Fig. Comparison of for and (Left), (Right), with exact solution, for Example 3

Fig. 3. Comparison of for and (Left) , (Right) , with exact solution, for Example 3 .

Fig. 4. Comparison of for and , for Example 3 .
Table 3
Absolute error for different values of and for for Example 3 .

0.2
0.4
0.6
0.8
1.2
1.4
1.6
1.8

Here, we have

Using Eq. (33) we have

Now we collocate Eq. (38) at the first root of , i.e.

Also by using Eq. (34) we get

By solving Eqs. (38) and (39) we obtain

Therefore

which is the exact solution of this problem.
It is clear that in Examples 1-4 the present method can be considered as an efficient method.
6. Conclusion
A general formulation for the Legendre operational matrix of fractional derivative has been derived. The fractional derivatives are described in the Caputo sense. This matrix is used to approximate numerical solution of a class of fractional differential equations. Our approach was based on the shifted Legendre tau and shifted Legendre collocation methods. In the limit, as approaches an integer value, the scheme provides solution for the integer-order differential equations. The solution obtained using the suggested method shows that this approach can solve the problem effectively. Moreover, only a small number of shifted Legendre polynomials is needed to obtain a satisfactory result.
Acknowledgments
The authors are very grateful to both reviewers of this paper for their constructive comments and nice suggestions which have improved the paper very much. The authors are also very much thankful to Professor Yong Zhou (School of Mathematics and Computational Science, Xiangtan University, P.R. China) for managing the review process for the current paper.
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