روش عددی برای یک کلاس از معادله کسری انتگرال- دیفرانسیل کسری مرتبه متغییر با مشتقات کسری آتانگانا- بالینو-کاپوتو

نوع مقاله : مقاله پژوهشی

نویسندگان

گروه ریاضی، دانشکده علوم، دانشگاه بیرجند، بیرجند، ایران.

چکیده

هدف اصلی ما در این مقاله، بررسی معادله انتگرال-دیفرانسیل کسری مرتبه متغیر شامل مشتقات کسری آتانگانا-بالینو-کاپوتو به‌صورت
\begin{align*}
&\mathfrak{D}_{\alpha(t)}^{ABC}\Big[u(x,t).g(x,t)\Big]+\frac{\partial u(x,t)}{\partial t}+\int_{0}^{t}u(x,Y)dY+\int_{0}^{t}u(x,Y).k(x,Y)dY\\
=&f(x,t),
\end{align*}
است. سعی کردیم با استفاده از یک روش عددی مبتنی بر عملگرهای ماتریسی شامل چند‌جمله‌ای چبیشف به حل عددی این معادله بپردازیم. این عملگرهای ماتریسی باعث تبدیل معادله انتگرال-دیفرانسیلی مرتبه کسری به یک سیستم‌ جبرخطی خواهد شد که با حل کردن این معادلات، جواب عددی معادله انتگرال-دیفرانسیل کسری فوق را به‌دست می‌آوریم. برای نشان دادن دقت و کارایی این روش چند مثال‌ عددی را که توسط نرم افزار متلب محاسبه شده است، بیان می‌کنیم. 

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Numerical solution for a class of variable order fractional integral-differential equation with Atangana-Baleanu-Caputo fractional derivative

نویسندگان [English]

  • Hajimohammad Mohammadinejad
  • Hassan Khosravi
Department of Mathematics, Faculty of Science, University of Birjand, Birjand, Iran.
چکیده [English]

In this paper we consider fractional integral-differential equations of variable order containing Atangana-Baleanu-Caputo fractional derivatives as follows:
begin{align*}
mathfrak{D}_{alpha(t)}^{ABC}&Big[u(x,t).g(x,t)Big]+frac{partial u(x,t)}{partial t}+int_{0}^{t}u(x,Y)dYnonumber\
&+int_{0}^{t}u(x,Y).k(x,Y)dY=f(x,t),
end{align*}
We try to solve this equation using a numerical method based on matrix operators including Chebyshev polynomials. By using these operational matrixes the fractional order integral-differential equation is transformed into an algebraic system which by solving them, we will obtain the numerical answer of the above fractional integral-differential equation. To show the accuracy and efficiency of this method, we have calculated some numerical examples by MATLAB software.

کلیدواژه‌ها [English]

  • Atangana-Baleanu-Caputo fractional derivative
  • Chebyshev polynomials
  • Operational matrixes
  • Fractional integral-differential equations
[1] Atangana A. and Baleanu D., New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, (2016).
[2] Chen Y., Yi M. and Yu C., Error analysis for numerical solution of fractional differential equation by Haar wavelets method Journal of Computational Science, 3(5) (2012), 367–373.
[3] El-Kalla I. L., Convergence of the Adomian method applied to a class of nonlinear integral equations, Applied Mathematics Letters, 21(4) (2008), 372–376.
[4] Garra R., Gorenflo R., Polito F. and Tomovski Ž., Hilfer-Prabhakar derivatives and some applications, Applied mathematics and computation, 242 (2014), 576–589.
[5] Han W., Chen Y. M., Liu D. Y., Li X. L. and Boutat D., Numerical solution for a class of multiorder fractional differential equations with error correction and convergence analysis, Advances in Difference Equations, (1) (2018), 241-253.
[6] Hasib Khan., Gómez Aguilar J. F., Aziz Khan. and Tahir Saeed Khan, Stability analysis for fractional order advection–reaction diffusion system, Physica A., 521 (2019), 737-751.
[7] Ichise M., Nagayanagi Y. and Kojima T., An analog simulation of non-integer order transfer functionsfor analysis of electrode processes, Journal of Electroanalytical Chemistry and Interfacial Electrochemistry, 33(2) (1971), 253–265.
[8] Jafari H., Tajadodi H. and Ganji R. M., A numerical approach for solving variable order differential equations based on Bernstein polynomials, Computational and Mathematical Methods, 1(5) (2019), e1055.
[9] Khubalkar S., Junghare A., Aware M. and Das S., Unique fractional calculus engineering laboratory for learning and research, International Journal of Electrical Engineering Education, (2018), 0020720918799509.
[10] Kilbas A. A., Srivastava H.M. and Trujillo J.J., Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, 2006.
[11] Lai J., Mao S., Qiu J., Fan H., Zhang Q., Hu Z. and Chen, J., Investigation progresses and applications of fractional derivative model in geotechnical engineering, Mathematical Problems in Engineering, 2016.
[12] Maleknejad D., Rashidinia J., Eftekhari T., Numerical Methods for Partial Differential Equations, Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach, 2020.
[13] Moghaddam B. P. and Machado J. A. T., A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels, Fractional Calculus and Applied Analysis, 20(4) (2017), 1023–1042.
[14] Momani S., Odibat Z. and Erturk V. S., Generalized differential transform method for solving a space and time fractional diffusion wave equation, Physics Letters A, 370(5–6) (2007), 379–387.
[15] Nikan O., Tenreiro Machado J. A., Golbabai A. and Rashidinia J., Numerical evaluation of the fractional Klein–Kramers model arising in molecular dynamics, Journal of Computational Physics, 6 November, (2020), 109983.
[16] Odibat Z. M., A study on the convergence of variational iteration method, Mathematical and Computer Modelling, 51(9-10) (2010), 1181–1192.
[17] Ortigueira M. D., Fractional calculus for scientists and engineers, (Vol. 84), Springer Science & Business Media, 2011.
[18] Podlubny I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, (Vol. 198), Elsevier, 1998.
[19] Prabhakar T. R., A singular integral equation with a generalized Mittag Leffler function in the kernel, 1971.
[20] Rashidinia J. and Mohmedi E., Convergence analysis of tau scheme for the fractional reactiondiffusion equation, The European Physical Journal Plus, 133, (2018), Article number: 402.
[21] Rashidinia J. and Mohmedi E., Approximate solution of the multi-term time fractional diffusion and diffusion-wave equations, Computational and Applied Mathematics, (39) (2020) Article number: 216.
[22] Snyder M. A., Chebyshev methods in numerical approximation, (Vol. 2). Prentice-Hall, 1966.
[23] Srivastava H. M., Saxena R. K., Pogany T. K. and Saxena R., Integral transforms and special functions, Applied Mathematics and Computation, 22(7) (2011), 487–506.
[24] Sun K. and Zhu M., Numerical algorithm to solve a class of variable order fractional integraldifferential equation based on chebyshev polynomials, Mathematical Problems in Engineering, 2015.
[25] Sun H., Zhang Y., Baleanu D., Chen W. and Chen Y., A new collection of real world applications of fractional calculus in science and engineering, Communications in Nonlinear Science and Numerical Simulation, 64 (2018), 213–231.
[26] Xu Y. and Ertürk V., A finite difference technique for solving variable order fractional integrodifferential equations, Bulletin of the Iranian Mathematical Society, 40(3) (2014).
[27] Xu Z. and Chen W., A fractional-order model on new experiments of linear viscoelastic creep of Hami Melon, Computers & Mathematics with Applications, 66(5) (2013), 677–681.
[28] Zayernouri M. and Karniadakis G. E., Fractional spectral collocation methods for linear and nonlinear variable order FPDEs, Journal of Computational Physics, 293 (2015), 312–338.