Numerical Solution of Nonlinear Fractional Volterra Integro-Differential Equations via Bernoulli Polynomials
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 162896, 7 pages
http://dx.doi.org/10.1155/2014/162896
Research Article
Numerical Solution of Nonlinear Fractional Volterra
Integro-Differential Equations via Bernoulli Polynomials
Emran Tohidi,1 M. M. Ezadkhah,2 and S. Shateyi3
1
Department of Mathematics, Aligoudarz Branch, Islamic Azad University, Aligoudarz, Iran
Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran
3
Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State,
Bloemfontein 9300, South Africa
2
Correspondence should be addressed to S. Shateyi;
Received 26 November 2013; Accepted 12 February 2014; Published 18 March 2014
Academic Editor: Hossein Jafari
Copyright © 2014 Emran Tohidi et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional
order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main problems to the
solution of algebraic equations systems by expanding the required approximate solutions as the linear combination of the Bernoulli
polynomials. Several examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method.
1. Introduction
In real world, for modeling and analysing a huge size of
problems we need fractional calculus. Fractional calculus
finds its application in many fields of sciences and engineering, including fluid flow, electrical networks, fractals theory,
control theory, electromagnetic theory, probability, statistics,
optics, potential theory, biology, chemistry, diffusion, and
viscoelasticity [1–4].
In recent years, fractional differential equations (FDEs)
and fractional integro-differential equations (FIDEs) have
become the focus of interest for many researchers in different
disciplines of science and technology because of the fact
that a realistic modeling of a physical phenomenon having
dependence not only on the time instant but also on the previous time history that can be successfully achieved by using
fractional calculus. However, besides modeling, the solution
techniques and their reliabilities are most important to catch
critical points at which a sudden divergence, convergence,
or bifurcation starts. Therefore, high accuracy solutions are
always needed. For this purpose several techniques were
proposed to solve the fractional order differential equations
(or integro-differential equations). The most commonly used
ideas are Adomian decomposition method (ADM) [5], variational iteration method (VIM) [6], fractional differential
transform method (FDTM) [7], fractional difference method
(FDM) [8], and power series method [9].
On the other hand, since the beginning of 1994, Laguerre,
Legendre, Taylor, Fourier, Hermite, and Bessel (matrix and
collocation) methods have been used in the works [10–15]
to solve linear differential, integral, and integro-differentialdifference equations and their systems. Also, the Bernoulli
(matrix and collocation) methods have been used to find the
approximate solutions of differential and integro-differential
equations [16–18]. To the best of our knowledge these polynomials have had no results for solving FIDEs. Moreover,
according to the discussions in [18], Bernoulli polynomials
have some certain properties that encourage us to use them
for solving any applied mathematics problem. These subjects
motivate us to present a new numerical scheme for solving
FIDEs.
In this paper, by using the Bernoulli polynomials as the
test functions and collocating the following FIDE (subject
to sufficient initial or boundary conditions) at the Legendre
2
Abstract and Applied Analysis
Gauss collocation points and also approximating the existing
integrals by the Gauss quadrature rule, we find the numerical
solution of the following FIDE:
𝑥
𝐷𝛼 𝑦 (𝑥) = 𝐹 (𝑥, 𝑦 (𝑥) , ∫ 𝐾 (𝑡, 𝑦 (𝑡)) 𝑑𝑡) ,
0
0 < 𝑥 < 1,
(1)
𝛼 > 0.
The rest of this paper is organized as follows. Some
preliminaries about the fractional calculus and also the
Bernoulli polynomials together with the Gauss quadrature
rule are provided in the next Section. Section 3 contains
the basic idea of the paper. In Section 4, several numerical
examples are given to show the robustness of the proposed
idea. The provided numerical examples show the efficiency
of the proposed idea with regard to some methods in the
literature. In the last section, we provide the conclusions.
operator of integer order. Some properties of the Caputo
fractional derivative, which are needed here, are as follows:
𝐷𝛼 𝐶 = 0,
(𝐶 is a constant)
0,
{
{
{
{
for 𝛽 ∈ N0 , 𝛽 < ⌈𝛼⌉ ,
{
{
𝐷𝛼 𝑥𝛽 = { Γ (𝛽 + 1) 𝛽−𝛼
{
𝑥 ,
{
{
{
{ Γ (𝛽 + 1 − 𝛼)
{ for 𝛽 ∈ N0 , 𝛽 ≥ ⌈𝛼⌉ or 𝛽 ∉ N, 𝛽 > ⌊𝛼⌋ ,
(5)
where the ceiling function ⌈𝛼⌉ denotes the smallest integer
greater than or equal to 𝛼 and the floor function ⌊𝛼⌋ denotes
the largest integer less than or equal to 𝛼.
Similar to the integer order differentiation, the Caputo
fractional differential operator is a linear operation; in other
words
𝐷𝛼 (𝜃𝑓 (𝑥) + 𝜆𝑔 (𝑥)) = 𝜃𝐷𝛼 𝑓 (𝑥) + 𝜆𝐷𝛼 𝑔 (𝑥) ,
2. Preliminaries
𝑛 − 1 < 𝛼 ≤ 𝑛,
In this section, we deal with several basic definitions and
properties of fractional calculus theory and also some useful
information about the Bernoulli polynomials together with
the Legendre Gauss quadrature rule which are further used
hereafter.
Definition 1. A real function 𝑓(𝑥), 𝑥 > 0, is said to be in the
space 𝐶𝜇 , 𝜇 ∈ R, if there exists a real number 𝑝, 𝑝 > 𝜇, such
that 𝑓(𝑥) = 𝑥𝑝 𝑓1 (𝑥), where 𝑓1 (𝑥) ∈ 𝐶[0, ∞), and it is said to
be in the space 𝐶𝜇𝑛 if and only if 𝑓(𝑛) ∈ 𝐶𝜇 , 𝑛 ∈ N0 = N ∪ {0}.
(2)
Definition 4. The Bernoulli polynomials play an important
role in different areas of mathematics, including number
theory and the theory of finite differences. The classical
Bernoulli polynomials 𝐵𝑛 (𝑥) are usually defined by means of
the following relations:
𝑑𝐵𝑛 (𝑥)
= 𝑛𝐵𝑛−1 (𝑥) ,
𝑑𝑥
1
∫ 𝐵𝑛 (𝑥) 𝑑𝑥 = 0,
0
𝑥
1
∫ (𝑥 − 𝑡)𝛼−1 𝑓 (𝑡) 𝑑𝑡,
Γ (𝛼) 0
(𝑛 ≥ 1) ,
(7)
Also the Bernoulli polynomials can be represented in the
form
𝑁
𝑛
𝐵𝑛 (𝑥) = ∑ ( ) 𝐵𝑛−𝑟 (0) 𝑥𝑟 .
𝑟
𝑟=0
(8)
Definition 5. The Legendre Gauss quadrature rule can be
defined as follows [11]:
𝛼 > 0,
(3)
1
∫ ℎ (𝑠) 𝑑𝑠 =
0
𝐽 𝑓 (𝑥) = 𝑓 (𝑥) .
0
(4)
1 1
1
∫ ℎ ( (𝑡 + 1)) 𝑑𝑡
2 −1
2
1
1𝑁
≈ ∑𝑤𝑖 ℎ ( (𝑡𝑖 + 1)) ,
2 𝑖=0
2
Definition 3. The fractional derivative of 𝑓(𝑥) in the Caputo
sense is defined as
𝐷𝛼 𝑓 (𝑥) = 𝐽𝑛−𝛼 𝑓(𝑛) (𝑥) ,
(𝑛 ≥ 1) ,
𝐵0 (𝑥) = 1.
Definition 2. The Riemann-Liouville fractional integral operator of order 𝛼 for a function in 𝐶𝜇 , where 𝜇 ≥ −1, is defined
as
𝐽𝛼 𝑓 (𝑥) =
(6)
where 𝜃 and 𝜆 are constants.
Clearly, 𝐶𝜇 is a vector space and the s (...truncated)
