Hybrid Rational Haar Wavelet and Block Pulse Functions Method for Solving Population Growth Model and Abel Integral Equations

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.


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][2][3][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][11][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.

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.

Mathematical Problems in Engineering
The Riemann-Liouville fractional integral operator   of order  ≥ 0 on the usual Lebesgue space  1 [0, ] is given by [21] The Riemann-Liouville fractional derivative of order  > 0 is normally defined as where  is an integer number.
The fractional derivative of order  > 0 in the Caputo sense is given by [21] where  is an integer,  > 0, and The useful relation between the Riemann-Liouville operator and Caputo operator is given by the following expression: where  is an integer,  > 0, and  () ∈  1 [0, ].
An -set of block pulse functions (BPFs) in the region of [0, ) is defined as follows: where  = 1, 2, . . ., −1 with positive integer values for  and ℎ = /.There are some properties for BPFs, for example, disjointness, orthogonality, and completeness.The set of BPFs may be written as an -vector as where  ∈ [0, 1).
A function () ∈  2 ([0, 1)) 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 Υ is called operational matrix of integration which can be represented by ) .

Implementation of the Method
In this section, we present a computational method for solving the nondimensional fractional population growth model and Abel integral equations.

Population Growth Model.
The Volterra model for nondimensional fractional population growth model is as follows: The analytical solution (26) for  = 1 is (see [24]) The exact values of  max were evaluated by using For solving (26), we first approximate   * () as where  is an unknown vector which should be found and Ψ() is the vector which is defined in (16).By using initial condition, (0) =  0 , 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   =   Υ.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 integrodifferential equation (26) with the initial condition (0) = 0.1; for more details, see Table 1 and Figures 1 and 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 () =   Ψ().Clearly, we can write (1) as follows:   (42) Finally, by solving this system and determining , we obtain the approximate solution of (41) as () =   Ψ().
(45) By solving this system, we obtain the approximate solution of the problem as () =   Ψ().Figure 3 shows the plot of error for presented method and the exact solution of this example.
To compare the numerical results and the exact solution, one can refer to Figure 4.

Error Analysis
In this section, we assume that () is a differentiable function and also   () is bounded on the interval [0, 1]; that is, If   () is the approximation of () as   where  = 2 +1 and  is a positive integer, the corresponding error function is denoted by   () = () −   ().Proof.See [26].
In other words, by increasing , the error function,   (), approaches to zero.If   () → 0 when  is sufficiently large enough, then the error decreases.

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  max 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.

Figure 1 :
Figure 1: Numerical solutions of the fractional population growth model for different values of .

Figure 2 :
Figure 2: Numerical solutions of the fractional population growth model with  = 0.1 for different values of .

Example 1 .
Consider the second kind Abel integral equation of the form  () =  2

Table 1 :
Comparison of exact value of  max with the proposed method (RHM), ADM and HPM.