Bernstein operational matrices for solving multiterm variable order fractional differential equations

Transcription

1 International Journal of Current Engineering and Technology E-ISSN P-ISSN INPRESSCO All Rights Reserved Available at Research Article Bernstein operational matrices for solving multiterm variable order fractional differential equations Alshaimaa A. Omar #* and Osama H. Mohammed # College of Science Al-nahrain University Baghdad Iraq Accepted 02 Jan 2017 Available online 04 Jan 2017 Vol.7 No.1 (Feb 2017) Abstract In this paper we use Bernstein polynomials to solve multiterm variable order fractional differential equations. The main idea of this paper is that we use Bernstein polynomials and operational matrices to solve such types of equations. The equation is transformed into the products of several dependent matrices which can also be viewed as an algebraic system by making use of the collocation points. By solving the algebraic system the numerical solutions will then be obtained. Finally numerical examples are presented to demonstrate the accuracy of the proposed method. Keywords: Variable order caputo fractional derivatives Bernstein polynomials operational matrices of variable fractional order derivative of Bernstein polynomials 1. Introduction 1 In recent years fractional calculus has attracted many researchers successfully in different disciplines of science and engineering. One of the main advantages of the fractional calculus is that the fractional derivatives provide a superior approach for the description of memory and hereditary properties of various materials and processes [Leda2007]. Many numerical methods using different kinds of fractional derivative operators for solving different types of fractional differential equations have been proposed. The most commonly used ones are Adomian decomposition method (ADM) [El-Kalla2011; Ahmed1998] generalized differential transform method (GDTM) [Shaher2007; Zaid2008; Shaher2008] wavelet method [Li Zhu2012; Mingxu2012] finite difference method [Deng2013] Laplace transform method [Gupta2014] variational iteration method [Odibat2006] fractional differential transform method [Arikoglu2009] operational approach [Luchko1999; Li2013; Bengochea2014] and other methods [Zaid2009; Kai2004; Javidi2009]. Moreover orthogonal functions also play an important role in finding numerical solutions for FDEs such as Block pulse functions [Yi2013] Bernstein polynomials [Saadatmandi2014] shifted Legendre polynomials [Khalil2014] Chebyshev wavelets [Zhu2012] Legendre wavelets [Heydari2014] etc. *Corresponding author: Alshaimaa A. Omar Recently the concepts of fractional derivatives of variable order have also been introduced and some research works of the relative practical applications have arisen [Sun2014; Sun2010; Sun2009]. Due to the fractional order exponents in differential operators analytical solutions of FDEs are usually difficult to obtain. Consequently different methods have been developed to give numerical solutions for FDEs such as viscoelasticity and damping diffusion and wave propagation and chaos [Rossikhinand1997; Sun; Chen2009; Chen2010; Chen2006]. In this paper a numerical solution of multiterm variable order fractional differential equations using Bernstein operational matrices will be represented. Bernstein polynomials play a prominent role in various areas of mathematics. These polynomials have frequently been used in the solution of integral equations differential equations and approximation theory. In recent years various operational matrices for the polynomials have been developed to cover the numerical solution of differential integral and integrodifferential equations. The operational matrices for Bernstein polynomials are introduced in order to solve different types of differential equations among them [Maleknejad2011] used the operational matrices for Bernstein polynomials to solve nonlinear Volterra Fredholm Hammerstein integral equations [Maleknejad2013] have been used operational matrices for solving physiology problems [Bataineh2016] have been used operational matrices for solving high order delay differential equations etc. This paper is organized as follows: 68 International Journal of Current Engineering and Technology Vol.7 No.1 (Feb 2017)

2 In section two the basic definitions and properties of the variable order fractional integrals and derivatives are given in section three we present Bernstein polynomials and it is operational matrices in section four main result of this paper is represented some illustrative examples are given in section five finally conclusions have been drawn in section six. 2. Basic definitions of the variable order fractional integrals and derivatives [Loreno2007] In this section some definitions of fractional integration and derivatives of variable order are listed. Definition (1) Riemann-Liouville fractional integral of the first kind of order By using the binomial expansion of equation (6) can be expressed as: ( Now we define Or in matrix form ) ( )( ) (7) ( ) ( )( ) ( )( ) ( )( ) ( )( ) (8) (9) (10) [ ( ) ] (1) [ And ( ) ] Definition (2) Riemann-Liouville fractional derivative of the first kind with order Definition (3) Caputo s fractional derivative with order α(t) ( ) (2) (3) 0 < 1. If we assume the starting time in a perfect situation we can get the definition as follows: Clearly (11) (12) 3.1 Function approximation using Bernstein polynomials A function can be expressed in terms of the Bernstein polynomials basis. In practice only the first (n+1) terms of Bernstein polynomials are considered. Hence Then we have (13) (4) With the definition above we can get the following formula : H is an the dual matrix of formula: (14) matrix which is called and it is given by the following { (5) 3. Bernstein polynomials and it s operational matrices [Jinsheng2014] ( )( ) ( ) Q is a Hilbert matrix given by (15) The Bernstein Polynomials of degree n are defined by Q= (16) ( ) (6) 69 International Journal of Current Engineering and Technology Vol.7 No.1 (Feb 2017)

3 3.2 Operational matrices of and Based on Bernstein polynomials Consider (17) and are fractional derivative in the caputo sense. When and are all constants (24) becomes (25); namely According to (5) we can get (25) =A (18) In order to solve equation (24) we define the approximate solution of equation (24) as (26) = Also recall that and M= (19) (27) ( ) (28) is called the operational matrix of. Hence Therefore ( ) Similarly we can get (20) (21) (22) Substituting equations (26) (27) and (28) into equation (24) therefore equation (24) will be transformed into the following form and as follows (29) By taking the collection points equation (29) becomes an algebraic system of equations in terms of the unknown vector c as follows (30) Thus is called the operational matrix of. ( ) (23) The vector can be found by solving the resulting algebraic system of equations. Finally the numerical solution is obtained by equation (26). 5. Illustrative examples 4. Main result The multiterm variable order linear fractional differential equation is given as follows: Example 1 Consider the following linear multiterm variable order fractional differential equation (24) (31) 70 International Journal of Current Engineering and Technology Vol.7 No.1 (Feb 2017)

4 2 ( ) ( ) ( ) 1.8 The analytic solution is given by For n=3 (32) ex t i u t i Figure 1: Analytical and numerical solution of example1 t i And Example 2 Consider the following linear variable order fractional differential equation (35) By applying the proposed method equation (31) can be transformed into the following form By taking (33) we get a linear system of algebraic equations with unknown c as follows (34) The analytic solution is given by For n=3 And (36) Solving the obtained system one can get the unknown and hence the approximate solution of equation (31). Following table 1 represent the approximate solution of example (1) using the proposed method compared with the method [Liu2016] and the exact solution. Table 1 Comparison between the exact solution with the proposed method and method [Liu2016] t Exact The proposed Method solution method [Liu2016] By applying the proposed method equation (35) can be transformed into the following form By taking (37) we get a linear system of algebraic equations with unknown c as follows (38) Following figure 1 represent the analytical solution and the numerical solution of equation (31). Solving the obtained system one can get the unknown and hence the approximate solution of equation (35). 71 International Journal of Current Engineering and Technology Vol.7 No.1 (Feb 2017)

5 Following table 2 represent the approximate solution of example (2) using the proposed method compared with the method [Liu2016] and the exact solution. Table 2 Comparison between the exact solution with the proposed method and method [Liu2016] t Exact The proposed Method solution method [Liu2016] Following figure 2 represent the analytical solution and the numerical solution of equation (35). ex t i u t i Figure 2: Analytical and numerical solution of example2 Conclusion In this paper the Bernstein Polynomials method and its operational matrices is used to solve multiterm variable order linear fractional differential equation. This technology reduces the original equation into a system of algebraic equations which greatly simplifies the problem. In order to confirm the efficiency of the proposed techniques two numerical examples are implemented. By comparing the numerical solution with the analytical solution and that of other methods in the literature [Liu2016] we demonstrate the high accuracy and efficiency of the proposed method. References Leda Galue S.L. Kalla B.N. Al-Saqabi (2007) Fractional extensions of the temperature field problems in oil strata Applied Mathematics and Computation Elsevier vol. 186 no.1 pp I. L. EI-Kalla (2011) Error estimate of the series solution to a class of nonlinear fractional differential equations Commun. Nonlinear Sci. Numer. Simulate Elsevier vol. 16 no.3 pp Ahmed. M. A. EI-Sayed (1998) Nonlinear functional differential equations of arbitrary orders Nonliear Analysis: Theory Methods & Applications Elsevier vol. 33 no.2 pp Shaher Momani Zaid. Odibat (2007) Generalized differential transform method for solving a space and time-fractional diffusion-wave equation Physics Letters A Elsevier vol.370 no. 5-6 pp t i Zaid. Odibat Shaher Momani (2008) A generalized differential transform method for linear partial differential equations of fractional order Applied Mathematics Letters Elsevier vol. 21 no. 2 pp Shaher Momani Zaid. Odibat (2008) Generalized differential transform method: Application to differential equations of fractional order Applied Mathematics and Computation Elsevier vol. 197 no.2 pp Li Zhu Qibin Fan (2012) Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet Communications in Nonlinear Science and Numerical Simulatation Elsevier vol. 17 no.6 pp Mingxu Yi Yiming Chen (2012) Haar wavelet operational matrix method for solving fractional partial differential equations Computer Modeling in Engineering &Sciences Tech Science vol. 88 no.3 pp W.H. Deng S.D. Du Y.J. W (2013) High order finite difference WENO schemes for fractional differential equations Applied Mathematics Letters. 26(2013) S. Gupta D. Kumar J. Singh (2014) Numerical study for systems of fractional differential equations via Laplace transforms Journal of the Egyptian Mathematical Society doi: /j.joems Z. Odibat S. Momani (2006) Application of variational iteration method to nonlinear differential equations of fractional order Int. J. Nonlinear Sci. Numer. Simul. 7 (2006) A. Arikoglu I. Ozkol (2009) Solution of fractional integrodifferential equations by using fractional differential transform method Chaos Solitons Fractals. 40 (2009) Y. Luchko R. Gorenflo (1999) An operational method for solving fractional differential equations with the Caputo derivatives Acta Mathematica Vietnamica. 24(2) (1999) M. Li W. Zhao (2013) Solving Abel s type integral equation with Mikusinski s operator of fractional order Advances in Mathematics Physics. Vol Article ID pages. G. Bengochea (2014) Operational solution of fractional differential equations Applied Mathematics Letters. 32 (2014) Zaid. Odibat Shaher Momani (2009) An algorithm for the numerical solution of differential equations of fractional order J. Appl. Math. Inform Elsevier vol. 26 no. 1-2 pp Kai Diethelm Neville J. Ford (2004) Multi-order fractional differential equations and their numerical solution Applied Mathematics and Computation Elsevier vol. 154 no.3 pp M. Javidi A. Golbabai (2009) Modified homotopy perturbation method for solving system of linear Fredholm integral equations Mathematical and Computer Modelling Elsevier vol. 50 no. 1-2 pp M.X. Yi J. Huang J.X. Wei (2013) Block pulse operational matrix method for solving fractional partial differential equation Applied Mathematics and Computation. 221 (2013) A. Saadatmandi (2014) Bernstein operational matrix of fractional derivatives and its applications Applied Mathematical Modeling. 38 (2014) H. Khalil R. A. Khan (2014) A new method based on Legendre polynomials for solutions of the fractional twodimensional heat conduction equation Computes & Mathematics with Applications. 67 (2014) L. Zhu Q.B. Fan (2012) Solving fractional nonlinear Fredholm integro-differential equations by the second kind 72 International Journal of Current Engineering and Technology Vol.7 No.1 (Feb 2017)

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