Numerical Solutions of Multi-order Fractional Antiperiodic Boundary Value Problems

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RESEARCH PAPER

Numerical Solutions of Multi-order Fractional Antiperiodic Boundary Value Problems HuiChol Choi1



KumSong Jong1 • KyongSon Jon1,2 • YongSim Sin1

Received: 2 May 2020 / Accepted: 11 September 2020 Ó Shiraz University 2020

Abstract In this paper, we propose the numerical scheme for solving nonlinear higher multi-order Caputo fractional differential equations with integral and antiperiodic boundary conditions, establish the existence of an approximate solution to our problem, and prove the convergence of our numerical method. The operational matrix of fractional integration based on hat functions is used to solve the given problem. Also, we present a numerical example to illustrate our main results. Keywords Multi-order fractional differential equation  Operational matrix method  Antiperiodic boundary condition  Integral boundary condition  Hat functions  Convergence analysis Mathematics Subject Classification 34A08  34B10  65L60

1 Introduction Fractional calculus which has a long history and has been utilized to mathematically model some complicated natural biological, physical, or industrial systems is the theory of integration and differentiation of arbitrary real or complex order. It was originated at the time of development of the classical calculus and first appeared in a letter from Leibniz to L’Hospital in 1695. In the following centuries, it was significantly developed within the scope of pure mathematics theory and has attracted the attention of many mathematicians and applied scientists in various fields of science and technology since the fact that hereditary properties and memory effects of numerous real-world processes could be explained by integrals and derivatives of non-integer or fractional order (Cattani 2018; Ilhan and Kiymaz 2020; Pak et al. 2020; Ravichandran et al. 2019). One of the valuable works on the subject of fractional calculus, i.e., the theory of derivatives and integrals of & KumSong Jong [email protected]; [email protected] 1

2

Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea Key Laboratory for Applied Statistics of MOE, School of Mathematics and Statistics, Northeast Normal University, Changchun, People’s Republic of China

fractional (non-integer) order, is the book of Podlubny (1999), which mainly dealt with fractional differential equations. In recent few decades, fractional differential equations (FDEs) have been often employed as mathematical models for many phenomena arisen in a variety of fields such as image processing, fluid mechanics, epidemiology, and so on (Danane et al. 2020; Gao et al. 2020; Hilfer 2000; Jong 2018; Kumar et al. 2018; Singh et al. 2018). Particularly, the applications of boundary value problems (BVPs) for FDEs have just emerged in diverse realms such as physics, chemistry, biology, mechanics, and engineering (Kilbas et al. 2006; Jong et al. 2019, 2020). Antiperiodic BVPs that recently attracted the attention of a large number o

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