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

Noratiqah Farhana binti Ismail, Chang Phang


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.


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

Full Text:



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.



  • There are currently no refbacks.

Journal of the Indonesian Mathematical Society
Mathematics Department, Universitas Gadjah Mada
Senolowo, Sinduadi, Mlati, Sleman Regency, Special Region of Yogyakarta 55281, Telp. (0274) 552243

p-ISSN: 2086-8952 | e-ISSN: 2460-0245

Journal of the Indonesian Mathematical Society is licensed under a Creative Commons Attribution 4.0 International License

web statistics
View My Stats