Numerical Solution for A Class of Fractional Variational Problem via Second Order B-Spline Function

Noratiqah Farhana binti Ismail; Chang Phang

doi:10.22342/jims.25.3.672.171-182

Submitted

August 30, 2017

Accepted

September 18, 2018

Published

October 31, 2019

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Abstract

In this paper, we solve a class of fractional variational problems (FVPs) by using operational matrix of fractional integration which derived from second order spline (B-spline) basis function. The fractional derivative is defined in the Caputo and Riemann-Liouville fractional integral operator. The B-spline function with unknown coefficients and B-spline operational matrix of integration are used to replace the fractional derivative which is in the performance index. Next, we applied the method of constrained extremum which involved a set of Lagrange multipliers. As a result, we get a system of algebraic equations which can be solve easily. Hence, the value for unknown coefficients of B-spline function is obtained as well as the solution for the FVPs. Finally, the illustrative examples shown the validity and applicability of this method to solve FVPs.

Keywords

fractional variational problems B-spline function operational matrix of integration Riemann-Liouville fractional integration Lagrange multiplier

How to Cite

Ismail, N. F. binti, & Phang, C. (2019). Numerical Solution for A Class of Fractional Variational Problem via Second Order B-Spline Function. Journal of the Indonesian Mathematical Society, 25(3), 171–182. https://doi.org/10.22342/jims.25.3.672.171-182

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References

Agrawal, O. P., "Generalized variational problems and EulerLagrange equations", Comput. Math. Appl., 59 (2010), 1852 - 1864.
Atanackovi, T.M., Konjik, S. and Pilipovi, S., "Variational problems with fractional derivatives: Euler Lagrange equations",J. Phys. A, 41 (2008), 095201.
Debnath, L., "Recent applications of fractional calculus to science and engineering", Int. J. Math. Math. Sci. 54 (2003), 3413 - 3442.
Dehghan, M., Hamedi, E.A. and Khosravian-Arab, H., "A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials", J. Vib. Control, 22 (2016), 1547 - 1559.
Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S., "An efficient Legendre spectral tau matrix formulation for solving fractional subdiusion and reaction subdiffusion equations", J. Comput. Dyn., 10 (2015), 021019.
Ezz-Eldien, S. S., et al., "New numerical approach for fractional variational problems using shifted Legendre orthonormal polynomials", J. Optim. Theory Appl., (2016), 1 - 26.
Ezz-Eldien, S.S., "New quadrature approach based on operational matrix for solving a class of fractional variational problems", J. Comput. Phys., 317 (2016), 362 - 381.
Goswami, J. C., Chan, A. K., and Chui, C. K., "On solving first-kind integral equations using wavelets on a bounded interval", IEEE Trans. Antennas and Propagation, 43 (1995), 614 - 622.
Ismaeelpour, T., Hemmat, A. A., and Saeedi, H., "B-spline operational matrix of fractional integration", Optik-International Journal for Light and Electron Optics, 130 (2017), 291 - 305.
Jafari, H., Tajadodi, H. and Baleanu, D., "A numerical approach for fractional order Riccati differential equation using B-spline operational matrix", Fract. Calc. Appl. Anal., 18 (2015), 387 - 399.
Khader, M.M. and Hendy, A.S., "A numerical technique for solving fractional variational problems", Math. Methods Appl. Sci., 36 (2013), 1281 - 1289.
Lewandowski, R., and Choryczewski, B., "Identification of the parameters of the KelvinVoigt and the Maxwell fractional models, used to modeling of viscoelastic dampers", Computers & structures, 88 (2010), 1 - 17.
Loverro, A., "Fractional calculus: history, definitions and applications for the engineer", Rapport technique, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering (2004), 1 - 28.
Machado, J.T., Kiryakova, V. and Mainardi, F., "Recent history of fractional calculus", Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140 - 1153.
Machado, J. T., and Mata, M. E., "Pseudo phase plane and fractional calculus modeling of western global economic downturn", Commun. Nonlinear Sci. Numer. Simul, 22 (2015), 396 - 406.
Magin, R. L., "Fractional calculus models of complex dynamics in biological tissues", Comput. Math. Appl., 59 (2010), 1586 - 1593.
Mainardi, F., "An historical perspective on fractional calculus in linear viscoelasticity", Fract. Calc. Appl. Anal., 15 (2012), 712 - 717.
Meral, F. C., Royston, T. J., and Magin, R., "Fractional calculus in viscoelasticity: an experimental study", Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 939 - 945.
Metzler, R., Schick, W., Kilian, H.G. and Nonnenmacher, T.F., "Relaxation in filled polymers: A fractional calculus approach." J. Chem. Phys., 103 (1995), 7180 - 7186.
Mohammed, O. H., "A direct method for solving fractional order variational problems by hat basis functions", Ain Shams Eng. J., (2016).
Riewe, F., "Nonconservative lagrangian and hamiltonian mechanics",Phys. Rev. E, 53 (1996), 1890 - 1899.
Saxena, R. K., Mathai, A. M., and Haubold, H. J., "On generalized fractional kinetic equations", Phys. A, 344 (2004), 657 - 664.
Scalas, E., Goren o, R., and Mainardi, F., "Fractional calculus and continuous-time finance." Phys. A, 284 (2000), 376 - 384.
West, B. J., "Fractional calculus in bioengineering", J. Stat. Phys., 126 (2007), 1285 - 1286.
Wony, P., "Construction of dual B-spline functions", J. Comput. Appl. Math., 260 (2014), 301 - 311.
Zhuang, P., Liu, F., Turner, I. and Gu, Y., "Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation", Appl. Math. Model., 38 (2014), 3860 - 3870.

References

Agrawal, O. P., "Generalized variational problems and EulerLagrange equations", Comput. Math. Appl., 59 (2010), 1852 - 1864.

Atanackovi, T.M., Konjik, S. and Pilipovi, S., "Variational problems with fractional derivatives: Euler Lagrange equations",J. Phys. A, 41 (2008), 095201.

Debnath, L., "Recent applications of fractional calculus to science and engineering", Int. J. Math. Math. Sci. 54 (2003), 3413 - 3442.

Dehghan, M., Hamedi, E.A. and Khosravian-Arab, H., "A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials", J. Vib. Control, 22 (2016), 1547 - 1559.

Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S., "An efficient Legendre spectral tau matrix formulation for solving fractional subdiusion and reaction subdiffusion equations", J. Comput. Dyn., 10 (2015), 021019.

Ezz-Eldien, S. S., et al., "New numerical approach for fractional variational problems using shifted Legendre orthonormal polynomials", J. Optim. Theory Appl., (2016), 1 - 26.

Ezz-Eldien, S.S., "New quadrature approach based on operational matrix for solving a class of fractional variational problems", J. Comput. Phys., 317 (2016), 362 - 381.

Goswami, J. C., Chan, A. K., and Chui, C. K., "On solving first-kind integral equations using wavelets on a bounded interval", IEEE Trans. Antennas and Propagation, 43 (1995), 614 - 622.

Ismaeelpour, T., Hemmat, A. A., and Saeedi, H., "B-spline operational matrix of fractional integration", Optik-International Journal for Light and Electron Optics, 130 (2017), 291 - 305.

Jafari, H., Tajadodi, H. and Baleanu, D., "A numerical approach for fractional order Riccati differential equation using B-spline operational matrix", Fract. Calc. Appl. Anal., 18 (2015), 387 - 399.

Khader, M.M. and Hendy, A.S., "A numerical technique for solving fractional variational problems", Math. Methods Appl. Sci., 36 (2013), 1281 - 1289.

Lewandowski, R., and Choryczewski, B., "Identification of the parameters of the KelvinVoigt and the Maxwell fractional models, used to modeling of viscoelastic dampers", Computers & structures, 88 (2010), 1 - 17.

Loverro, A., "Fractional calculus: history, definitions and applications for the engineer", Rapport technique, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering (2004), 1 - 28.

Machado, J.T., Kiryakova, V. and Mainardi, F., "Recent history of fractional calculus", Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140 - 1153.

Machado, J. T., and Mata, M. E., "Pseudo phase plane and fractional calculus modeling of western global economic downturn", Commun. Nonlinear Sci. Numer. Simul, 22 (2015), 396 - 406.

Magin, R. L., "Fractional calculus models of complex dynamics in biological tissues", Comput. Math. Appl., 59 (2010), 1586 - 1593.

Mainardi, F., "An historical perspective on fractional calculus in linear viscoelasticity", Fract. Calc. Appl. Anal., 15 (2012), 712 - 717.

Meral, F. C., Royston, T. J., and Magin, R., "Fractional calculus in viscoelasticity: an experimental study", Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 939 - 945.

Metzler, R., Schick, W., Kilian, H.G. and Nonnenmacher, T.F., "Relaxation in filled polymers: A fractional calculus approach." J. Chem. Phys., 103 (1995), 7180 - 7186.

Mohammed, O. H., "A direct method for solving fractional order variational problems by hat basis functions", Ain Shams Eng. J., (2016).

Riewe, F., "Nonconservative lagrangian and hamiltonian mechanics",Phys. Rev. E, 53 (1996), 1890 - 1899.

Saxena, R. K., Mathai, A. M., and Haubold, H. J., "On generalized fractional kinetic equations", Phys. A, 344 (2004), 657 - 664.

Scalas, E., Goren o, R., and Mainardi, F., "Fractional calculus and continuous-time finance." Phys. A, 284 (2000), 376 - 384.

West, B. J., "Fractional calculus in bioengineering", J. Stat. Phys., 126 (2007), 1285 - 1286.

Wony, P., "Construction of dual B-spline functions", J. Comput. Appl. Math., 260 (2014), 301 - 311.

Zhuang, P., Liu, F., Turner, I. and Gu, Y., "Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation", Appl. Math. Model., 38 (2014), 3860 - 3870.

Numerical Solution for A Class of Fractional Variational Problem via Second Order B-Spline Function

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