Conformable Kesirli Diferansiyel Denklemlerinin Taylor ve Sonlu Farklar Metodu ile Sayısal Çözümleri

Yıl 2019, Cilt: 23 Sayı: 3, 850 - 863, 25.12.2019

Şuayip Toprakseven

https://doi.org/10.19113/sdufenbed.579361

Cited By: 7

Öz

Bu çalışmada yeni
tanımlanan conformable kesirli türevli denklemler için güvenilir ve etkili bir metot türettik. Kesirli Taylor açılımından ilk önce Euler ve
Taylor metodunu geliştirdik. Bu Taylor açılımı başlangıç noktasından farklı bir noktada
açılmış genelleştirilmiş Taylor serisiridir. Öngörülen metotlar daha etkili ve hızlı olduğunu
birinci dereceden kesirli diferansiyel denklemlere ve ikinci dereceden salınımlı kesirli
diferansiyel denklemlere  uygulayarak gösterdik. İkinci
metodumuz ise kesirli diferansiyel denklemi zayıf tekil integral denklemine dönüştürüp, çarpım intagrasyon kuralını
uygulayarak çözmek olacaktır. Bu yeni tanımda özel tanımlı fonksiyonlar olmadığı için,
metotlar daha doğru sonuç verecek ve bilgisayar programlaması daha kolay olacaktır. Bu
öngörülen metotların kararlılık ve yakınsaklıkları
ispatlanmış olup, teorik sonuçları destekleyen sayısal örnekler verilmiştir.

Anahtar Kelimeler

Kesirli diferansiyel denklemler, Kesirli euler metodu, Kesirli adams metodu, Riemann-louville ve caputo türevi, Conformable kesirli türev, Taylor metodu

Numerical Solutions of Conformable Fractional Differential Equations by Taylor and Finite Difference Methods

Öz

We drive efficient and
reliable finite difference methods for fractional differential equations (FDEs)
based on recently defined conformable fractional derivative. We first derive
fractional Euler and fractional Taylor methods based on the fractional Taylor
expansion. This fractional Taylor series are the generalized fractional Taylor series
that are independent of initial point. We show that the proposed methods are
more efficient and faster by applying these methods on first order FDEs and
second order oscillatory FDEs. Our second approach is based on inverting FDEs
to a weakly singular integral equation that is approximated by product
integration rule. This new definition has no special functions and thus the
proposed numerical methods will be more accurate and easier to implement than
existing methods for FDEs. We prove the stability and convergence of the
proposed methods. Numerical examples are given to support the theoretical
results.

Anahtar Kelimeler

Fractional differential equations, Fractional euler methods, Fractional adams methods, Riemman-liouville and caputo derivative, Conformal fractional derivative, Taylor methods

Kaynakça

• [1] Zaslavsky, G. M., 2012. Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2012), 461–580.
• [2] Miller, K., Ross, B. 1993. An Introduction to the Fractional Calculus and Fractional Differential Equations, 1st Edition. Wiley, New York, 1993.
• [3] Podlubny, I. 1999. Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press Inc., San Diego, CA, 1999.
• [4] Khalil, R., Horani, M. ,Yousef, A., Sababheh, M. 2014. A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264 (2014), 65–70.
• [5] Abdeljawad, T. 2015. On conformable fractional calculus. Journal of Computational and Applied Mathematics, 279 (2015), 57–66.
• [6] Chung,W. S. 2015. Fractional Newton mechanics with conformable fractional derivative. Journal of Computational and Applied Mathematics, 290 (2015), 150–158.
• [7] Benkhettou, N., Hassani, S., Torres, D. 2015. A conformable fractional calculus on arbitrary time scales. J.King Saud Univ. Sci., 28 (2015), 93–98.
• [8] Yang, S., Wang, L., Zhang, S. 2018. Conformable derivative: application to non-Darcian flow in lowpermeability porous media. Appl. Math. Lett. 79 (2018), 105–110.
• [9] Zhou, H., Yang, S., Zhang, S. 2018. Conformable derivative approach to anomalous diffusion. Physica A, 491(2018), 1001–1013.
• [10] Zhao, D., Luo, M. 2017. General conformable fractional derivative and its physical interpretation. Calcolo, 54(2017) 903–917.
• [11] Li, C. P., Chen, A., Ye, J. J. 2012. Numerical approaches to fractional calculus and fractional ordinary differential equation. J. Comput. Phys., 230 (2012), 3352–3368.
• [12] Lin, R., Liu, F. 2007. Fractional high order methods for the nonlinear fractional ordinary differential equation. Nonlinear Analysis, 66 (2007), 856–869.
• [13] Li, C., Deng, W. 2007. Remarks on fractional derivatives. Appl. Math. Comput., 187 (2007), 777–784.
• [14] Deng, W. H. 2008. Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal., 47 (2008), 204–226.
• [15] Lubich, C. 1986. A stability analysis of convolution quadratures for Abel-Volterra integral equations. MA Journal of Numerical Analysis, I6 (1986), 87–101.
• [16] Diethelm, K., Ford, N., Freed, A. 2002. A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn., 290 (2002), 3–22.
• [17] Li, C., Zeng, F. 2012. Finite difference methods for fractional differential equations. Int. J. Bifurc. Chaos Appl. Sci. Eng., 22 (4) (2012), 1230014.
• [18] Li, C., Zeng, F. 2015. Numerical Methods for Fractional Calculus., 1st Edition. Chapman and Hall/CRC Press,USA.
• [19] Young, A. 1954. Approximate product-integration. Proc. Roy. Soc. London Ser. A., 224 (1954), 552–561.
• [20] Caccioppoli, R. 1930. Un teorema generale sull esistenza de elemente uniti una transformazione funzionale. Acad. Naz. Linzei., 11 (1930), 31–49.
• [21] Usero, D. 2008. Fractional Taylor series for Caputo fractional derivatives. Construction of numerical schemes. http://http://www.fdi.ucm.es/profesor/lvazquez/calcfrac/docs/paper_Usero.pdf (Eri¸sim Tarihi: 23.01.2008).
• [22] Deng, W. H. 2007. Numerical algorithm for the time fractional Fokker–Planck equation. J. Comput. Phys., 227 (2007), 1510–1522.
• [23] Huang, Y. , Lau, T. 2003. A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equation of the second kind. J. Math. Anal. Appl., 282 (2003), 56–62.