Document Details
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Document Type |
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Thesis |
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Document Title |
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FRACTIONAL LANGEVIN EQUATIONS WITH MULTI-POINT BOUNDARY CONDITIONS معادلات لانجفن الكسرية بشروط حدية متعددة النقاط |
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Subject |
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faculty of science |
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Document Language |
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Arabic |
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Abstract |
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Fractional calculus has appeared as an important area in a sundry scientific fields for instance, but not exclusive, physics, mathematics, chemistry, engineering, etc. Fractional differential equations also attracted many authors and researchers in all different scientific disciplines such as engineering, mathematics, physics, chemistry, etc. For recent development. Fractional derivatives give a clear method for the description of memory and heredity characteristic of different materials and processes. These properties of the fractional derivative make the fractional order models more logical and authentic than the original integer order models. Over the past few years, fractional calculus has been used extensively and increasingly in the modeling of many complex physical and biological systems. This science may become the calculus of the 21st century.
The Langevin equation which have formulated at the first time by Langevin in 1908, has been derived to be an influential a way to characterize the evolution of physical phenomena in fluctuating environments[10]. As the powerful growth of fractional derivative, the generalized Langevin equations of fractional orders have been offered by Mainardi and Pironi [32]. At the recent times, sundry contributions involved about the existence and uniqueness of solution for generalized Langevin equations, have been issued, see[2, 6, 9, 14, 20-22, 26, 27, 30, 33, 39, 46, 48, 50, 52-55]and the references mentioned here.
They presented a fractional Langevin equation as a special case of a generalized Langevin equation, and for the first time represented the velocity and displacement correlation functions in terms of the Mittag-Leffler functions.
Eab and Lim [12] studied the possibility of application of fractional Langevin equation of distributed order for modeling single file diffusion and ultraslow diffusion. Also, they used fractional generalized Langevin equation to model anomalous diffusive processes including single file-type diffusion. Sandev et al[41,42]provided expressions for variances and mean squared displacement for fractional generalized Langevin equations for a free particle represented in the presence of the cases of internal and external noise. They discussed its application to model anomalous diffusive processes in complex media including phenomena similar to single file diffusion or possible generalizations thereof.
Recently, several contributions mindful with the uniqueness and existence results for fractional generalized Langevin equations, have been published, see [6,14,21,27,30,48,52-55]and the references given therein.
Ahmad et al [2] find out the existence of solution for three-point nonlocal boundary value problem
where and are the Caputo's fractional derivatives of orders and is a continuously differentiable function.
Li et al [26] investigated the following fractional Langevin equation with infinite-point boundary value conditions
Fixed point theorems contribute with a substantial and great role in the study of the uniqueness and existence of integral, differential and integro differential equations. Although there are a large number of these theorems, but a limited number of them have been focused by the authors in this area such as Krasnoselskii's, nonlinear alternative Leray-Schauder, Banach contraction principle and Leray-Schauder degree. Krasnoselskii-Zabreiko's fixed point theorem for asymptotically linear mappings is one of the immutable point theorems that give important and accurate results in the existence of solutions for differential equations. However, it did not adequately draw the attention of many authors in their applications. Of contributions that used Krasnoselskii-Zabreiko's fixed point theorem [1, 24, 47]and it is worth pointing out that this theorem was provided at the first time by [23].
In Chapter 1, we render requisite definitions of the generalized integral and derivative and preparatory results that are necessity to accomplish this paper.
In Chapter 2, we examine the existence and uniqueness of solution for generalized Langevin equation that has two distinct fractional orders with a three-point boundary value problem.
In Chapter 3, we investigate the existence and uniqueness of solutions for Langevin equation which has Caputo fractional derivatives of two different orders that has anti-periodic and a new class of multi-point boundary conditions.
In chapter 4, we discuss the existence and uniqueness of solution for generalized Langevin equation included two different order with multi-point and multi nonlocal integral conditions. |
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Supervisor |
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Prof. Dr. Ahmed Salem El-Sharif |
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Thesis Type |
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Master Thesis |
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Publishing Year |
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1441 AH
2020 AD |
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Co-Supervisor |
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Dr. Faris S. Alzahrani |
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Added Date |
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Monday, June 22, 2020 |
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Researchers
| بلقيس سعود الغامدي | Alghamdi, Balqees Saud | Researcher | Master | |
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