A numerical method for solving nonlinear Fredholm integral equations of second kind is proposed. The Fredholm-type equations, which have many applications in mathematical physics, are then considered. The method is based upon hybrid function approximate. The properties of hybrid of block-pulse functions and Chebyshev series are presented and are utilized to reduce the computation of nonlinear Fredholm integral equations to a system of nonlinear. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method.

Over the last years, the fractional calculus has been used increasingly in different areas of applied science. This tendency could be explained by the deduction of knowledge models which describe real physical phenomena. In fact, the fractional derivative has been proved reliable to emphasize the long memory character in some physical domains especially with the diffusion principle. For example, the nonlinear oscillation of earthquake can be modeled with fractional derivatives, and the fluid-dynamic traffic model with fractional derivatives can eliminate the deficiency arising from the assumption of continuum traffic flow [

In this paper, we study the numerical solution of a nonlinear fractional integrodifferential equation of the second:

During the last decades, several methods have been used to solve fractional differential equations, fractional partial differential equations, fractional integrodifferential equations, and dynamic systems containing fractional derivatives, such as Adomian’s decomposition method [

The paper is organized as follows: in Section

We give some basic definitions and properties of the fractional calculus theory, which are used further in this paper.

The Riemann-Liouville fractional integral operator of order

The Caputo definition of fractal derivative operator is given by

Hybrid functions

A function

If

The integration of the vector

Our purpose is to derive the hybrid functions operational matrix of the fractional integration. For this purpose, we consider an

Similarly, hybrid function may be expanded into an

In [

Next, we derive the hybrid function operational matrix of the fractional integration. Let

Using (

The following property of the product of two hybrid function vectors will also be used.

Let

Consider (

We can easily verify the accuracy of the method. Given that the truncated hybrid function in (

In this section, we applied the method presented in this paper for solving integral equation of the form (

Let us first consider fractional nonlinear integro-differential equation:

The numerical results for

The approximate solution of Example

As the second example considers the following fractional nonlinear integro-differential equation:

Absolute error for

0.1 | 5.1985e-003 | 1.5106e-004 | 2.8496e-006 |

0.2 | 1.1372e-003 | 2.4887e-004 | 3.9120e-006 |

0.3 | 7.4698e-004 | 3.2711e-004 | 4.6808e-006 |

0.4 | 1.2729e-003 | 7.0337e-005 | 3.1231e-006 |

0.5 | 4.6736e-003 | 4.3451e-004 | 3.2653e-006 |

0.6 | 1.2160e-003 | 2.5000e-006 | 2.6369e-006 |

0.7 | 6.0767e-004 | 6.2935e-005 | 4.7123e-007 |

0.8 | 6.0442e-004 | 3.2421e-004 | 4.8631e-006 |

0.9 | 1.2039e-003 | 6.2276e-005 | 2.0707e-006 |

Exact and numerical solutions of Example

Absolute error of Example

We have solved the nonlinear Fredholm integro-differential equations of fractional order by using hybrid of block-pulse functions and Chebyshev polynomials. The properties of hybrid of block-pulse functions and Chebyshev polynomials are used to reduce the equation to the solution of nonlinear algebraic equations. Illustrative examples are given to demonstrate the validity and applicability of the proposed method. The advantages of hybrid functions are that the values of

The method can be extended and applied to the system of nonlinear integral equations, linear and nonlinear integro-differential equations, but some modifications are required.

The authors are grateful to the reviewers for their comments as well as to the National Natural Science Foundation of China which provided support through Grant no. 40806011.