ ؇݁ލٺگ۰ Ⴄ၍ިّިً و݁ލٺگ۰ ر୷ஓ؇نܳ-٭ިڣ٭ܭ وذዻዧ ሒᇭ إޗ؇ر ཇ๤وط ༡ڎل۰ ݆݁ ஁ஓޔ ܋ިཇ ๴ودߌߵ لၯၽ٭۬ و಻ਃ݁ި؇ن আॻ༟৖৑؇෠੼ ت ݁ۇٴዛው٭.۰ ّأٺ݄ڎ ا৊ዛዊৎ۠٭۰ ا৊ৎٺٴأ۰ আॻ༟ ູොި لܭ ا৊ৎأ؇دᄭᄟ اܳٺڰ؇ݪܹ٭۰ اܳـܝཏ ๤ل۰ إሌᇿ ݁أ؇دᄭᄟ ّႤၽܹ݁٭۰ ݁Ⴄၽڣ۰٪ ً؇ݿٺ༱ڎام دوال ؗદઊਵ ا৊ৎݠਊಾޚ۰ ً႟ၽ ݁ފ؊ᄭᄟ,቉ ቕ ّޚٴ٭ݑ َޙݠ ل؇ت اܳۇٴگޚ۰ اܳݱ؇݁ڎة اܳఈ႙ၽݿ٭ܝ٭۰ (ً؇َ؇خ, ނ؇ودر, ނ٭ڰݠ, ܳଫଃي-ނ؇ودر This research investigates the existence and uniqueness of solutions to fractional-order boundary value

problems, addressing two types of fractional derivatives: the Caputo derivative and the Riemann- Liouville derivative, under Cauchy, Dirichlet, and Neumann-type boundary conditions on finite in- tervals. The methodological approach consists of transforming the fractional differential equation

into an equivalent integral formulation using the corresponding Green’s functions, followed by the application of classical fixed-point theorems (Banach, Schauder, Schaefer, and Leray Schauder) to establish existence and uniqueness results, which are supported by illustrative examples provided at the end of each chapter. boundary conditions, finite interval, fixed point theorem, Green’s function