Study Boundary Problem with Integral condition for Fractional Differential Equations

Nawal Aziz Abdulkader, N. Adnan
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引用次数: 1

Abstract

Many years ago there was a great interest in studying the existence of positive solutions for fractional differential equations. Many researchers have considered the existence of positive solutions of non-linear differential equations of non-integer order with integral boundary value conditions using fixed point theorems. G.wang et al.(2012) investigated the following fractional differential equations DW(t) + f(t,W(t)) = 0, 0 < t < 1 with integral boundary value condition W(0) = W(0) = 0 , W (1) = λ∫ W(s)ds 1 0 were 2 < α ≤ 3 λ is a positive number (0 < λ < 2), Dis the standard fractional derivative equation of Caputo who obtained his results by means of Guo-krosnosel'skii theorem in a cone also A.Cabada et at. (2013) established the following non-linear fractional differential equation with integral boundary value conditions DW(t) + f(t,W(t)) = 0 , 0 < t < 1 W(0) = W(0) = 0 ,W(1) = λ ∫ W(s)ds , were 2 < α ≤ 3 , λ > 0 , λ ≠ α , 1 0 Dis Riemann –Liovuville standard fractional derivative. and f is a continuous function. The results were based on Guo-krasnosel'skii fixed point theorem in a cone . In this paper we investigate the existent results of a positive solution for the integral boundary value conditions of the following system of equations: D h(t) + k(t, h(t)) = 0 , t ∈ (0,1) h(0) = h(0) = h(0) = 0 , h(1) = δ∫ h(n)dn 1 0 where 3< β ≤ 4 , δ is a positive number , δ ≠ 3 , D denotes Caputo standard derivative and k is a continuous function.Our work based on Banach's and Schauder's theorem. Keyword: Fractional Differential Equation, Integral boundary value Conditions, Schauder's theorem, Green function. 1.Introduction In the last few decades, fractional order calculus has been of the most rapidly developing areas of mathematical analysis. In fact, a natural phenomenon may depend not only on the time instant but also on the previous time history, which can be successfully modeled by fractional calculus. Also fractional differential equations Study boundary problem with Integral condition for Fractional Differential Equations 239 play an important role because of their application in various fields of science such as mathematics, physics, mechanism, economics, engineering and biological sciences etc.(see[1-4]). Integral boundary conditions have several applications in real-life problem such as population dynamics, underground water flow, and blood flow problems. For a detailed description of the integral boundary conditions, we refer in this study to some recent papers [5,6,7]. Our establishment is the existence of positive solutions of the integral boundary conditions of the following fractional differential equation:D h(t) + k(t, h(t)) = 0 , 0 < t < 1 (1) h(0) = h(0) = h(0) = 0 , h(1) = δ ∫ h(n)dn 1 0 (2) where 3 < β ≤ 4 , 0 < δ < 3 , D denotes Caputo standard derivative and k: [0,1] × [0,∞) → [0,∞) is a continuous function. Firstly we obtain the exact expression of the Green's function related to the liner equation Dh(t) + g(t) = 0 , 0 < t < 1 (3) corresponded to the integral condition h(0) = h(0) = h(0) = 0 , h(1) = δ ∫ h(ρ)dρ 1 0 Our work is based on Banach's and Schauder's fixed point theorems. 2.Preliminaries In this article , we present some definitions, notations, lemmas and theorems from fractional calculus theorem. Definition 2.1.[1] Capote derivative fractional order α of the function J: [0,∞) → R is defined as : DJ(t) = 1 Γ(ρ − α) ∫ Jρ(c) (t − c)1+α−n t
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研究分数阶微分方程具有积分条件的边界问题
许多年前,人们对研究分数阶微分方程正解的存在性产生了极大的兴趣。许多研究者利用不动点定理研究了具有整边值条件的非整数阶非线性微分方程正解的存在性。g.w wang et al.(2012)研究了分数阶微分方程DW(t) + f(t,W(t)) = 0, 0 < t < 1,积分边值条件W(0) = W(0) = 0, W(1) = λ∫W(s),当2 < α≤3 λ为正数(0 < λ < 2)时,得到分数阶微分方程,并给出了Caputo的标准分数阶微分方程,Caputo利用锥上的Guo-krosnosel’skii定理得到了结果。(2013)建立了具有积分边值条件的非线性分数阶微分方程DW(t) + f(t,W(t)) = 0, 0 < t < 1 W(0) = W(0) = 0,W(1) = λ∫W(s)ds,分别为2 < α≤3,λ > 0, λ≠α, 1 0 Dis Riemann -Liovuville标准分数阶导数。f是一个连续函数。结果基于锥上的Guo-krasnosel不动点定理。本文研究了下列方程组积分边值条件的正解的存在性结果:D h(t) + k(t, h(t)) = 0, t∈(0,1)h(0) = h(0) = h(0) = 0, h(1) = δ∫h(n) dn10,其中3< β≤4,δ为正数,δ≠3,D为Caputo标准导数,k为连续函数。我们的研究基于巴拿赫定理和绍德定理。关键词:分数阶微分方程,积分边值条件,Schauder定理,Green函数。1.在过去的几十年里,分数阶微积分已经成为数学分析中发展最快的领域之一。事实上,一种自然现象可能不仅依赖于时间瞬间,而且还依赖于以前的时间历史,这可以通过分数微积分成功地建模。分数阶微分方程的积分条件边界问题在数学、物理、机械、经济、工程和生物科学等科学领域有着重要的应用(参见[1-4])。积分边界条件在人口动态、地下水流动和血流问题等现实问题中有许多应用。对于积分边界条件的详细描述,我们在本研究中参考了最近的一些论文[5,6,7]。我们建立了以下分数阶微分方程积分边界条件的正解的存在性:D h(t) + k(t, h(t)) = 0,0 < t < 1 (1) h(0) = h(0) = 0, h(1) = δ∫h(n)dn 1 0(2)其中3 < β≤4,0 < δ < 3, D表示Caputo标准导数,k: [0,1] ×[0,∞)→[0,∞)是连续函数。首先,我们得到了线性方程Dh(t) + g(t) = 0,0 < t < 1(3)对应的积分条件h(0) = h(0) = h(0) = 0, h(1) = δ∫h(ρ)dρ 10的格林函数的精确表达式。2.本文给出了分数阶微积分定理中的一些定义、记号、引理和定理。定义2.1。[1]函数J:[0,∞)→R的Capote导数分数阶α定义为:DJ(t) = 1 Γ(ρ−α)∫Jρ(c) (t−c)1+α−n t
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