Cubic Spline Solution of Fractional Bagley-torvik Equation

Fractional calculus is a natural extension of the integer order calculus and recently, a large number of applied problems have been formulated on fractional di¤erential equations. Analytical solution of many applications, where the fractional di¤erential equations appear, cannot be established. Therefore, cubic polynomial spline function is considered to …nd approximate solution for fractional boundary value problems (FBPs). Convergence analysis of the method is considered. Some illustrative examples are presented and the obtained results reveal that the proposed technique is very e¤ective, convenient and quite accurate to such considered problems. 1. Introduction In the last few decades, many phenomena in science and engineering are described within the framework of the theory of fractional di¤erential equations. Boundary value problems of fractional order occur in the description of many physical processes of stochastic transport, the investigation of liquid …ltration in a strongly porous medium, cellular systems, di¤usion wave, control theory, signal processing and oil industries, [9]. In particular, the 1/2-order derivative or 3/2-order derivative describe the frequency-dependent damping materials quite satisfactorily, and the Bagley-Torvik equation with 1/2-order derivative or 3/2-order derivative describes motion of real physical systems, an immersed plate in a Newtonian ‡uid and a gas in a ‡uid, respectively. For details we may refer to ([1]-[5],[15],[17]-[18],[23][26]). It is these successful applications of fractional-order derivatives that draw the researchers attention to fractional calculus such as in [33], the authors considered the numerical solution of the fractional boundary value problem (FBVP) D y 00 (x)+p(x)y = g(x); 0 < 1; x 2 [a; b]; with Dirichlet boundary conditions using quadratic polynomial spline, also in [34] the authors used cubic polynomial spline function based method in combined with shooting method to …nd approximate solution of second order FBVP with Dirichlet boundary conditions, (see [6]-[8],[12]-[13],[15]-[16],[19]-[22],[27]-[29]). Key words and phrases. Cubic polynomial spline, Fractional boundary value problem, BagleyTorvik equation, Dirichlet boundary conditions, Error bound. Submitted Sept. 7, 2012. 230 EJMAA-2013/1(2) CUBIC SPLINE SOLUTION 231 In this paper, we consider the numerical solution of the following Bagley-Torvik equation: y 00 (x) + ( D + )y = f(x); m 1 < m; x 2 [a; b] (1) Subject to boundary conditions: y(a) A1 = y(b) A2 = 0 (2) where Ai(i = 1; 2); ; are real constants and m = 1 or 2 . The function f(x) is continuous on the interval [a; b] and the operator D represents the Caputo fractional derivative. When = 0 , Equation (1) is reduced to the classical second order boundary value problem. The main objective of this work is to use cubic polynomial spline function to establish a new numerical method for the FBVP (1-2). This approach has its own advantage that it not only provides continuous approximations to y(x) , but also y(x); j = 1; 2 for at every point of the range of integration ([25],[30]-[32]). This paper is organized as follows: In section 2, we introduce some de…nitions and theorem necessary to our work. Derivation of our method is established in section 3. Convergence analysis of the new method is presented in section 4. In section 5, we apply our method to singular boundary value problem of fractional order. In section 6, numerical results are included to show the applications and advantages of our method. 2. Preliminaries In this section, de…nitions of fractional derivative and integral, used in our work, will be presented. There are di¤erent de…nitions for fractional derivatives, the most commonly used ones are the Riemann-Liouville and the Caputo derivatives. Let f(x) be a function de…ned on (a; b), then De…nition 1 [16] The Riemann-Liouville fractional derivative D f(x) = 1 (m ) d dxm x Z 0 (x t) f(t)dt; > 0;m 1 < < m where is the gamma function. De…nition 2 [16]The Riemann-Liouville fractional integral D a f(x) = 1 ( ) x Z