Abstract
We use a computational method based on rational Haar wavelet for solving nonlinear fractional integro-differential equations. To this end, we apply the operational matrix of fractional integration for rational Haar wavelet. Also, to show the efficiency of the proposed method, we solve particularly population growth model and Abel integral equations and compare the numerical results with the exact solutions.
1. Introduction
Fractional calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders (including complex orders). It is also known as generalized integral and differential calculus of arbitrary order. Fractional differential equations are generalized from classical integer-order ones, which are obtained by replacing integer-order derivatives by fractional ones. In recent years, fractional calculus and differential equations have found enormous applications in mathematics, physics, chemistry, and engineering [1–4]. A large class of dynamical systems appearing throughout the field of engineering and applied mathematics is described by fractional differential equations. For that reason, reliable and efficient techniques for the solution of fractional differential equations are indeed required. The most frequently used methods are Walsh functions [5], Laguerre polynomials [6], Fourier series [7], Laplace transform method [8], the Haar wavelets [9], Legendre wavelets [10–12], and the Chebyshev wavelets [13, 14]. Kronecker operational matrices have been developed by Kilicman and Al Zhour for some applications of fractional calculus [15]. Recently, in [16], the authors proposed a new method based on operational matrices to solve fractional Volterra integral equations.
Recently, many authors applied operational matrices of integration and derivative to reduce the original problem into an algebraic one. According to this fact that the orthogonal polynomials play an important role to solve integral and differential equations, many researchers constructed operational matrix of fractional and integer derivatives for some types of these polynomials, such as Flatlet oblique multiwavelets [17, 18], B-spline cardinal functions [19], Legendre polynomials, Chebyshev polynomials, and CAS wavelets [20]. The main aim of this paper is to use an operational matrix of fractional integration to reduce a nonlinear fractional integro-differential equation to nonlinear algebraic equations.
The rest of the paper is organized as follows: In Section 2, we introduce some basic mathematical preliminaries that we need to construct our method. Also, we recall the basic definitions from block pulse functions and fractional calculus. In Section 3, we recall definition of rational Haar wavelet. In Section 4, we apply proposed method to solve fractional population growth model and Abel integral equations. Section 5 is devoted to convergence and error analysis. Finally, in Section 6, conclusion of numerical results is presented.
2. Preliminaries
In this section, we recall some basic definitions from fractional calculus and some properties of integral calculus which we shall apply to formulate our approach.
The Riemann-Liouville fractional integral operator of order on the usual Lebesgue space is given by [21] The Riemann-Liouville fractional derivative of order is normally defined as where is an integer number.
The fractional derivative of order in the Caputo sense is given by [21] where is an integer, , and .
The useful relation between the Riemann-Liouville operator and Caputo operator is given by the following expression: where is an integer, , and .
An -set of block pulse functions (BPFs) in the region of is defined as follows: where with positive integer values for and . There are some properties for BPFs, for example, disjointness, orthogonality, and completeness.
The set of BPFs may be written as an -vector as where .
A function may be expanded by the BPFs as where is given by (6) and is an -vector given by and the block pulse coefficients are obtained as
The integration of vector defined in (6) may be obtained as where is called operational matrix of integration which can be represented by Kilicman and Al Zhour (see [15]) have given the block pulse operational matrix of fractional integration as follows: where and .
3. Rational Haar Wavelets
The Haar functions are an orthogonal family of switched rectangular waveforms where amplitudes can differ from one function to another. The orthogonal set of Haar wavelet functions is defined in [0, 1) as follows (see [22]): where , and is a positive integer, and and represent the integer decomposition of the index , that is, , , . Also we have . This set of functions is complete, since any function can be expanded into Haar wavelets by where
Operational Matrix of Fractional Integration. Equation (7) implies that rational Haar wavelets can be also expanded into -term BPFs as for . Clearly we have where and , for (see [15, 23]).
For example, for , the Haar operational matrix into BPFs can be expressed as Let where matrix is called the Haar wavelet operational matrix of fractional integration. Using (18) and (12), we have By (21) and (22), we get therefore the Haar wavelet operational matrix of fractional integration is as follows: For example, with , the Haar operational matrix into BPFs can be expressed as
4. Implementation of the Method
In this section, we present a computational method for solving the nondimensional fractional population growth model and Abel integral equations.
4.1. Population Growth Model
The Volterra model for nondimensional fractional population growth model is as follows: The analytical solution (26) for is (see [24]) The exact values of were evaluated by using For solving (26), we first approximate as where is an unknown vector which should be found and is the vector which is defined in (16). By using initial condition, , and (4), we have By using (18) and (30), we conclude that Let By using (31) and (32), we have . From (5) we have Also, from (10) we have where . By using (5), (31), and (34), we have where Now by substituting (29), (31), (33), and (35) into (26), we obtain By replacing by , we obtain the following system of nonlinear algebraic equations: Finally, by solving this system, we obtain the approximate solution of the problem as . As a numerical example, we consider the nonlinear fractional integro-differential equation (26) with the initial condition ; for more details, see Table 1 and Figures 1 and 2.
4.2. Abel Equations
Consider the generalized linear Abel integral equations of the first and second kinds, respectively, as [25] where and are differentiable functions. Here, we apply fractional integration operational matrix of rational Haar wavelet to solve Abel integral equations as fractional integral equations.
Let . Clearly, we can write (1) as follows:
Now, by using (40), we obtain fractional form of Abel integral equations of the first and second kind, respectively, as the following form:
By collocating (41) in , for , we obtain the following systems of algebraic equations:
Finally, by solving this system and determining , we obtain the approximate solution of (41) as .
Example 1. Consider the second kind Abel integral equation of the form The exact solution is .
Let and , for .
From (1), (18), and (21), we have So, we get the algebraic equations as follows: By solving this system, we obtain the approximate solution of the problem as . Figure 3 shows the plot of error for presented method and the exact solution of this example.
Example 2. Consider Abel integral equation of the first kind The exact solution is .
To compare the numerical results and the exact solution, one can refer to Figure 4.
5. Error Analysis
In this section, we assume that is a differentiable function and also is bounded on the interval ; that is, If is the approximation of aswhere and is a positive integer, the corresponding error function is denoted by .
Theorem 3. Suppose that with bounded first derivative, , and . Then we have the error bound as follows: where .
Proof. See [26].
In other words, by increasing , the error function, , approaches to zero. If when is sufficiently large enough, then the error decreases.
6. Conclusion
In this paper, we presented a numerical scheme for solving fractional population growth model and Abel integral equations of the first and second kinds. The method which is employed is based on the rational Haar wavelet. In Figures 1 and 2, the solutions of fractional population growth model for different values of and are shown. Table 1 represents the exact value of and comparison of our used rational Haar method (RHM) with ADM (Adomian Decomposition Method [27]) and HPM (Homotopy-Padé Approximation Method [28]). By considering Abel integral equations of the first and second kinds as a fractional integral equation, we use fractional calculus properties for solving these singular integral equations. The fractional integration is described in the Riemann-Liouville sense. This matrix is used to approximate the numerical solution of the generalized Abel integral equations of the first and second kinds. Presented approach was based on the collocation method. Figures 3 and 4 show the plot of the error of presented method and the exact solution of Abel equations (Examples 1 and 2, resp.). The obtained results show that the used technique can be a suitable method to solve the fractional integro-differential equations.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.