During the last few decades, fractional differential equations (FDEs) have been applied to describe mathematical phenomenon in physics, chemistry, damping laws, rheology, control theory, signal processing, viscoelastic materials, polymers and so on [4, 10, 13, 17, 19]. Since most FDEs do not have exact analytic solutions, many researchers have tried to find solutions of FDEs using approximate and numerical techniques. For example see [3, 12, 16, 20–24]. Our aim in this work is the following type of multi-order FDE:
where n − 1 < α ≤ n, the coefficients yi (i = 1, …, k + 1) are constant, 0 < β1 < β2 < ... < βk < α and g is a known function. Moreover, Dα y(x) denotes the Caputo fractional derivative of order α. It is defined as [4, 7, 17]:
In (3) I(k − α) denotes the Riemann-Liouville fractional. It is generally defined as follows:
Here we list the few properties of these two operators as follow:
The Boubaker polynomials were established for the first time by Boubaker [1, 5, 6] as a guide for solving a one-dimensional heat transfer equation and second order differential equations. Kumar used these polynomials to solve Love’s equation in a particular physical system . We will generalize the operational matrix for fractional integration using Boubaker polynomials [1, 5, 6, 14].
In this study, we want to solve the multi-order FDE using the operational matrix for fractional integration based on Boubaker Polynomials. The aim of this approach is converting the multi-order FDE into a set of algebraic equations by expanding the unspecified function within Boubaker polynomials.
This work is organized into six sections. Section 2 deals with some properties of Boubaker polynomials. The operational matrix is computed for fractional order integration in Section 3. We convert the multi-order FDE to system of algebraic equations in Section 4. Following this we solve a selection of numerical examples in Section 5 by using the proposed technique. A brief conclusion is presented in Section 6.
2 Boubaker polynomials
The symbol ⌊ ⌋ denotes the floor function.
The Boubaker polynomials could be calculated by following recursive formula:
2.1 Approximation of function
The function f is approximated by Boubaker polynomials as following:
where B(x)T = [B0, B1, …, BN], Bi(x), i = 0, 1, 2, ···, N denote the Boubaker polynomials, CT = [c0, c1, …, cN] are unknown Boubaker cofficients and N is chosen as any positive integer.
Then CT can be obtained by
and 〈B(x), B(x)〉 is called the dual matrix of ϕ denoted by Q, and Q is obtained as:
By using the expression (7) and taking n = 0, …, N, we find the corresponding matrix relation as we can express Boubaker polynomials in terms of power basis functions
and if N is odd,
(default) and if N is even,
3 Operational matrix for fractional order integration
For a vector B(Mx), we can approximate the operational matrices of fractional order integration as:
where Pα is the (N + 1) × (N + 1) Riemann-Liouville fractional operational matrix of integration for Boubaker polynomials. We compute Pα as follows:
By substituting B(x) = ZX(x), we get:
where the matrix D(N + 1) × (N + 1) is given by
Now we approximate xk + α by N + 1 terms of the Boubaker basis
where Ei = [Ei, 0, Ei, 1, …, Ei, N] and
where E is an (N + 1) × (N + 1) matrix with Ei as its column. Therefore, we can write
Finally, we obtain
is called the operational matrix of fractional integration Boubaker polynomials.
4 Operational matrix for multi-order FDE
where ni − 1 < βi ≤ ni, ni ∈ N. This implies that
Now, we approximate y and g by Boubaker polynomials B(x) as follows:
where GT and CT are known and unknown vectors, respectively.
We also approximate the fractional order integrals by using (26) as follow:
Finally, we get:
Finally, by solving the above system of algebraic equations we find vector C can be obtained . Consequently y(x) can be approximated by (31).
Here, we use the presented numerical approach to solve several illustrative examples.
In Fig. 1, we plotted the exact solution and the approximate solutions of y for N = 3 and N = 5. Definitely, by increasing the value of N, the approximate value of y(x) will close to the exact values.
We applied Boubaker polynomials approach to solve (38) with N = 3. In this case we obtain y(x) = x2
The exact solution of (40) is:
We solved this equation by Boubaker polynomials with N = 3 and obtained the exact solution 1 + x.
We applied this method for N = 3, 4, 6, and the result is plotted in Fig. 2. It can be seen that by increasing the value of N, the approximate solution will converge to the exact values.
In this work we applied Boubaker polynomials for solving multi-order FDE. The Boubaker polynomials operational matrices of fractional integration was used. Illustrative examples were presented to show the applicability and validity of the approach.
Mathematica was used for computation in this paper.
Akkaya T., Yalcinbas S., Sezer M., Numerical solutions for the pantograph type delay differential equation using First Boubaker polynomials, Applied Mathematics and Computation, 2013, 219, 9484–9492. Google Scholar
Bagley R.L., Torvik P.J., Fractional calculus: a different approach to the analysis of viscoelastically damped structures, AIAA J. 1983, 21 (5), 741–748. Google Scholar
Bhrawy A.H., Taha T.M., An operational matrix of fractional integration of the Laguerre polynomials and its application on a semi-infinite interval, Mathematical Sciences, 2012, 6(41), 1–7. Google Scholar
Baleanu D., Diethelm K., Scalas E., Trujillo J.J., Fractional Calculus Models and Numerical Methods (Series on Complexity, Nonlinearity and Chaos, World Scientific, (2012). Google Scholar
Boubaker K., On modified Boubaker polynomials: some differential and analytical properties of the new polynomials issued from an attempt for solving bi-varied heat equation, Trends Appl. Sci. Res., 2007, 2 (6), 540–544. Google Scholar
Boubaker K., The Boubaker polynomials, a new function class for solving bi-varied second order differential equations. FEJ Appl. Math. 2008, 31 (3), 273–436. Google Scholar
Caputo M., Linear models of dissipation whose Q is almost frequency independent. Part II, J. Roy. Austral. Soc. 13 (1967) 529–539. Google Scholar
Daftardar-Gejji V., Jafari H., Solving a multi-order fractional differential equation using Adomian decomposition, Applied Mathematics and Computation, 2007, 189 (1), 541–548. Google Scholar
Ghanouchi J., Labiadh H., An Attempt to solve the heat transfer equation in a model of pyrolysis spray using 4q-order Boubaker polynomials. Int. J. Heat Technol., 2008, 26(1), 49–53.\newpageGoogle Scholar
Gorenflo R., Mainardi F., Scalas E., Raberto M., Fractional calculus and continuous-time finance III. The diffusion limit, in Mathematical Finance, Trends Math., 2001, 171-180. Google Scholar
Jafari H., Das S., Tajadodi H., Solving a multi-order fractional differential equation using homotopy analysis method, Journal of King Saud University-Science, 2011, 23, 151–155. Google Scholar
Jafari H., Numerical Solution of Time-Fractional Klein–Gordon Equation by Using the Decomposition Methods. ASME. J. Comput. Nonlinear Dynam. 2016;11(4):041015-041015-5. . 10.1115/1.4032767Google Scholar
Kilbas A.A., Srivastava H.H., Trujillo J.J., Theory and Applications of Fractional Differential Equations, (Elsevier, The Netherlands), (2006). Google Scholar
Kumar A.S., An analytical solution to applied mathematics-related Love’s equation using the Boubaker polynomials expansion scheme, Journal of the Franklin Institute, 2010, 347, 1755–1761. Google Scholar
Labiadh H., Boubaker K., A Sturm-Liouville shaped characteristic differential equation as aguide to establish a quasi-polynomial expression to the Boubaker polynomials, Diff.Eq.Cont.Proc.2(2007)117–133.Google Scholar
Li X., Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method, Commun Nonlinear Sci Numer simulat, 2012, 17, 3934–3946. Google Scholar
Podlubny I., Fractional Differential Equations, Academic press, New York, 1999. Google Scholar
Rostamy D., Alipour M., Jafari H., Baleanu D., Solving multi-term orders fractional differential equations by operational matrices of BPs with convergence analysis, Romanian Reports in Physics, 2013, 65 (2), 334–349.Google Scholar
Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives Theory and Applications, (Gordon and Breach, New York, 1993). Google Scholar
Saadatmandi A., Dehghan M., A new operational matrix for solving fractional-order differential equations, Computers and Mathematics with Applications, 2010, 59, 1326–1336. Google Scholar
Yousefi S.A., Behroozifar M., Operational matrices of Bernstein polynomials and their applications, International Journal of Systems Science, 2010, 41, 709–716.Google Scholar
Hafez Ramy M., Ezz-Eldien Samer S., Bhrawy Ali H., et al., A Jacobi Gauss-Lobatto and Gauss-Radau collocation algorithm for solving fractional Fokker-Planck equations, Nonlinear Dynamics, 2015, 82 (3), 1431–1440.Google Scholar
Hosseini V.R., Shivanian E., Chen W., Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping, Journal of Computational Physics, 2016, 312, 307–332.Google Scholar
Avazzadeh Z., Chen W., Hosseini V.R., The Coupling of RBF and FDM for Solving Higher Order Fractional Partial Differential Equations, Applied Mechanics and Materials, 2016, 598, 409–413. Google Scholar
About the article
Published Online: 2016-07-12
Published in Print: 2016-01-01
Citation Information: Open Physics, Volume 14, Issue 1, Pages 226–230, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2016-0028.
© 2016 A. Bolandtalat et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0