# یک اسپلاین غیر چند جمله ای برای تقریب معادله بگلی-تورویک کسری

نوع مقاله : مقاله پژوهشی

نویسندگان

گروه ریاضی، دانشکده علوم پایه، دانشگاه رازی، کرمانشاه، ایران

چکیده

در این تحقیق به کمک یک تابع غیر چند جمله ای و عملگر گرانوالد جواب معادله کسری بگلی-توریک را تقریب خواهیم زد. روش های ارائه شده یک سیستم از معادلات را نتیجه می دهد. سپس در ادامه تحلیل خطا برپایه این اسپلاین نمایی و همچنین تحلیل همگرایی روش مورد بحث قرار می گیرد و یک دسته روش از مرتبه دوم بدست می آید. این روش پیشنهادی نه تنها برای حل معادلاتبگلی-ترویک کسری بلکه برای انواع معادلات کسری می تواند مورد استفاده قرار بگیرد. بواسطه مثالهایی کارایی این روش های عددی را با سایر روش های موجود مقایسه می گردد.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

### An non-polynomial spline approximation for fractional Bagley-Torvik equation

نویسندگان [English]

• Sara Ivani
• Reza Jalilian
Department of Mathematics, Razi University, Kermanshah, Iran
چکیده [English]

In this paper, we approximate the solution of fractional

Bagley–Torvik equation by using the non-polynomial spline function and

the shifted Gr"{u}nwald difference operator. The proposed methods

reduce to the system of algebraic equations. The convergence

analysis of the methods has been discussed. The numerical examples

are presented to illustrate the applications of the methods and to

compare the computed results with the other methods.

کلیدواژه‌ها [English]

• Caputo derivative
• Grunwald-Letnikov operator. Non-polynomial spline
• Convergence analysis
• Fractional Bagley-Torvik equation

#### مراجع

 Akram G. and Tariq H., An exponential spline technique for solving fractional boundary
value problem, Calcolo., 53 (2016), 545-558.
 Akgül A., A new method for approximate solutions of fractional order boundary value problems,
Neural Parallel Sci. Comput., 22(1–2) (2017), 223–237.
 Akgül A., Mustafa I. and Baleanu D., On solutions of variable-order fractional differential
equations, Int. J. Optim. Control., 7 (2017), 112–116.
 Al-Mdallal Q. M., Syam M. I. and Anwar M. N., A collocation shooting method for solving
fractional boundary value problems, Commun. Nonlinear Sci Numer. Simul., 15(12)
(2010), 3814–3822.
 Bagley R. L. and torvic P. J., On the appearance of the fractional derivative in the behavior
of real materials, J. Appl Mech., 51(2) (1984), 294-298.
 Diego Murio A., Implicit finite difference approximation for time fractional diffusion equations,
J. Comput. Appl. Math., 56 (2008), 1138-1145.
 Diethelm K. and Ford J., Numerical solution of the Bagley-Torvik equation, BIT Numer.
Math., 42 (2002), 490-507.
 Emadifar H. and Jalilian R., An exponential spline approximation for fractional Bagly-
Torvic equation, Boundary value Problems, 20 (2020), https://doi.org/10.1186/s13661-020-
01327-2.
 Esmaeili S., The numerical solution to the Bagley-Torvik equation by exponential integrators,
Scientia Iranica, 24(6) (2017), 2941-2951.
 Fazli H. and Juan J., An investigation of fractional Bagley-Torvik equation, Open Mathematics,
(2019), 499-512.
 Gülsu M., Öztürk Y. and Anapali A., Numerical solution of the fractional Bagley-Torvik
equation arising in fluid mechanics, Int. J. Comput. Math., 94(1) (2017), 173-184.
 Jalilian R. and Tahernezhad T., Exponential spline method for approximation solution of
Fredholm integro-differential equation, Int. J. Comput. Math., 97(3) (2019), 1-17.
 Maleknejad Kh., Rashidinia J. and Jalilian H., Nonpolynomial spline functions and Quasilinearization,
FILOMAT, 32(11) (2018).
 Maleknejada K. and Torkzadeh L., Hybrid functions approach for the fractional riccati
differential equation, Filomat., 30(9) (2016), 2453-2463.
 Oldham K. B. and Spanier J., The Fractional Calculus, Academic Press, New York, 1974.
 Podlubny I., Fractional Differential Equations, Academic Press, New York, 1990.
 Podlubny I., Matrix approach to discrete fractional calculus, Fract. Calc. Appl. Anal., 3(4)
(2000), 359-386.
 Rashidinia J. and Jalilian R., Non-polynomial spline for solution of boundary-value problems
in plate deflection theory, I. J. Comput Math., 84 (2007), 1483-1494.
 Ray S. S. and Bera R. K., Analytical solution of the Bagley-Torvik equation by Adomian
decomposition method, Appl. Math. Comput., 168 (2005), 398-410.
 Saadatmand A. and Dehghan M., A new operational matrix for solving fractional-order
differential equations, Comput. Math. Appl., 59 (2010), 1326-1336.
 Sakar M. G., Sald O. and Akgül A., A novel technique for fractional Bagley-Torvik equation,
Proc. Natl. Acad.Sci.India, Sect. A Phys. Sci., https://doi.org/10.1007/s40010-018-
0488-4.
 Saw V. and Kumar S., Numerical scheme for solving two point fractional Bagley-Torvik
equation using Chebyshev collocation method, WSEAS Transactions on Systems, 17
(2018), 166-177.
 Sayevand Kh. and Mirzaee F., A unique continuous solution for the Bagley-Torvik equation,
CJMS., 1(1) (2012), 47-51.
 Staněk S., Two-point boundary value problems for the generalized Bagley-Torvik fractional
differential equation, Cent. Eur. J. Math., 11(3) (2013), 574–593.
 Tian W., Zhou H. and Deng W., A class of second order difference approximations for
solving space fractional diffusion equations, Math Computat., 84 (2015), 1703-1727.
 Van Daele M., Vanden Berghe G. and De Meyer H., A smooth approximation for the solution
of a fourth-order boundary value problem based on nonpolynomial splines, J. Comput.
Appl. Math., 51 (1994), 383-394.
 Yüzbaş Ş., Numerical solution of the Bagley-Torvik equation by the Bessel collocation
method, Math. Methods Appl. Sci., 36 (2013), 300-312.
 Zahra W. K. and Elkholy S. M., Quadratic spline solution for boundary value problem of
fractional order, Numer. Algorithms, 59 (2012), 373-391.
 Zahra W. K. and Elkholy S. M., Cubic spline solution of fractional Bagley-Torvik equation,
Electron J. Math. Anal. Appl., 1(2) (2013), 230-241.

### سابقه مقاله

• تاریخ دریافت: 06 مرداد 1398
• تاریخ بازنگری: 16 مرداد 1399
• تاریخ پذیرش: 23 خرداد 1401
• تاریخ اولین انتشار: 01 تیر 1401