Using the Euler's method to solve ordinary differential equations of higher order with a mixture of integer And Caputo derivatives

In this paper we present an application of the Euler's method to the numerical solution of fractional ordinary differential equations. These equations include both a classical differential operator of integer order and the fractional one defined in the Caputo sense. Our previous work was limited to the order of fractional derivative ) 0,1 α ∈ . This study considers numerical schemes for higher orders of a fractional derivative. We then compare our schemes with analytical solutions in order to show their good numerical precision. Introduction A contemporary mathematical modeling uses fractional ordinary differential equations as an alternative approach to the ordinary differential equations of integer order. Here we start with a class of ordinary differential equations defined as: ( ) ( ) ( ) ( ) ( ) ( ) 1 1 , , ,. , , ,. , 0 m p f x y x D y x D y x D y x D y x α α = (1) where ( ) y x is a continuous function being a solution of above equation, ( ) ( ) 1 ,. , p D y x D y x are derivatives of integer order and ( ) ( ) 1 ,. , m D y x D y x α α are fractional derivatives of real orders 1,. , m R α α ∈ . It should be noted that an analytical solution of such equation is limited to its linear form and includes some special functions. However, numerical techniques presented in literature [1-5] have also many disadvantages, i.e. introduction of the initial conditions included in the Riemann-Liouville derivative [8], the unreasonable assumption that a method applied to a single term equation is proper for solving a multi-term equation [3, 7] etc. In our work [8] we introduced a novel numerical technique in order to omit these disadvantages. However the technique has considered six types of equations in which the integer order belonged to the range { } 0,1,2 p∈ and the fractional one Please cite this article as: Tomasz Blaszczyk, Jacek S. Leszczynski, Using the Euler's method to solve ordinary differential equations of higher order with a mixture of integer and Caputo derivatives, Scientific Research of the Institute of Mathematics and Computer Science, 2007, Volume 6, Issue 1, pages 31-40. The website: http://www.amcm. pcz.pl/ T. Blaszczyk, J.S. Leszczynski 32 was ) 0,1 α ∈ . We also checked three numerical techniques applied to solution of Eqn. (1) the Adams method [9], the Gear's method [10] and the Euler's method [9] and four discrete forms of the left-side Caputo derivative. According to [11] we define such operator as: ( ) ( ) ( ) ( ) ( ) 0 0 1 1 n x C x x n x y D y x d n x α α τ τ α τ − + = Γ − − ∫ (2) where [ ] 1 n α = + and [⋅] indicates an integer part of a real number. In literature [2, 6, 12] one may find another discrete forms of the Caputo derivative (2). To extend our considerations we will focus on ordinary differential equations with a mixture of integer and fractional derivatives where the real order α of a fractional derivative is greater than ) 0,1 α ∈ . We then will show and test numerical algorithms adopted to fractional differential equations where real order of the fractional derivative satisfies the range ) 1,2 α ∈ . 1. Problem statement Here we focus on the following types of equations: • p n > for 3 p = , ) 1,2 α ∈ , 2 n = ( ) ( ) 0 3 0 C x x D y x D y x α λ + = (3) • p n = for 2 p = , ) 1,2 α ∈ , 2 n = ( ) ( ) 0 2 0 C x x D y x D y x α λ + = (4) It should be noted that the function ( ) y x being solution of such equations belongs to the class of continuous functions. To solve above equations numerically we use the Euler's method where a discrete form of the Caputo derivative (2) is proposed. In the next section we show a way how to discretize this derivative. 1.1. Analytical solution In order to compare numerical results we solve the above system of equations analytically. In literature [12-14] one may find a way how to generate the analytical solution for some types of fractional differential equations. Here we can apply the Laplace transform [15] that to solve the equations (3) and (4) respectivelly. Using the Euler's method to solve ordinary differential equations of higher order with . 33 Notice that the classical transform of any differential operator of integer order m N ∈ has the following form: ( ) ( ) ( ) 1 1 0 0 m m m k m k k D y x s F s s D y x − − + =   = −   ∑ (5) However the Laplace transform of the Caputo derivative (2) is: ( ) ( ) ( ) 0 1 1 0 0 n C k k x x k D y x s F s s D y x α α α − − − =   = −   ∑ (6) Assuming initial conditions as: ( ) ( ) ( ) 1 2 0 0 0 0 0 0 , ' , " y x y D y x y D y x y = = = (7) and using the Laplace transform and retransforming results we obtain analytical solution of Eqn. (3) in the following form: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 2 3 0 0 0 0 0 3 ,3 0