# Solving Fractional Variational Problem Via an Orthonormal Function

### Abstract

In the present paper, a direct numerical technique for solving fractional optimal control problems based on an orthonormal wavelet, is introduced. First we approximate the involved functions by Sine-Cosine wavelet basis; then, an operational matrix is used to transform the given problem into a linear system of algebraic equations, which is easier. In fact operational matrix of Riemann-Liouville fractional integration and derivative of Sine-Cosine wavelet are employed to achieve a linear algebraic equation. The mentioned matrices are derived via hat functions. The solution of transformed system, gives us the solution of original problem. Two numerical examples are also given. Finally, the paper is ended with conclusion### References

T. Akbarian, and M. Keyanpour A new approach to the numerical solution of fractional order optimal control problems Applications and Applied Mathematics, vol. 8, no. 2, pp. 523–534, 2013.

M. H. Akrami, M. H. Atabakzadeh, and G. H. Erjaee, The operational matrix of fractional integration for shifted Legendre polynomials, Iranian Journal of Science and Technology, vol. 37, no. 4, pp. 439-444, 2013.

A. Arikoglu, and I. Ozkol, Solution of fractional integro-differential equations by using fractional differential transform method,Chaos, Solitons & Fractals, vol. 40, no. 2, pp. 521–529, 2009.

R. L. Bagley, and P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology,vol. 27, no. 3, pp. 201–210, 1983.

R. T. Baillie, Long memory processes and fractional integration in econometrics, Journal of Econometrics, vol. 73, no.1, pp. 5–59,1996.

N. R. Bastos, Calculus of variations involving Caputo-Fabrizio fractional differentiation, Statistics, Optimization and Information Computing, vol. 6, no. 1, pp. 12–21, 2018.

A.H.Bhrawy,D.Baleanu,andL.Assas, Efficient generalized Laguer respectral methods for solving multi-term fractional differential equations on the half line, Journal of Vibration and Control, Vol. 20,no. 7, pp. 973–985, 2014.

A. H. Bhrawy, E. H. Doha, D. Baleanu, and S. S. Ezz-Eldien, A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations, Journal of Computational Physics, vol. 293, pp. 142–156, 2015.

A. Biswas, A. H. Bhrawy, M. A. Abdelkawy, A. A. Alshaery, and E. M. Hilal, Symbolic computation of some nonlinear fractional differential equations, Romanian Journal of Physics, vol.59, no. 5-6, pp. 433–442, 2014.

G. W. Bohannan, Analog fractional order controller in temperature and motor control applications, Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1487–1498, 2008.

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamic, Springer Berlin Heidelberg, 2012.

S. Das, Analytical solution of a fractional diffusion equation by variational iteration method, Computers & Mathematics with Applications, vol. 57, no. 3, pp. 483–487, 2009.

M. Dehghan, J. Manafian, and A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numerical Methods for Partial Differential Equations: An International Journal, vol. 26, no. 2, pp. 448–479, 2010.

M. Dehghan, J. Manafian Herris, and A. Saadatmandi, The Solution of the linear fractional partial differential equations using the homotopy analysis method, Zeitschrift für Naturforschung-A, vol. 65, no. 11, pp. 935–949, 2010.

E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, An efficient Legendre spectral tau matrix formulation for solving fractional sub-diffusion and reaction sub-diffusion equations, Journal of Computational and Nonlinear Dynamics, vol. 10, no. 2, pp. 021019, 2015.

V. S. Erturk, and S. Momani, Solving systems of fractional differential equations using differential transform method, Journal of Computational and Applied Mathematics, vol. 2015, no. 1, pp. 142-159, 2008.

M. R. Eslahchi, M. Dehgh, and M. Parvizi, Application of the collocation method for solving nonlinear fractional integro-differential equations, Journal of Computational and Applied Mathematics, vol. 257, no.1, pp. 105-128, 2014.

B. Ghazanfari, and A. Sepahvandzadeh, Homotopy perturbation method for solving fractional Bratu-type equation, Journal of Mathematical Modeling, vol. 2, no. 2, pp. 143–155, 2015.

J. S. Gu, and W. S. Jiang, The Haar wavelets operational matrix of integration, International Journal of Systems Science, vol. 27,no. 7, pp. 623–628, 1996.

J. He, Nonlinear oscillation with fractional derivative and its applications, International Conference on Vibrating Engineering, Dalian, China, vol. 98, pp. 288–291, 1998 .

J. He, Some applications of nonlinear fractional differential equations and their approximations, Bulletin of Science, Technology& Society, vol. 15, no. 2, pp. 86–90, 1999.

C. H. Hsiao, Haar wavelet direct method for solving variational problem, Mathematics and Computers in Simulation, vol. 64, no.5, pp. 569–585, 2004.

M. T. Kajani, M. Ghasemi, and E. Babolian, Numerical solution of linear integro-differential equation by using SineCCosine wavelets, Applied Mathematics and Computation, vol. 180, no. 2, pp. 569–574, 2006.

A. Kheirabadi, A. Mahmoudzadeh Vaziri, and S. Effati, A new approach for solving optimal control problem by using orthogonal function, Springer, International Conference on Management Science and Engineering Management, pp. 223–232, 2017.

A. Lotfi, M. Dehghan, and S. A. Yousefi, A numerical technique for solving fractional optimal control problems, Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1055–1067, 2011.

I.Malmir, Optimalcontrolof lineartime-varying systemswith stateand inputdelays byChebyshevwavelets, Statistics,Optimization and Information Computing, vol. 5, no. 4, pp. 302–324, 2017.

M. Meerschaert, and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations,Applied Numerical Mathematics, vol. 56, no. 1, pp. 80–90, 2006.

F. Mohammadi, Wavelet Galerkin method for solving stochastic fracthional differential equations, Journal of Fractional Calculus and Applications, vol. 7, no. 1, pp. 73–86, 2016.

S. Momani, and Z. Odibat, Numerical approach to differential equations of fractional order, Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 96–110, 2007.

S. Momani, and Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Solitons & Fractals, vol. 31, no. 5, pp. 1248–1255, 2007.

Z. Odibat, and N. Shawagfeh, Generalized Taylors formula, Applied Mathematics and Computation, vol. 186, no. 1, pp. 286–293,2007.

R. Panda, and M. Dash, Fractional generalized splines and signal processing, Signal Process, vol. 86, no. 9, pp. 2340–2350, 2006.

I. Podlubny, The Laplace transform method for linear differential equations of the fractional order, arXiv preprint funct-an/9710 005, 1997.

I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, vol. 198, 1998.

Y. A. Rossikhin, and M. V. Shitikova , Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Applied Mechanics Reviews, vol. 50, no. 1, pp. 15–67, 1997.

A. Saadatmandi, and M. Dehghan, A new operational matrix for solving fractional order differential equations, Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1326–1336, 2010.

N. T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Applied Mathematics and Computation, vol. 131, no. 2, pp. 517–529, 2002.

S.Sohrabi,Comparision Chebyshev wavelets method with BPFS method for solving Abels integral equation, AinShams Engineering

Journal, vol. 2, no. 3–4, pp. 249–254, 2011.

X. W. Tangpong, and O. Agrawal, Fractional optimal control of a continuum system, Journal of Vibration and Acoustics, vol. 131,no. 2, p. 021012, 2009.

C. Tricaud, and Y. Chen, An approximation method for numerically solving fractional order optimal control problems of general form, Computers & Mathematics with Applications, vol. 59, pp. 1644–1655, 2010.

M. P. Tripathi, V. K. Baranwal, R. K. Pandey, and O. P. Singh, A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions, Communications in Nonlinear Science and Numerical Simulation, vol. 18,no. 6, pp. 1327–1340, 2013.

G. W. Wang, and T. Z. Xu, The modified fractional sub-equation method and its applications to nonlinear fractional partial differential equations, Romanian Journal of Physics, vol. 59, no. 7-8, pp. 636-645, 2014.

X. J. Yang, D. Baleanu, Y. Khan, and S. T. Mohyud-Din, Local fractional variational iteration method for diffusion and wave equations on Cantor sets Romanian Journal of Physics, vol.59, no. 1-2, pp. 36-48, 2014.

*Statistics, Optimization & Information Computing*,

*7*(2), 447-455. https://doi.org/10.19139/soic.v7i2.502

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).