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# Numerical Solution of Fractional Variational Problems Using Direct Haar Wavelet Method

 Osama H. M.1, Fadhel S. F.1 , Zaid A. M.2 Department of Mathematics and Computer Applications, College of Science, Al-Nahrain University, Baghdad, Iraq Economic and Administration College, the Iraqi University, Baghdad, Iraq Related article at Pubmed, Scholar Google

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## Abstract

This paper presents a clear procedure for the fractional variational solution via Haar wavelet technique. The fractional derivative is defined in the Riemann-Liouville sense. The fractional variational problem is solved by means of the direct method using the Haar wavelet and the problem will be reduced to the solution of an algebraic equations. The numerical solution for the class of problem considered can be obtained directly from the functional and there is no need to solve the fractional Euler-Lagrange equation. The examples are included in order to demonstrate the validity and applicability of the suggested approach.

### Keywords

Haar wavelet method, Fractional calculus, Calculus of variation, Fractional calculus of variation.

### INTRODUCTION

The use of fractional calculus of modeling physical system has been widely considered in the last decades, [1]. Although, the concept of the fractional derivatives was introduced already in the middle of the 19th century by Riemann and Liouville, [2]. The first work, devoted exclusively to the subject of fractional calculus, is the book by Oldham and Spanier [3] published in 1974. After that, the number of publications about the fractional calculus has rapidly increased. The reason for this is that the same physical processes as a anomalous diffusion, complex viscoelastisity, behaviour of mehatronic and biological systems, rheology, etc. can be described adequately by classical models, [2]. A fractional calculus of variations problem is a subtopic of fractional calculus and it is a problem in which either the objective functional or the constraint equation or both contain at least one fractional derivative term, [4]. This occurs naturally in many problems of physics, mechanics and engineering in order to provide more accurate models of physical phenomena (see [5-13]), However, the fractional calculus of variations is a new field, ;Its starting point appear to be the references [14], [15] where Riewe developed the nonconcentrative Lagrangian, Hamiltonian, and other concepts of classical mechanics using fractional derivative, [16]. Particularly, a fractional calculus of variation concerns the variational principles on functionals involving fractional derivative as we mention above and this leads to the statement of fractional Euler-Lagrange equations (see [14], [4], [17]). Fractional Euler-Lagrange equations are difficult to solve explicitly and consequently it is of interest to develop efficient numerical schemes for such dynamical systems. In this paper, we shall use the direct Haar wavelet method for a class of fractional variational problems. Haar wavelet theory has been innovated and applied to various fields in engineering ([18]-[25]), and have proved to be a wonderful mathematical tool.
The idea of this paper is to introduce Haar wavelets, then present a direct method for solving fractional problems via Haar wavelets. The procedure begins by assuming the admissible functions by Haar wavelets with coefficients to be determined, then establishing an operational matrix for performing integration and finding the necessary condition for exterimization, solving the resulting algebraic equation yields the Haar coefficients. This indicates that for the class of problems that will be considered, the numerical solution can be obtained directly from the functional, and there is no need to solve the fractional Euler-Lagrange equations.

### FRACTIONAL DERIVATIVE AND INTEGRATION

In this section, we shall review the basic definitions and properties of fractional integral and derivative, which are used further in this paper, [1].
Definition (1):
The Riemann-Liouville fractional integral operator of order α > 0, is defined as:

### CONCLUSION AND DISCUSSION

Direct Haar wavelet method has been presented for a fractional variational problem. The procedure considered in this paper can be considered as a generalization to the results given in [28]. From the illustrative examples, it ca be seen that this operational matrix approach can obtain accurate and satisfying results. All computational results are made by MATLAB program.

### References

[1] I. Podlubny, Fractional Differential Equations, Academic press, San Diego, 1999.

[2] Ü. Lepik, “Solving fractional integral equations by the Haar wavelet method”, Applied Mathematics and Computation, vol.214, pp.468-478, 2007.

[3] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.

[4] O. P. Agrawal, "Formulation of Euler-Lagrange equations for fractional variational problems", J. Math. Anal. Appl., Vol.272, No.1, pp. 368- 379, 2002.

[5] R. A. El-Nabulsi and D. F. M. Torres, “Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann-Liouville derivatives of order (α,β)”, Math. Methods Appl. Sci., Vol.30, No.15, pp.1931-1939, 2007.

[6] G. S. F. Frederic and D. F. M. Torres, “A formulation of Neother’s theorem of fractional problems of the calculus of vrariations”, J. Math. Anal. Appl., Vol.334, No.2, pp.834-846, 2007.

[7] R. A. El-Nabulsi and D. F. M. Torres, “Fractional action like variational problems”, J. Math. Phys., Vol.49, No.5, pp.053521-7, 2008.

[8] G. S. F. Frederico and D. F. M. Torres, “Fractional conservation law in optimal control theory”, Nonlinear Dynam., Vol.53, No.3, pp.215-222, 2008.

[9] P. Almedia, A. B. Malinowska and D. F. M. Torres, “A fractional calculus of variation of multiple integrals with applications to vibrating string”, J. Math. Phys., Vo.51, No.3, pp.033503-12, 2010.

[10] P. Almedia and D. F. M. Torres, “Leitwann’s direct method for fractional optimization”, Appl. Math. Compute., Vol.217, No.3, pp.956-962, 2010.

[11] N. R. O. Bastes, R. A. C. Ferreira and D. F. M. Torres, “Discrete-time fractional variational problems”, Signal Process., Vol.91, No.3, pp.513- 524, 2011.

[12] R. A. C. Ferreira and D. F. M. Torres, “Fractional h-difference equation arising from the calculus of variations”, Appl. Anal. Discrete Math., Vol.5, doi:10.2298/AADM 110131002F, 2011.

[13] D. Mozyrska and D. F. M. Torres, “Modified optimal energy and initial memory of fractional continuous-time linear systems”, Signal Process., Vol.91, No.3, 2011.

[14] F. Riewe, “Non concentrative Lagrangian and Hamiltonian mechanics”, Phys. Rev., Vol.E53, No.2, pp.1890-1899, 1996.

[15] F. Riewe, “Mechanic with fractional derivatives”, Phys. Rev., Vol.E55, No.3, pp.3589-3592, 1997.

[16] O. P. Agrawal, “A general finite element formulation for fractional variational problems”, J. Math. Anal. Appl., Vol.337, pp. 1-12, 2008.

[17] D. Baleanu and S. I. Muslih, “Lagrangian formulation of classical fields within Riemann-Liouville fractional derivative”, Physc. Scripta, Vol.72, No.2-3, pp. 119-121, 2005.

[18] G. Strang, “Wavelets and dilation equations: a brief introduction”, SIAM Rev., Vol.31, pp. 614-627, 1989.

[19] I. Danbechies, “The wavelet transform, time-frequency location and signal analysis”, IEEE Trans. Inform. Theory, Vol.36, pp. 961-1005, 1990.

[20] C. Chen and C. Hsiao, “Haar wavelet method for solving lumped and distributed-parameter systems”. IEEE Proc Control Theory Appl., Vol.144, No.1, pp. 87-94, 1997.

[21] C. H. Hsiao and W. J. Wang, “State analysis and optimal control of linear time varying systems via Haar wavelets”, Optimal Control Appl. Methods, Vol.19, pp. 423-433, 1998.

[22] C. F. Chen and C. H. Hsiao, “Wavelets approach to optimizing dynamic systems”, IEEE Proc. Control Theory, Appl., Pt D146, pp. 213-219, 1998.

[23] C. H. Hsiao and W. J. Wang, “State analysis and optimal control of time varying discrete systems via Haar wavelets”, J. Optimization Theory Appl., Vol.103, pp. 523-640, 1999.

[24] C. H. Hsiao and W. J. Wang, “Optimal control of linear time-varying systems via Haar wavelets”, J. Optimization Theory Appl., Vol.103, pp. 641-655, 1999.

[25] C. H. Hsiao and W. J.Wang, “State analysis and parameter estimation of bilinear systems via Haar wavelets”, IEEE Trans. Circuits Syst. In Foundation Theory Appl., Vol.47, pp. 246-20, 2000.

[26] L. Yuanlu and Z. Weiwi, "Haar wavelet operational matrix of a fractional order integration and its applications in solving the factional order differential equations''. Appl. Math. Comput., Vol.216, pp. 2276-2285, 2010.

[27] O. P. Agrawal, “Fractional variational calculus and the transversality conditions”, J. Phys. A.: Math. Gen., Vol.39, pp. 10375-10389, 2006.

[28] C. H. Hsiao, “Haar wavelet direct method for solving variational problems”, Mathematics and Computers in Simulation, Vol.64, pp.569-585, 2004.