The construction of operational matrix of fractional derivatives using B-spline functions

Abstract

Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. Here we construct the operational matrix of fractional derivative of order α in the Caputo sense using the linear B-spline functions. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus we can solve directly the problem. The method is applied to solve two types of fractional differential equations, linear and nonlinear. Illustrative examples are included to demonstrate the validity and applicability of the new technique presented in the current paper.

Introduction

Fractional differential operators have a long history, having been mentioned by Leibniz in a letter to L’Hospital in 1695. A history of the development of fractional differential operators can be found in [26], [28], [32]. One of the most recent works on the subject of fractional calculus, i.e. the theory of derivatives and integrals of fractional (non-integer) order, is the book of Podlubny [33], which deals principally with fractional differential equations. Today, there are many works on fractional calculus (see for example [4], [20]). Also the interested reader can refer to the interesting and comprehensive recent work [42] for the history of fractional calculus.

In the present paper we intend to extend the application of the linear B-spline functions to solve the fractional differential equations. Our main aim is to generalize the B-spline operational matrix to fractional calculus. It is worthy to mention here that, the method based on using the operational matrix of a cardinal function for solving differential equations is computer oriented.

The aim of this work is to present a numerical method for approximating the solution of multi-order fractional differential as:

$$F\left(y(x)\text{,}{D}^{{\beta }_1}y(x)\text{,}\dots \text{,}{D}^{{\beta }_m}y(x)\right)=g(x)\text{,}$$

with boundary or supplementary conditions

$$H_i(y({\xi }_i)\text{,}{y}^{\prime }({\xi }_i)\text{,}\dots \text{,}{y}^{(n)}({\xi }_i))=d_i\text{,}i=0\text{,}1\text{,}\dots \text{,}n\text{,}$$

where F is a multivariable function and g is known function, ξ_i ∈ [0, 1], i = 0,1, … , n, 0 ⩽ n < max{β_i, i = 1, 2, … , m} ⩽ n + 1, H_i are linear combinations of y(x), y′(x), … , y⁽ⁿ⁾(x), and y(x), g(x) ∈ L²[0, 1] and $D^{{\beta }_i}$ denotes the Caputo fractional derivative of order β_i and fractional partial differential equation (FPDE) as

$$a_1\frac{{\partial }^{\alpha }y}{\partial x^{\alpha }}+a_2\frac{{\partial }^{\beta }y}{\partial t^{\beta }}=f(x\text{,}t)$$

with the boundary conditions

$$\begin{array}{cc}\hfill & y(0\text{,}t)=g_0(t)\text{,}\hfill \\ \hfill & y(x\text{,}0)=g_1(x)\text{,}\hfill \end{array}$$

where a₁ and a₂ are given constant, y(x, t) ∈ L²([0, 1] × [0, 1]), 0 < α ⩽ 1 and 0 < β ⩽ 1.

There are several techniques for solving such equations like Adomian decomposition method [14], [27], [30], [35], [43], He’s variational iteration method [17], [29], [31], homotopy perturbation method [29], [40], homotopy analysis method [16], collocation method [34], Galerkin method [12] and other methods [21], [24]. Also author of [46] has developed the finite difference methods [5] for solving the fractional diffusion equations. Most of the methods have been utilized in linear problems and a few number of works have considered nonlinear problems. Also we refer the interested reader to [8], [9], [39], [41] for some basic ideas in Adomian decomposition method, the variational iteration method, homotopy perturbation method and the homotopy analysis method, respectively.

In this paper, we introduce a new operational method to solve multi-order fractional differential and partial differential equations. The method is based on reducing the equation to a system of algebraic equations by expanding the solution as linear B-spline functions with unknown coefficients. The main characteristic of an operational method is to convert a differential equation into an algebraic one. It not only simplifies the problem but also speeds up the computation. It is considerable that, for $\alpha \text{,}\beta \text{,}{\beta }_i\in \mathbb{N}\text{,}i=1\text{,}\dots \text{,}m$, Eqs. (1.1), (1.2), (1.3), (1.4) are multi-order differential and partial differential equations and the method can be easily applied for them.

Additionally, in the present work, the fractional derivatives are considered in the Caputo sense. The reason for adopting the Caputo definition, as pointed by Momani and Noor [27], is as follows: to solve differential equations (both classical and fractional), we need to specify additional conditions in order to produce a unique solution. For the case of the Caputo fractional differential equations, these additional conditions are just the traditional conditions, which are akin to those of classical differential equations, and are therefore familiar to us. In contrast, for the Riemann–Liouville fractional differential equations, these additional conditions constitute certain fractional derivatives (and/or integrals) of the unknown solution at the initial point x = 0, which are functions of x. These initial conditions are not physical; furthermore, it is not clear how such quantities are to be measured from experiment, say, so that they can be appropriately assigned in an analysis. For more details see Appendix A and [33]. We also refer the interested reader to [6], [7], [10], [13], [18], [19], [25], [37], [38], [45].

This paper is organized as follows:

In Section 2, the formulation of the linear B-spline scaling functions on [0, 1] is given, and then we derive the operational matrix of fractional-order derivative required for our subsequent development. In Sections 3 Numerical solution of multi-order fractional differential equations, 4 Numerical solution of the fractional partial differential equations the proposed method is used to approximate the multi-order fractional differential equations and fractional partial differential equations respectively. As a result a set of algebraic equations is formed and the solution of the considered problem is introduced. Numerical simulations are reported in Section 5. In Section 6, we give a brief conclusion. An Appendix A is given which consists some definitions of fractional derivatives. Note that we have computed the numerical results by Maple programming.