{"title":"Quadratic B-Spline Galerkin Method for Numerical Solutions of Fractional Telegraph Equations","authors":"O. Tasbozan, A. Esen","doi":"10.18052/WWW.SCIPRESS.COM/BMSA.18.23","DOIUrl":null,"url":null,"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","PeriodicalId":252632,"journal":{"name":"Bulletin of Mathematical Sciences and Applications","volume":"16 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of Mathematical Sciences and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.18052/WWW.SCIPRESS.COM/BMSA.18.23","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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