Abstract
In this paper, a robust, effective, and accurate numerical approach is proposed to obtain the numerical solution of fractional differential equations. The principal characteristic of the approach is the new orthogonal functions based on shifted Legendre polynomials to the fractional calculus. Also the fractional differential operational matrix is driven. Then the matrix with the Tau method is utilized to transform this problem into a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via some examples. It is shown that the FLF yields better results. Finally, error analysis shows that the algorithm is convergent.
1. Introduction
Fractional-order calculus has a long history. Compared with integer order, fractional differential equations show many advantages over the simulation of physical phenomena [1]. In recent decades, fractional partial differential equations (FPDEs) [2] are widely used to describe engineering processes and dynamical systems; more and more researchers have devoted to study a variety of methods for solving fractional differential equations. The most commonly used methods are generalized differential transform method [3], Adomian decomposition method [4], finite difference method [5], and so on. Recently, the authors have proposed wavelet methods for solving a class of fractional convection-diffusion equations. In [6], different families of wavelets [7–11] have been widely applied for solving problems of partial differential equations. In [12], the authors have obtained the approximate solutions of fractional differential equations using the generalized block pulse operational matrix.
The methods based on the orthogonal functions [13–15] are powerful and wonderful for solving FPDEs and have achieved great success in this field. Now the operational matrices of fractional derivative and integral [14–16] for Bernstein polynomials and Jacobi polynomials [15] have been derived. However, these polynomials use integer power series to approximate fractional ones; it cannot accurately represent properties of fractional integral and differential. Recently, in [17], Rida and Yousef have proposed a fractional extension of classical Legendre polynomials by replacing the integer order derivative in Rodrigues formula [18] with fractional-order derivatives. These functions are complex, so they are not unsuitable for solving FPDEs. Subsequently, Kazem et al. put forward the orthogonal fractional-order Legendre functions based on shifted Legendre polynomials. In [19], they used these functions to obtain the numerical solutions of fractional partial differential equations. Numerical results show that their methods are effective, accurate, and easy to implement. In this paper, FLF is suitable to characterize the properties of fractional partial differential equations; thus we attempt to use fractional-order Legendre functions to acquire numerical solution of the FPDEs.
The article is organized as follows. In Section 2, we introduce some basic definitions of fractional derivatives and integrals. In Section 3, we construct the FLFs and introduce some properties. In Section 4, the fractional operational matrices of FLFs are obtained. In Section 5, we use the proposed method to solve FPDEs. In Section 6, the proposed method is tested through several numerical examples. In Section 7, the error analysis is given. Finally Section 8 concludes the paper.
2. Preliminaries and Notations
In this section, we recall the essentials of the fractional calculus theory that will be used in this article.
Definition 1. The Riemann-Liouville fractional integral operator of order is defined as
Definition 2. The Riemann-Liouville definition of fractional differential operator is given as follows:
Definition 3. The Caputo definition of fractional differential operator is defined as
The relation between the Riemann-Liouville operator and Caputo operator is given by the following expressions:
For , and constant , Caputo fractional derivative has some basic properties which are needed here as follows:
Definition 4 (generalized Taylor’s formula). Suppose that for , then one haswhere . Also, one haswhere
In case , the generalized Taylor’s formula (6) is the classical Taylors formula.
3. Definitions of Fractional-Order Legendre Definition and Function Approximation
3.1. Shifted Legendre Polynomials
The well-known Legendre polynomials are orthogonal with the weight function , which is defined on the interval ; namely,
In order to use these polynomials on the interval , let , and then we derived the shifted Legendre polynomials which are denoted by . The shifted Legendre polynomials are orthogonal with the weight function over , with the orthogonally property
Then can be obtained as follows:The analytic form of the shifted Legendre polynomial of degree is given bywhere
3.2. Fractional-Order Legendre Definition
We define the fractional-order Legendre functions (FLFs) by introducing the change of variable and on shifted Legendre polynomials. The fractional-order Legendre function is denoted by . The fractional-order Legendre functions are a particular solution of the normalized eigenfunctions of the singular Sturm-Liouville problem.
Then can be obtained as follows:
We can derive the analytic form of of degree as follows:where and .
Theorem 5. The FLFs are orthogonal with the weight function on the interval ; then the orthogonally condition is
Proof. With , where is the Kronecker function, let , and then we have
3.3. Function Approximation
Suppose ; it can be expanded in terms of the fractional-order Legendre functions as follows:where the coefficients are obtained by
If we consider truncated series in (17), we obtainwhere and are vector given by
Theorem 6. Suppose for . and If is the best approximation to from , then the error bound is presented as follows:where
Proof. Consider the generalized Taylor’s formulawhere .
With Definition 4Since is the best approximation to from and , henceNow by taking the square roots, then the theorem is proved.
For arbitrary function , they can be expanded as the following formula:where
Theorem 7. If the series converges uniformly to on the square , then one has
Proof. By multiplying on both sides of (25), where and are fixed and integrating term wise with regard to and on , then
Theorem 8. If the function is a continuous function on and the series converges uniformly to , then is the FLFs expansion of .
Proof. Using contradiction, letThen there is at least one coefficient such that ; however
Theorem 9. If two continuous functions defined on have the identical FLFs expansions, then these two functions are identical.
Proof. Suppose that and can be expended by FLFs as follows:By subtracting the above two equations with each other, we have
Theorem 10. If the sum of the absolute values of the FLFs coefficients of a continuous function forms a convergent series, then the FLFs expansion is absolutely uniformly convergent and converges to the function .
Proof. Consider If we consider truncated series in (25), we obtainwhereIn this paper, we use the Tau method [20] to compute the coefficients .
4. Fractional Differential Operational Matrix of FLFs
The derivative of the can be approximatedwhere is called the FLFs operational matrix of derivative.
Theorem 11. Suppose is the operational matrix of Caputo fractional derivatives of order , when ; then the elements of are obtained aswhere
Proof. With the properties of the derivative (ii) and the orthogonally of FLFs, we haveLetMultiplying on both sides of (40), we have Substituting (40) and (41) into (39), we have Hence,
5. Description of the Proposed Method
We consider a class of fractional partial differential equations as follows:subject to the initial conditionswhere and are fractional derivative in Caputo sense, is the known continuous function, and is the unknown function.
If we approximate function by a terms of the FLFs, it can be written as (34). Then we havewhere
Substituting (46) into (44), we have
Similar to the typical Tau method [20], then we generate algebraic equations by multiplying (for ), integrating from 0 to1 and using the orthogonal property, we have
For condition. (45), we havewith
So we obtain
Equations (49) and (52) generate linear algebraic equations. By solving the linear algebraic equations, the unknown coefficients can be obtained.
6. Error Analysis
We consider as the error function of the approximate solution for , where is the exact solution of (44) [21].
Therefore, satisfies the following problem:where is the residual function,
We find an approximation to the error function in the same way as we did before for the solution of the problem, the error function satisfies the problem
We should note that, in order to construct the approximate to , only (55) needs to be recalculated in the same way as we did before for the solution of (44).
7. Numerical Examples
Example 1. We consider the following FPDE:where ; using the method presented in Section 5, the equation is transformed as follows:By solving (57), we can obtain the numerical solution of the problem.
When , the matrix is get as follows:When , the matrix is get as follows:When , Figures 1, 2, and 3 show the numerical solution of (56) for with . The exact solution is which is shown in Figure 4. We can see clearly that the numerical solutions are very good agreement with the exact solution. Particularly, when the numerical solution matches the exact solution very well.
Example 2. We consider the following equation:The exact solution of this problem is , where The equation is transformed as follows:when , and . The results obtained using our proposed method and ADM at are shown in Figures 5 and 6. Comparing the two figures, it can be concluded that the FLFM has a better advantage than ADM for solving fractional partial differential equations and the numerical results have achieved a good convergence precision when small and are selected.
Example 3. Consider the following fractional Poisson equation: () When , The exact solution is When , the absolute errors for are shown in Figures 7, 8, and 9.
() When The exact solution is unknown.
Therefore, in order to show the efficiency of the present method, we define the error function as follows:Let with , and ; the absolute error is shown in Figures 10, 11, and 12.
When , we can have the coefficients matrix . Then we have the numerical solution .
8. Conclusions
In this paper, we define a basis of FLFs and have derived its fractional differential operational matrix. The fractional derivatives are described in the Caputo sense. The matrix together with Tau method is used to effectively approximate the exact solution of fractional partial differential equations. Compared with other numerical approach, the FLFs-Tau method can accurately represent properties of fractional calculus. Moreover, only a small number of FLFs are needed to obtain a satisfactory result. Numerical results which are given in the previous section demonstrate good accuracy of the algorithm. Finally, error analysis shows that the algorithm is convergent.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by Dr. Startup funds of Taiyuan University of Science and Technology (20122054).