On Efficient Method for System of Fractional Differential Equations
© Najeeb Alam Khan et al. 2011
Received: 14 December 2010
Accepted: 5 February 2011
Published: 9 March 2011
The present study introduces a new version of homotopy perturbation method for the solution of system of fractional-order differential equations. In this approach, the solution is considered as a Taylor series expansion that converges rapidly to the nonlinear problem. The systems include fractional-order stiff system, the fractional-order Genesio system, and the fractional-order matrix Riccati-type differential equation. The new approximate analytical procedure depends only on two components. Comparing the methodology with some known techniques shows that the present method is relatively easy, less computational, and highly accurate.
Fractional differential equations have received considerable interest in recent years and have been extensively investigated and applied for many real problems which are modeled in different areas. One possible explanation of such unpopularity could be that there are multiple nonequivalent definitions of fractional derivatives . Another difficulty is that fractional derivatives have no evident geometrical interpretation because of their nonlocal character. However, during the last 12 years fractional calculus starts to attract much more attention of scientists. It was found that various, especially interdisciplinary, applications [2–6] can be elegantly modeled with the help of the fractional derivatives.
The homotopy perturbation method is a powerful devise for solving nonlinear problems. This method was introduced by He [7–9] in the year 1998. In this method, the solution is considered as the summation of an infinite series that converges rapidly. This technique is used for solving nonlinear chemical engineering equations , time-fractional Swift-Hohenberg (S-H) equation , viscous fluid flow equation , Fourth-Order Integro-Differential equations , nonlinear dispersive equations , Long Porous Slider equation , and Navier-Stokes equations . It can be said that He's homotopy perturbation method is a universal one, which is able to solve various kinds of nonlinear equations. The new homotopy perturbation method (NHPM) was applied to linear and nonlinear ODEs .
In this paper, we construct the solution of system of fractional-order differential equations by extending the idea of [17, 18]. This method leads to computable and efficient solutions to linear and nonlinear operator equations. The corresponding solutions of the integer-order equations are found to follow as special cases of those of fractional-order equations.
2. Basic Definitions
We give some basic definitions, notations, and properties of the fractional calculus theory used in this work.
It has the following properties:
3. Analysis of New Homotopy Perturbation Method
Our aim is to study the mathematical behavior of the solution and for different values of . This goal can be achieved by forming Pade' approximants, which have the advantage of manipulating the polynomial approximation into a rational function to gain more information about and . It is well known that Pade' approximants will converge on the entire real axis, if and are free of singularities on the real axis. It is of interest to note that Pade' approximants give results with no greater error bounds than approximation by polynomials. To consider the behavior of solution for different values of , we will take advantage of the explicit formula (4.9) available for and consider the following two special cases.
5. Concluding Remarks
M. Jamil is highly thankful and grateful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan, the Department of Mathematics & Basic Sciences, NED University of Engineering & Technology, Karachi-75270, Pakistan, and also the Higher Education Commission of Pakistan for generously supporting and facilitating this research work.
- Kilbas HM, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, The Netherlands; 2007.MATHGoogle Scholar
- Bagley RL, Calico RA: Fractional order state equations for the control of viscoelastically damped structures. Journal of Guidance, Control, and Dynamics 1991,14(2):304–311. 10.2514/3.20641View ArticleGoogle Scholar
- Mahmood A, Parveen S, Ara A, Khan NA: Exact analytic solutions for the unsteady flow of a non-Newtonian fluid between two cylinders with fractional derivative model. Communications in Nonlinear Science and Numerical Simulation 2009,14(8):3309–3319. 10.1016/j.cnsns.2009.01.017View ArticleMATHGoogle Scholar
- Mahmood A, Fetecau C, Khan NA, Jamil M: Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders. Acta Mechanica Sinica 2010,26(4):541–550. 10.1007/s10409-010-0353-4MathSciNetView ArticleMATHGoogle Scholar
- He JH: Nonlinear oscillation with fractional derivative and its applications. Proceedings of the International Conference on Vibrating Engineering, 1998, Dalian, China 288–291.Google Scholar
- Khan NA, Khan N-U, Ara A, Jamil M: Approximate analytical solutions of fractional reaction-diffusion equations. Journal of King Saud University—Science. In pressGoogle Scholar
- He J-H: Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering 1999,178(3–4):257–262. 10.1016/S0045-7825(99)00018-3MathSciNetView ArticleMATHGoogle Scholar
- He J-H: A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics 2000,35(1):37–43. 10.1016/S0020-7462(98)00085-7MathSciNetView ArticleMATHGoogle Scholar
- He J-H: Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and Computation 2003,135(1):73–79. 10.1016/S0096-3003(01)00312-5MathSciNetView ArticleMATHGoogle Scholar
- Khan NA, Ara A, Mahmood A: Approximate solution of time-fractional chemical engineering equations: a comparative study. International Journal of Chemical Reactor Engineering 2010., 8, article A19: Google Scholar
- Khan NA, Khan N-U, Ayaz M, Mahmood A: Analytical methods for solving the time-fractional Swift-Hohenberg (S-H) equation. Computers and Mathematics with Applications. In pressGoogle Scholar
- Khan NA, Ara A, Ali SA, Jamil M: Orthognal flow impinging on a wall with suction or blowing. International Journal of Chemical Reactor Engineering. In pressGoogle Scholar
- Yıldırım A: Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method. Computers & Mathematics with Applications 2008,56(12):3175–3180. 10.1016/j.camwa.2008.07.020MathSciNetView ArticleMATHGoogle Scholar
- Koçak H, Öziş T, Yıldırım A: Homotopy perturbation method for the nonlinear dispersive K(m,n,1) equations with fractional time derivatives. International Journal of Numerical Methods for Heat & Fluid Flow 2010,20(2):174–185. 10.1108/09615531011016948MathSciNetView ArticleMATHGoogle Scholar
- Khan Y, Faraz N, Yildirim A, Wu Q: A series solution of the long porous slider. Tribology Transactions 2011,54(2):187–191. 10.1080/10402004.2010.533818View ArticleGoogle Scholar
- Khan NA, Ara A, Ali SA, Mahmood A: Analytical study of Navier-Stokes equation with fractional orders using He's homotopy perturbation and variational iteration methods. International Journal of Nonlinear Sciences and Numerical Simulation 2009,10(9):1127–1134. 10.1515/IJNSNS.2009.10.9.1127Google Scholar
- Aminikhah H, Biazar J: A new HPM for ordinary differential equations. Numerical Methods for Partial Differential Equations 2010,26(2):480–489.MathSciNetMATHGoogle Scholar
- Aminikhah H, Hemmatnezhad M: An efficient method for quadratic Riccati differential equation. Communications in Nonlinear Science and Numerical Simulation 2010,15(4):835–839. 10.1016/j.cnsns.2009.05.009MathSciNetView ArticleMATHGoogle Scholar
- Khan Y, Faraz N: Modified fractional decomposition method having integral w.r.t . Journal of King Saud University—Science. In pressGoogle Scholar
- Bataineh AS, Noorani MSM, Hashim I: Solving systems of ODEs by homotopy analysis method. Communications in Nonlinear Science and Numerical Simulation 2008,13(10):2060–2070. 10.1016/j.cnsns.2007.05.026MathSciNetView ArticleMATHGoogle Scholar
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