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Chebyshev Wavelets Method for Solution of Nonlinear Fractional Integrodifferential Equations in a Large Interval
An efficient Chebyshev wavelets method for solving a class of nonlinear fractional integrodifferential equations in a large interval is developed, and a new technique for computing nonlinear terms in such equations is proposed. Existence of a unique solution for such equations is proved. Convergence and error analysis of the proposed method are investigated. Moreover in order to show efficiency of the proposed method, the new approach is compared with some numerical methods.
Fractional integrodifferential equations (FIDEs) arise in modelling processes in applied sciences such as physics, engineering, and biology. The nonlinear fractional integro-differential equation (NFIDE) of the type where is Caputo fractional derivative, is a parameter describing the order of the fractional derivative, , , and are fixed constants, and is a nonlinear continuous function, arise in the mathematical modelling of various physical phenomena, such as heat conduction in materials with memory. Moreover, these equations are encountered in combined conduction, convection, and radiation problems [1–3]. Therefore in recent years, numerous works have been focusing on the solution of these problems. Some of these methods are Adomian decomposition method (ADM) , fractional differential transform method (FDTM) , and the collocation method . Most of these methods have been utilized in linear problems, and a few number of works have considered nonlinear problems. In  Rawashdeh applied Legendre wavelet method for solving fractional Voltera integro-differential equations in the form Also in  Awawdeh et al. applied homotopy analysis method (HAM) for solution of (2). Mittal and Nigam  applied the ADM for (1) in the form In view of successful application of wavelets in approximation theory [10–17], we will use the Chebyshev wavelets for solving a generalized form of the previous described equations of the form with high nonlinearity in a large interval under the initial condition .
Here, for simplicity, we assume that and are continuous functions on , is continuous on , and also and are analytic functions.
The method is based on reducing the equation to a system of nonlinear algebraic equations by expanding the solution as Chebyshev wavelet bases with unknown coefficients. We note that, for , (4) is an ordinary integro-differential equation and the method can be easily applied to it. Also the method is fast and mathematically simple and guarantees the necessary accuracy for a small number of basic functions. Moreover in order to show the efficiency of the proposed method the new approach is compared with the ADM, VIM [18–21] and ITM which has been proposed by Daftardar-Gejji and Jafari in 2006 [22, 23].
The outline of this paper is as follows: in Section 2, some basic definitions of the fractional calculus, Chebyshev wavelets, functional approximation, and operational matrix of fractional integration are described. In Section 3 the existence of a unique solution for the problem is established. In Section 4, Chebyshev wavelets method and error investigation are described. In Section 5, some numerical examples are solved by applying the methods of this article. Finally a conclusion is drawn in Section 6.
2. Basic Definitions
2.1. Fractional Calculus
Here, we give some basic definitions and properties of the fractional calculus theory which will be used further in this paper.
Definition 1. A real function , is said to be in the space , if there exists a real number such that , where and it is said to be in the space if .
Definition 2. The Riemann-Liouville fractional integration operator of order of a function , is defined as
It has the following properties:
where , and .
Riemann-Liouville fractional derivative operator of order is defined as where is an integer and .
The Riemann-Liouville derivatives have certain disadvantages when trying to model real-world phenomena with fractional differential equations. Therefore, we will now introduce a modified fractional differential operator proposed by Caputo .
Definition 3. The fractional derivative operator of order in the Caputo sense is defined as
where is an integer, , and .
The Riemann-Liouville and Caputo operators have a useful property as where is an integer, , and .
For more details on the mathematical properties of fractional derivatives and integrals see .
2.2. Chebyshev Wavelets
Chebyshev wavelets have four arguments; , and ; moreover is the degree of the Chebyshev polynomial of the first kind and is the normalized time, that is, . They are defined on the interval as  where , and is a fixed positive integer. The coefficients in (11) are used for orthonormality of the system. Here, is the set of well-known Chebyshev polynomials of degree which are orthogonal with respect to the weight function on the interval and satisfy the following recursive formula: We should note that in dealing with Chebyshev polynomials the weight function has to be dilated and translated as , to get orthogonal wavelets.
2.3. Function Approximation
An arbitrary function defined over may be expanded into Chebyshev wavelet basis as where in which (,) denotes the inner product.
If the infinite series in (13) is truncated, then (13) can be written as where and are matrices given by Taking the collocation points where , we define the wavelet matrix as: Indeed has a diagonal form (see ).
2.4. The Operational Matrix of Fractional Integration
The fractional integration of order of the vector function defined in (15) can be expressed as where is the operational matrix of fractional integration of order . In  it is shown that the matrix can be approximated as where is the operational matrix of fractional integration of order of the Block Pulse functions (BPFs) that has the following form : where .
3. Existence and Uniqueness
For investigating existence of a unique solution for initial value problem (4), we reconsider this equation in the following operator form: where we define Applying operator , the inverse of , on both sides of (22), and considering initial condition yield Equation (24), can be written as a fixed point equation , where the map is defined as Now let be the Banach space of all continues functions on , with the norm . Moreover, suppose that the nonlinear terms and satisfy Lipschitz conditions on as where and are fixed positive constants. Then we have the following.
Theorem 4. If , then the initial value problem (4) has a unique solution .
Proof. There are different approaches in applying the Banach fixed point theorem to prove the existence of a unique solution for (4). We give a sketch of two approaches below. For this purpose we define the nonlinear integral operator
Approach 1. Let , then we have
Then for we have
Since , by contraction mapping theorem, the initial value problem (4) has a unique solution in .
Approach 2. Let us introduce the norm
on the space , which is equivalent to the standard norm on . The parameter is chosen to satisfy . Then for we have
Therefore, the operator is a contraction on the Banach space , and so the initial value problem (4) has a unique solution in .
4. Chebyshev Wavelets Method
Here, the Chebyshev wavelets expansion together with their operational matrices of fractional order integration are used for numerical solution of nonlinear NFIDE (4). For solving this problem we assume that where is an unknown vector and is the expansion of (15) by BPFs, that is, , and is defined in Appendix A. By using initial conditions and (9), we have From (35) we have This equation can also be written as and from Appendix B, we have where By substituting (34) and (38) into (4), we obtain Now, from (37) and (40) we have This is a nonlinear algebraic equation. Now, by taking collocation points, expressed in (16), this equation is transformed into a nonlinear system of algebraic equations with unknowns . By solving this system and determining , we get the numerical solution of problem (4).
4.1. Error Investigation of the Chebyshev Wavelets Method
In this section we investigate error of the Chebyshev wavelets method. The representation error can be obtained when a function is presented on interval .
Theorem 5 (see ). A function defined on with bounded second derivative, say , can be expanded as an infinite sum of Chebyshev wavelets, and the series converges uniformly to , that is where ’s are defined in (13).
Theorem 6 (see ). Let be a continuous function defined on , with bounded second derivative , and let be the approximate solution using Chebyshev wavelets method, respectively. Then for the error bound one has: where
Lemma 7. Suppose that is approximated on by and moreover suppose that by solving some problems one has found as an approximation of and put: Then for each one has
Proof . We have This completes the proof.
Corollary 8. Suppose that by solving some problems one obtains as the approximation of . Then one has:
In real problems, we often tend to solve equations with unknown exact solutions. Hence, when we apply our method to these problems, we cannot say that this approximate solution is good or bad unless we are able to calculate the error function . Then it is necessary to introduce a process for estimating the error function when the exact solution is unknown . Here, we introduce a method to estimate the error function. Let be an approximate solution of (4). Then from (24), it is concluded that satisfies the following equation: where is the perturbation function that depends only on and can be obtained by substituting the computed solution into the equation Subtracting (53) from (24), we obtain in which we define as the error function. Since and are analytic functions, then we can write and similarly where, by Taylor’s Theorem, , and . Thus, by substituting (55), and (56) in the error equation (54), we get a nonlinear fractional integral equation in which the error function is unknown. Obviously, we can apply the proposed method for this equation to find an approximation of the error function .
5. Numerical Examples
In this section, for the sake of comparison, we have selected some examples that their exact solutions exist, which will ultimately show the simplicity, effectiveness, and exactness of the proposed method. All programs are performed by Maple 15 and digits 20. We consider the following test problems.
Example 1. Let us first consider the following NFIDE: where The exact solution of this problem for is . Figure 1 shows the behavior of the numerical solution for . From Figure 1, it is seen that the numerical solution is in a very good agreement with the exact solution for . Therefore, we hold that the solutions for and are also credible. Figure 1 also shows that as , the approximate solution tends to , which is the exact solution of this equation in the case .
Example 2. Consider the following NFIDE: where The exact solution of this problem is .
This NFIDE is now solved by the Chebyshev wavelet method for . Numerical solutions of this problem for some values of are shown in Figure 2. It is obviously seen that computing via the standard ADM and ITM is very difficult because analytic computation of is impossible. Also computing other components , via the standard ADM, VIM, and ITM, is very difficult in each iteration and also requires a large amount of computational work. We can avoid such integrations by taking the truncated Maclaurin expansion for the exponential term in (59), that is, . In order to show efficiency of the proposed method, a comparison between the numerical solution given by the proposed method for , of the ADM, ITM, and VIM, and the exact solution is performed in Figure 3. From Figure 3, it can be concluded that the Chebyshev wavelet method for the numerical solution of this problem is very efficient and accurate in comparison with other presented numerical methods.
Example 3. Consider the following NFIDE: where The exact solution of this problem is .
This problem is now solved by the Chebyshev wavelet method for . Numerical solutions of this problem for some values of are shown in Figure 4. A comparison between the numerical solution given by the proposed method for , of the ADM and ITM and of the VIM and the exact solution is performed in Figure 5. From Figure 5, it can be concluded that the Chebyshev wavelet method for the numerical solution of this problem is very efficient and accurate in comparison with other presented numerical methods. Moreover, the solution has been derived in a large domain. It can be mentioned that computing the components via the ADM, VIM and ITM require a large amount of computational work and memory, such that computing other components for is very difficult.
Example 4. Finally, consider the following NFIDE: where The exact solution of this problem is .
This NFIDE is now solved by the proposed method for . Numerical solutiones of this problem for some selected values of are shown in Figure 6. It is clearly seen that computing the components via the standard ADM, VIM, and ITM is very difficult and also requires a large amount of computational work. We can avoid such fractional integrations by taking the truncated Maclaurin expansion for the exponential term in (63). A comparison between the numerical solution given by the proposed method for , of the ADM, ITM, and VIM for and and the exact solution is performed in Figure 7. From Figure 7, it can be concluded that the Chebyshev wavelet method for the numerical solution of this problem is very efficient and accurate in comparison with the ADM, VIM, and ITM.
6. Discussion and Conclusions
In this paper, we proposed the Chebyshev wavelets method for the numerical solution of nonlinear fractional integro-differential equations in a large interval. Existence of a unique solution for such equations is proved. Convergence and error analysis of the Chebyshev wavelets expansion is discussed. Efficient approximate solutions have derived for NFIDEs, and the results have been shown remarkable performance. Moreover the new approach has been compared with the ADM, VIM, and ITM in solving various nonlinear fractional integro-differential equations. There are five important points to be noted here.(1)The proposed method uses a new technique for computation of nonlinear terms in such equations.(2)The proposed approach can provide the suitable approximate solution in a large interval by using only a few number of collocation points, as shown in the solved examples.(3)The proposed method is very simple and requires less computational work in comparison with ADM, VIM, and ITM.(4)The proposed method overcomes the probable difficulties arising in calculating integrals.(5)The proposed method does not need to use Adomian polynomials and also has no need for the Lagrange multiplier, correction functional, stationary conditions, the variational theory, and so forth, which eliminates the complications that exist in VIM.
So in comparison with the ADM, VIM, and ITM, the proposed method is proved to be simpler in principle and more convenient for computer algorithms.
A. Some Properties of BPFs
An -set of BPFs is defined as where .
The most important properties of BPFs are disjointness, orthogonality, and completeness.
Disjointness. This property can be clearly obtained from the definition of BPFs as follows:
Orthogonality. It is clear that where is the Kroneker delta.
Completeness. The sequence is complete in , namely, , and , results in almost everywhere. Because of completeness of , Parseval’s identity holds; that is, we have for every real bounded function , and Consider the first -terms of BPFs and write them concisely as the -vector as follows: Then from the above representation and the disjointness property, it follows that
B. Expanding the Nonlinear Terms via Chebyshev Wavelet Bases
From (37) and (A.6), we have and in general by induction we have For an analytic function , by Maclaurin’s expansion, , we have and from (B.2) and (B.3) we have Since the series in the left hand side is absolutely and uniformly convergent to , each series in the right hand side is also absolutely convergent to the corresponding ; that is, Now, from (B.4) and (B.5), we have
This work was supported in part by the National Natural Science Foundation of China under the Project Grant nos. 61272402, 61070214, and 60873264.
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