ارائه دو مدل برای تحلیل عددی جواب معادلات دیفرانسیل-انتگرال کسری و مقایسه آنها
محورهای موضوعی : آمار
محمود بهروزی فر
1
(استاد، دانشکده علوم پایه ، دانشگاه صنعتی نوشیروانی بابل، مازندران، ایران.)
کلید واژه: Operational matrix, Collocation technique, Riemann-Liouville integral and derivative, Fractional Caputo derivative, Fractional-order integro-differential equations,
چکیده مقاله :
در این مقاله، دو مدل برای تحلیل عددی جواب معادلات دیفرانسیل-انتگرال کسری ارائه می دهیم و سپس به مقایسه نتایج بکارگیری آنها بر روی مسائل متنوع می پردازیم. برای این منظور ابتدا ماتریس‌های عملیاتی چند جمله‌ای‌های ژاکوبی را بیان کرده و سپس هر مسئله را با دو روش حل می‌نماییم: روش ماتریس عملیاتی مشتق مرتبه کسری کاپوتو و روش ماتریس عملیاتی انتگرال مرتبه کسری ریمان-لیوویل. در هر دو روش با استفاده از تکنیک نقطه گذاری به یک دستگاه معادلات جبری خطی یا غیرخطی خواهیم رسید با کمک روش تکراری نیوتن حل می شوند. روش‌های ارائه شده روی چند مثال پیاده سازی شده است و نتایج عددی حاصل بیانگر کارایی بالای هر دو روش است. لازم به ذکر است که تمامی محاسبات با کمک نرم افزار متمتیکا انجام شده است. نتایج عددی نشان می‌دهد که برای معادله دیفرانسیلی که جواب آن بصورت چند جمله ای می باشد بهتر است از روش اول و در معادله غیر خطی که جواب آن بصورت تابع متعالی است بهتر است که روش دوم استفاده شود.
In this paper, we exhibit two methods to numerically solve the fractional integro differential equations and then proceed to compare the results of their applications on different problems. For this purpose, at first shifted Jacobi polynomials are introduced and then operational matrices of the shifted Jacobi polynomials are stated. Then these equations are solved by two methods: Caputo fractionalderivative method and the Riemann-Liouville fractional integral method. In the both method, a set of linear or nonlinear algebraic equations are achieved using collocation technique. Tow presented methods are implemented on some test problems. Numerical results explain the high performance of tow methods. Note that all calculations have been done by Mathematica software. Numerical results show that it should be used the first method when the exact solution of differential equation is a polynomial and the second method should be used when the exact solution of differential equation is a transcendental function.
[1] M.R. Eslahchi , Mehdi Dehghan , M. Parvizi, Application of the collocation method for solving nonlinear fractional integro-differential equations, Applied Mathematics 257 (2014) 105–128.
[2] K. Oldham, J. Spanier, The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary Order, in: Mathematics in Science and Engineering, Academic Press, 1974.
[3] I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
[4] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, Calif, USA, 2006.
[5] S. Momani, N.T. Shawagfeh, Decomposition method for solving fractional Riccati differential equations, Appl. Math. Comput. 182, 2006, 1083-1092.
[6] S. Momani, M.A. Noor, Numerical methods for fourth-order fractional integro-differential equations, Appl. Math. Comput. 182, 2006, 754-76
[7] V. D. Gejji, H. Jafari, Solving a multi-order fractional differential equation, Appl. Math. Comput. 189, 2007, 541-548.
[8] S. S. Ray, K. S. Chaudhuri, R. K. Bera, Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method, Appl. Math. Comput. 182, 2006, 544-552.
[9] Q. Wang, Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method, Appl. Math. Comput. 182, 2006, 1048-1055.
[10] M. Inc, The approximate and exact solutions of the space - and time-fractional Burgers equations with initial conditions by variational iteration method, J. Math. Anal. Appl. 345, 2008, 476-484.
[11] S. Momani, Z. Odibat, Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A 355, 2006, 271-279.
[12] Z. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul. 7, 2006, 271-279.
[13] S. Momani, Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A 365, 2007, 345-35
[14] N. H. Sweilam, M. M. Khader, R. F. Al-Bar, Numerical studies for a multi-order fractional differential equation, Phys. Lett. A 371, 2007, 26-33.
[15] I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul. 14, 2009, 674-684.
[16] E. A. Rawashdeh, Numerical solution of fractional integro-differential equations by collocation method, Appl. Math. Comput. 176, 2006, 1-6.
[17] E. H. Doha, A. H. Bhrawy, S. S. Ezz-Eldien, A new Jacobi operational matrix :an application for solving fractional differential equations, Appl. Math. Model. 36, 2012, 4931-4943.
[18] A. H. Bhrawy, M. M. Tharwat, M. A. Alghamdi, A new operational matrix of fractional integration for shifted Jacobi polynomials, Bull. Malays. Math. Sci. Soc, 2013.
[19] R. Hilfer, Applications of fractional calculus in physics, World Scientific Publishing Co. Inc. River Edge, NJ, 200
[20] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, Hackensack, NJ, USA, 2012.
[21] M. Alipour, Numerical solution for fractional differential equations and optimal control problems, PhD Thesis, 2013.
[22] E. Kreyszig, Introductory functional analysis with applications, John Wiley and Sons Incorporated, 1978.
[23] E. H. Doha, On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials, J. Phys. A Math. Gen. 37, 2004, 657-675.
[24] Y. Luke, The special functions and their approximations, Academic Press, New York, 1969.
[25] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamic, Prentice-Hall, Englewood Cliffs, NJ, 1988.
[26] Li Zhu , Qibin Fan, Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet, Commun Nonlinear Sci. Numer Simulat. 17 ,2012, 2333–2341.
[27] Li Zhu , Qibin Fan, Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW, Commun Nonlinear Sci Numer Simulat. 18 ,2013, 1203–1213.
[28] Y. Yang,Y. Chen, Y. Huang, Convergence Analysis of the Jacobi Spectral-Collocation Method for Fractional Integro-Differential Equations, Acta Mathematica Scientia. 2014,673–69
[29] S.Mashayekhi,M.Razzaghi, Numerical solution of nonlinear fractional integro-differential equations by hybridfunctions,Engineering Analysis with Boundary Elements. 56,2015,81–89.
