Numerical Solution of Nonlinear Fredholm Integrodifferential Equations of Fractional Order by Using Hybrid of Block-Pulse Functions and Chebyshev Polynomials

Abstract

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.

1. Introduction

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 [1]. In the fields of physics and chemistry, fractional derivatives and integrals are presently associated with the application of fractals in the modeling of electrochemical reactions, irreversibility, and electromagnetism [2], heat conduction in materials with memory, and radiation problems. Many mathematical formulations of mentioned phenomena contain nonlinear integrodifferential equations with fractional order. Nonlinear phenomena are also of fundamental importance in various fields of science and engineering. The nonlinear models of real-life problems are still difficult to be solved either numerically or theoretically. There has recently been much attention devoted to the search for better and more efficient solution methods for determining a solution, approximate or exact, analytical or numerical, to nonlinear models [3–5].

In this paper, we study the numerical solution of a nonlinear fractional integrodifferential equation of the second:𝐷^𝛼𝑓(𝑥)−𝜆¹⁰𝑘(𝑥,𝑡)[𝑓(𝑡)]^𝑚𝑑𝑡=𝑔(𝑥),𝑚>1,(1.1) with the initial condition𝑓^(𝑖)(0)=𝛿_𝑖,𝑖=0,1,…,𝑟−1,𝑟−1<𝛼≤𝑟,𝑟∈𝑁(1.2) by hybrid of block-pulse functions and Chebyshev polynomials. Here, 𝑔∈𝐿²([0,1)),𝑘∈𝐿²([0,1)²)are known functions; 𝑓(𝑥) is unknown function. 𝐷^𝛼 is the Caputo fractional differentiation operator and mis a positive integer.

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 [6–11], He’s variational iteration method [12–14], homotopy perturbation method [15, 16], homotopy analysis method [3], collocation method [17], Galerkin method [18], and other methods [19–21]. But few papers reported application of hybrid function to solve the nonlinear fractional integro-differential equations.

The paper is organized as follows: in Section 2, we introduce the basic definitions and properties of the fractional calculus theory. In Section 3, we describe the basic formulation of hybrid block-pulse function and Chebyshev polynomials required for our subsequent. Section 4 is devoted to the solution of (1.1) by using hybrid functions. In Section 5, we report our numerical finding and demonstrate the accuracy of the proposed scheme by considering numerical examples.

2. Basic Definitions

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

Definition 2.1. The Riemann-Liouville fractional integral operator of order 𝛼≥0 is defined as [22] 𝐽^𝛼𝑓(𝑥)=1Γ(𝛼)^𝑥0𝐽(𝑥−𝑡)^𝛼−1𝑓(𝑡)𝑑𝑡,𝛼>0,𝑥>0,₀𝑓(𝑥)=𝑓(𝑥).(2.1) It has the following properties: 𝐽^𝛼𝑥^𝛾=Γ(𝛾+1)Γ(𝛼+𝛾+1)𝑥^𝛼+𝛾,𝛾>−1.(2.2)

Definition 2.2. The Caputo definition of fractal derivative operator is given by 𝐷^𝛼𝑓(𝑥)=𝐽^𝑚−𝛼𝐷^𝑚𝑓(𝑥)=1Γ(𝑚−𝛼)^𝑥0(𝑥−𝑡)^{𝑚−𝛼−1}𝑓^(𝑚)(𝑡)𝑑𝑡,(2.3) where𝑚−1≤𝛼≤𝑚,𝑚∈𝑁,𝑥>0. It has the following two basic properties: 𝐽𝐷^𝛼𝐽^𝛼𝑓(𝑥)=𝑓(𝑥),_𝛼𝐷^𝛼𝑓(𝑥)=𝑓(𝑥)−^𝑚−1_𝑘=0𝑓^(𝑘)0⁺𝑥^𝑘𝑘!,𝑥>0.(2.4)

3. Properties of Hybrid Functions

3.1. Hybrid Functions of Block-Pulse and Chebyshev Polynomials

Hybrid functions ℎ_𝑛𝑚(𝑥),𝑛=1,2,…,𝑁,𝑚=0,1,2,…,𝑀−1,are defined on the interval [0,1) asℎ_𝑛𝑚(𝑥)=𝑇^𝑚(2𝑁𝑥−2𝑛+1),𝑥∈𝑛−10,otherwise𝑁,𝑁𝑛(3.1) and 𝜔_𝑛(𝑡)=𝜔(2𝑁𝑡−2𝑛+1), where 𝑛 and 𝑚are the orders of block-pulse functions and Chebyshev polynomials.

3.2. Function Approximation

A function 𝑦(𝑥) defined over the interval 0 to 1 may be expanded as𝑦(𝑥)=^∞_∞𝑛=1_𝑚=0𝑐_𝑛𝑚ℎ_𝑛𝑚(𝑥),(3.2) where 𝑐_𝑛𝑚=𝑦(𝑥),ℎ_𝑛𝑚(𝑥),(3.3) in which (⋅,⋅)denotes the inner product.

If 𝑦(𝑥) in (3.2) is truncated, then (3.2) can be written as𝑦(𝑥)=^𝑁_{𝑛=1𝑀−1𝑚=0}𝑐_𝑛𝑚ℎ_𝑛𝑚(𝑥)=𝐶^𝑇𝐻(𝑥)=𝐻^𝑇(𝑥)𝐶,(3.4) where 𝐶and 𝐻(𝑥), given by𝐻(𝑥)=ℎ₁₀𝐶=𝑐₁₀,𝑐₁₁,…,𝑐_1𝑀−1,𝑐₂₀,…,𝑐_2𝑀−1,…𝑐_𝑁0,…,𝑐_𝑁𝑀−1^𝑇,(3.5)(𝑥),ℎ₁₁(𝑥),…,ℎ_1𝑀−1(𝑥),ℎ₂₀(𝑥),…,ℎ_2𝑀−1(𝑥),…ℎ_𝑁0(𝑥),…,ℎ_𝑁𝑀−1(𝑥)^𝑇.(3.6) In (3.4) and (3.5), 𝑐_𝑛𝑚,𝑛=1,2,…,𝑁,𝑚=0,1,…,𝑀−1,are the coefficients expansions of the function 𝑦(𝑥) and ℎ_𝑛𝑚(𝑥),𝑛=1,2,…,𝑁,𝑚=0,1,…,𝑀−1,are defined in (3.1).

3.3. Operational Matrix of the Fractional Integration

The integration of the vector 𝐻(𝑥)defined in (3.6) can be obtained as ^𝑥0𝐻(𝑡)𝑑𝑡≈𝑃𝐻(𝑥),(3.7) see [23], where 𝑃is the 𝑀𝑁×𝑀𝑁operational matrix for integration.

Our purpose is to derive the hybrid functions operational matrix of the fractional integration. For this purpose, we consider an m-set of block pulse function as𝑏_𝑛(𝑥)=⎧⎪⎨⎪⎩10,,𝑚𝑖o≤th𝑡er≤wi𝑖s+𝑚e,1,𝑖=0,1,2,…,𝑚−1.(3.8) The functions 𝑏_𝑖(𝑥) are disjoint and orthogonal. That is,𝑏_𝑖(𝑥)𝑏_𝑗(𝑥)=𝑏0,𝑖≠𝑗,_𝑗(𝑥),𝑖=𝑗.(3.9) From the orthogonality of property, it is possible to expand functions into their block pulse series.

Similarly, hybrid function may be expanded into an NM-set of block pulse function as𝐻(𝑥)=Φ𝐵(𝑥),(3.10) where 𝐵(𝑥)=[𝑏₁(𝑡),𝑏₂(𝑡),…,𝑏_𝑁𝑀(𝑡)] and Φ is an 𝑀𝑁×𝑀𝑁 product operational matrix.

In [24], Kilicman and Al Zhour have given the block pulse operational matrix of the fractional integration 𝐹^𝛼as follows:𝐽^𝑎𝐵(𝑥)≈𝐹^𝛼𝐵(𝑥),(3.11) where 𝐹^𝛼=1𝑙^𝛼1Γ(𝛼+2)⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1𝜉¹𝜉₂𝜉₃⋯𝜉_𝑙−101𝜉₁𝜉₂⋯𝜉_𝑙−2001𝜉₁⋯𝜉_𝑙−3⋮0⋮0⋮0⋮0⋱⋯𝜉⋮₁0000⋯1⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(3.12) with 𝜉_𝑘=(𝑘+1)^𝛼+1−2𝑘^𝛼+1+(𝑘−1)^𝛼+1.

Next, we derive the hybrid function operational matrix of the fractional integration. Let 𝐽^𝛼𝐻(𝑥)≈𝑃^𝛼𝐻(𝑥),(3.13) where matrix 𝑃^𝛼is called the hybrid function operational matrix of fractional integration.

Using (3.10) and (3.11), we have𝐽^𝛼𝐻(𝑥)≈𝐽^𝛼Φ𝐵(𝑥)=Φ𝐽^𝛼𝐵(𝑥)≈Φ𝐹^𝛼𝐵(𝑥).(3.14) From (3.10) and (3.13), we get𝑃^𝛼𝐻(𝑥)=𝑃^𝛼Φ𝐵(𝑥)=Φ𝐹^𝛼𝐵(𝑥).(3.15) Then, the hybrid function operational matrix of fractional integration 𝑃^𝛼 is given by𝑃^𝛼=Φ𝐹^𝛼Φ⁻¹.(3.16) Therefore, we have found the operational matrix of fractional integration for hybrid function.

3.4. The Product Operational of the Hybrid of Block-Pulse and Chebyshev Polynomials

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

Let 𝐻(𝑥)𝐻^𝑇(𝑥)𝐶≅𝐶𝐻(𝑥),(3.17) where 𝐶=⎛⎜⎜⎜⎜⎜⎜⎝𝐶¹0𝐶0⋯0₂⋮0⋮0⋱⋯⋯𝐶⋮0_𝑁⎞⎟⎟⎟⎟⎟⎟⎠(3.18) is an 𝑀𝑁×𝑀𝑁 product operational matrix. And, 𝐶_𝑖𝑖=1,2,3,…𝑁 are 𝑀×𝑀 matrices given by𝐶_𝑖=12⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝2𝑐^𝑖02𝑐_𝑖12𝑐_𝑖22𝑐_𝑖3⋯2𝑐_{𝑖,𝑀−2}2𝑐_{𝑖,𝑀−1}𝑐_𝑖12𝑐_𝑖0+𝑐_𝑖2𝑐_𝑖1+𝑐_𝑖3𝑐_𝑖2+𝑐_𝑖4⋯𝑐_{𝑖,𝑀−3}+𝑐_{𝑖,𝑀−1}𝑐_{𝑖,𝑀−2}𝑐_𝑖2𝑐_𝑖1+𝑐_𝑖32𝑐_𝑖0+𝑐_𝑖4𝑐_𝑖1+𝑐_𝑖5⋯𝑐_{𝑖,𝑀−4}𝑐_{𝑖,𝑀−3}⋯⋮⋯⋮⋯⋮2𝑐_𝑖0⋮⋱⋮⋮+𝑐_𝑖𝑢𝑐_𝑖1+𝑐_𝑖,𝑢+1⋯𝑐_𝑖𝑣⋯⋯⋯⋯⋯⋯⋯⋯⋯𝑐_𝑖1⋯⋯+𝑐_𝑖𝑢2⋯⋯𝑐_𝑖02⋯⋮𝑐_𝑖0⋯⋮𝑐_𝑖1𝑐_{𝑖,𝑀−1}𝑐_{𝑖,𝑀−2}𝑐_{𝑖,𝑀−3}𝑐_{𝑖,𝑀−4}⋯𝑐_𝑖12𝑐_𝑖0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.19) We also define the matrix 𝐷 as follows:𝐷=¹⁰𝐻(𝑥)𝐻^𝑇(𝑥)𝑑𝑥.(3.20) For the hybrid functions of block-pulse and Chebyshev polynomials, 𝐷 has the following form:𝐷=⎛⎜⎜⎜⎜⎜⎜⎝⋮𝐿00⋮𝐿00⋱⋯⋯⋯⋮𝐿00⎞⎟⎟⎟⎟⎟⎟⎠,(3.21) where 𝐿is 𝑀×𝑀 nonsingular symmetric matrix given in [23].

4. Nonlinear Fredholm Integral Equations

Consider (1.1); we approximate 𝑔(𝑥),𝑘(𝑥,𝑡) by the way mentioned in Section 3 as𝑘(𝑔𝑥(,𝑥𝑡))==𝐻𝐻^𝑇_𝑇(𝑥)𝐺,(𝑥)𝐾𝐻(𝑡).(4.1) (see [25]), Now, let𝐷^𝛼𝑓(𝑥)≈𝐴^𝑇𝐻(𝑥).(4.2) For simplicity, we can assume that 𝛿_𝑖=0 (in the initial condition). Hence by using (2.4) and (3.13), we have 𝑓(𝑥)≈𝐴^𝑇𝑃^𝛼𝐻(𝑥).(4.3) Define[𝑓(𝑡)]_𝑚𝐶=𝑐₀,𝑐₁,𝑐₂,…,𝑐_𝑙−1^𝑇=𝐴^𝑇𝑃^𝛼,=𝐻^𝑇(𝑡)𝐶^𝑚=𝐶^𝑇𝐻(𝑡)^𝑚=𝐶^𝑇𝐻(𝑡)⋅𝐻^𝑇(𝑡)𝐶𝐻^𝑇(𝑡)𝐶^𝑚−2.(4.4) Applying (3.17) and (4.4), [𝑓(𝑡)]^𝑚=𝐴^𝑇𝐶𝐻[(𝑓𝑡)(𝑡𝐻)]^𝑇_𝑚(𝑡)𝐶^𝑚−2=𝐴^𝑇𝐶𝐻(𝑡)⋅𝐻^𝑇(𝑡)𝐶𝐵^𝑇(𝑡)𝐶^𝑚−3,=𝐶^𝑇𝐶^𝑚−1𝐻(𝑡)=𝐶^∗𝐻(𝑡).(4.5) With substituting in (1.1), we have𝐻𝐻^𝑇(𝑥)𝐴−𝜆¹⁰𝐻^𝑇(𝑥)𝐾𝐻(𝑡)𝐻^𝑇(𝑡)𝐶^∗𝑇𝑑𝑡=𝐻^𝑇(𝑥)𝐺,_𝑇(𝑥)𝐴−𝜆𝐻^𝑇(𝑥)𝐾¹⁰𝐻(𝑡)𝐻^𝑇(𝑡)𝑑𝑡⋅𝐶^∗𝑇=𝐻^𝑇(𝑥)𝐺(4.6) Applying (3.20), we get𝐴−𝜆𝐾𝐷⋅𝐶^∗𝑇=𝐺,(4.7) which is a nonlinear system of equations. By solving this equation, we can find the vector𝐶.

We can easily verify the accuracy of the method. Given that the truncated hybrid function in (3.4) is an approximate solution of (1.1), it must have approximately satisfied these equations. Thus, for each 𝑥_𝑖∈[0,1],𝐸𝑥_𝑖=𝐴^𝑇𝐻𝑥_𝑖−𝜆¹⁰𝑘𝑥_𝑖,𝑡𝐶^∗𝐻(𝑡)𝑑𝑡−𝑔𝑥_𝑖≈0.(4.8) If max 𝐸(𝑥_𝑖)=10^−𝑘 (𝑘is any positive integer) is prescribed, then the truncation limit 𝑁is increased until the difference 𝐸(𝑥_𝑖) at each of the points 𝑥_𝑖 becomes smaller than the prescribed10^−𝑘.

5. Numerical Examples

In this section, we applied the method presented in this paper for solving integral equation of the form (1.1) and solved some examples.

Example 5.1. Let us first consider fractional nonlinear integro-differential equation: 𝐷^𝛼𝑓(𝑥)−¹⁰𝑥𝑡[𝑓(𝑡)]²𝑑𝑡=1−𝑥4,0≤𝑥<1,0<𝛼≤1,(5.1) (see [26]), with the initial condition 𝑓(0)=0.
The numerical results for 𝑀=1,𝑁=2,and 𝛼=1/4,1/2,3/4,and1 are plotted in Figure 1. For𝛼=1, we can get the exact solution𝑓(𝑥)=𝑥. From Figure 1, we can see the numerical solution is in very good agreement with the exact solution when𝛼=1.

Figure 1 

The approximate solution of Example 5.1 for 𝑁=1, 𝑀=2.

Example 5.2. As the second example considers the following fractional nonlinear integro-differential equation: 𝐷^1/2𝑓(𝑥)−¹⁰𝑥𝑡[𝑓(𝑡)]⁴𝑑𝑡=𝑔(𝑥),0≤𝑥<1,(5.2) with the initial condition 𝑓(0)=0 and 𝑔(𝑥)=(1/Γ(1/2))((8/3)√𝑥³−2√𝑥)−(𝑥/1260), the exact solution is𝑓(𝑥)=𝑥²−𝑥. Table 1 shows the numerical results for Example 5.2.

Table 1 

Absolute error for𝛼=1/2and different values of M, N for Example 5.2.

Example 5.3. 𝐷^5/3𝑓(𝑥)−¹⁰(𝑥+𝑡)²[𝑓(𝑡)]³𝑑𝑡=𝑔(𝑥),0≤𝑥<1,(5.3) (see [12]), where 𝑔(𝑥)=6Γ(1/3)³√𝑥−𝑥²7−𝑥4−19,(5.4) and with these supplementary conditions 𝑓(0)=𝑓^(0)=0. The exact solution is𝑓(𝑥)=𝑥². Figures 2 and 3 illustrates the numerical results of Example 5.3 with𝑁=2,𝑀=3.

Figure 2 

Exact and numerical solutions of Example 5.3 for 𝑁=2, 𝑀=3.

Figure 3 

Absolute error of Example 5.3 for 𝑁=2, 𝑀=3.

6. Conclusion

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 𝑁 and 𝑀 are adjustable as well as being able to yield more accurate numerical solutions. Also hybrid functions have good advantage in dealing with piecewise continuous functions.

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.

Acknowledgments

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.