Quadratic B-Spline Galerkin Method for Numerical Solutions of Fractional Telegraph Equations

O. Tasbozan, A. Esen
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引用次数: 8

Abstract

Letters which were written by two famous mathematicians G.W. Leibnitz and L’Hospital to each other in 1695 can be considered as the beginning of the fractional calculus’ taking part in literature. After that time, a huge contribution has been made for the development of arbitrary order differentiation and integration by a lot of famous mathematicians such as Euler, Laplace, Fourier, Lacroix, Abel, Riemann, Liouville, Caputo [7]. Fractional order differentiation concept helps to state the many physical problems and lead to huge amount of applications [8]. Recently scientists have shown the effectiveness of the fractional order differential equations on expressing the complex events and modelling many physical, engineering phenomenons in their studies [4]. The huge amount of applications on these type equations can bee seen frequently in various fields such as viscoelastic, biology, signal process, electromagnetic, chaos and fractals, traffic system, chemistry, control system, economics, finance and etc. [9]. Riemann-Liouville fractional concept which proposed by two famous mathematicians Riemann and Liouville is composed of fractional integral and fractional derivative. In this concept, in the fractional initial value problems, the initial conditions are comprised of limit values of Rieman-Liouville derivative in the initial point. This situation causes a big problem in solving fractional initial value problems. There isn’t any physical meaning of these initial points. Caputo’s fractional derivative which is presented byM. Caputo in 1967 involves the limit values in the initial points of integer order derivatives with initial values that are given with fractional order equation. Due to this user friendliness of Caputo’s definition, many scientists prefer Caputo’s derivative as fractional derivative operator in many in their studies [4]. In last decades, the importance of fractional order differential equations increased rapidly. Due to this increasement investigating the analytical and numerical solutions of fractional order differential equations(FDEs) becomes an attractive topic for scientists. So a lot of methods are used to obtain analytical and numerical solutions of FDEs such as Laplace transform method [4], power series method [4], finite element method [10, 11], Adomian decomposition method [12], variational iteration method [13], differential transform method [14], homotopy perturbation method [15], homotopy analysis method [16, 17], finite difference methods [18], and etc. Finite elements method which is used commonly in various fields of physics and engineering, first arise in 1960. Argyris, Clough and Zienkiewicz made contribution to this method [19]. After the development of computer technology in recent 50 years, this method has been occupied in an important place in solving many problems which arise in physics and engineering [20]. Bulletin of Mathematical Sciences and Applications Submitted: 2016-09-21 ISSN: 2278-9634, Vol. 18, pp 23-39 Revised: 2016-12-09 doi:10.18052/www.scipress.com/BMSA.18.23 Accepted: 2016-12-09 2017 SciPress Ltd, Switzerland Online: 2017-05-31
分数阶电报方程数值解的二次b样条伽辽金法
1695年,两位著名数学家莱布尼茨和洛必达的书信可以被认为是分数微积分参与文学的开始。此后,欧拉、拉普拉斯、傅立叶、拉克鲁瓦、阿贝尔、黎曼、Liouville、卡普托等众多著名数学家对任意阶微分与积分的发展做出了巨大贡献[7]。分数阶微分概念有助于描述许多物理问题,并导致大量的应用[8]。近年来,科学家们已经在研究中证明了分数阶微分方程在表达复杂事件和模拟许多物理、工程现象方面的有效性[4]。在粘弹性、生物学、信号过程、电磁学、混沌与分形、交通系统、化学、控制系统、经济、金融等各个领域,都能频繁地看到这类方程的大量应用[9]。Riemann-Liouville分数概念由著名数学家Riemann和Liouville提出,由分数积分和分数导数组成。在这个概念中,分数初值问题的初始条件由黎曼-刘维尔导数在初始点处的极限值构成。这种情况给解决分数初值问题带来了很大的困难。这些初始点没有任何物理意义。卡普托分数阶导数由m。Caputo(1967)研究了用分数阶方程给出初值的整阶导数的初值点极限值问题。由于Caputo定义的这种用户友好性,许多科学家在许多研究中更倾向于将Caputo导数作为分数阶导数算子[4]。近几十年来,分数阶微分方程的重要性迅速增加。由于这种增长,研究分数阶微分方程(FDEs)的解析解和数值解成为科学家的一个有吸引力的话题。因此,得到fde的解析解和数值解的方法很多,如拉普拉斯变换法[4]、幂级数法[4]、有限元法[10,11]、Adomian分解法[12]、变分迭代法[13]、微分变换法[14]、同伦摄动法[15]、同伦分析法[16,17]、有限差分法[18]等。有限元法产生于1960年,广泛应用于物理和工程的各个领域。Argyris、Clough和Zienkiewicz对该方法做出了贡献[19]。经过近50年计算机技术的发展,该方法在解决物理和工程中出现的许多问题中占有重要地位[20]。数学科学与应用通报提交日期:2016-09-21 ISSN: 2278-9634 Vol, pp 23-39修回日期:2016-12-09 doi:10.18052/www.scipress.com/BMSA.18.23接受日期:2016-12-09 2017 SciPress Ltd, Switzerland在线日期:2017-05-31
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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