Approximate Technique for Solving Class of Fractional Variational Problems

Affiliation(s)

^{1}
Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic, University (IMSIU), Riyadh, Saudi Arabia.

^{2}
Department of Mathematics, College of Science, Beni-Suef University, Beni-Suef, Egypt.

^{3}
Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt.

ABSTRACT

This paper is devoted to implementing the Legendre spectral collocation method to introduce numerical solutions of a certain class of fractional variational problems (FVPs). The properties of the Legendre polynomials and Rayleigh-Ritz method are used to reduce the FVPs to the solution of system of algebraic equations. Also, we study the convergence analysis. The obtained numerical results show the simplicity and the efficiency of the proposed method.

This paper is devoted to implementing the Legendre spectral collocation method to introduce numerical solutions of a certain class of fractional variational problems (FVPs). The properties of the Legendre polynomials and Rayleigh-Ritz method are used to reduce the FVPs to the solution of system of algebraic equations. Also, we study the convergence analysis. The obtained numerical results show the simplicity and the efficiency of the proposed method.

KEYWORDS

Fractional Variational Problems, Caputo Fractional Derivatives, Legendre Spectral Collocation Method, Rayleigh-Ritz Method, Convergence Analysis

Fractional Variational Problems, Caputo Fractional Derivatives, Legendre Spectral Collocation Method, Rayleigh-Ritz Method, Convergence Analysis

Cite this paper

Solouma, E. , Khader, M. (2015) Approximate Technique for Solving Class of Fractional Variational Problems.*Applied Mathematics*, **6**, 837-846. doi: 10.4236/am.2015.65078.

Solouma, E. , Khader, M. (2015) Approximate Technique for Solving Class of Fractional Variational Problems.

References

[1] Agrawal, O.P. (2002) Formulation of Euler-Lagrange Equations for Fractional Variational Problems. Journal of Mathematical Analysis and Application, 272, 368-379.

http://dx.doi.org/10.1016/S0022-247X(02)00180-4

[2] Agrawal, O.P. (2008) A General Finite Element Formulation for Fractional Variational Problems. Journal of Mathematical Analysis and Application, 337, 1-12.

http://dx.doi.org/10.1016/j.jmaa.2007.03.105

[3] Dehghan, M. and Tatari, M. (2006) The Use of Adomian Decomposition Method for Solving Problems in Calculus of Variations. Mathematical Problems in Engineering, 2006, 12 p.

http://dx.doi.org/10.1155/MPE/2006/65379

[4] Elsgolts, L. (1977) Differential Equations and the Calculus of Variations. Translated from the Russian by G. Yankovsky, Mir, Moscow.

[5] Agrawal, O.P. (2001) A New Lagrangian and a New Lagrange Equation of Motion for Fractionally Damped Systems. Journal of Applied Mechanics, 68, 339-341.

http://dx.doi.org/10.1115/1.1352017

[6] Lotfi, A. and Yousefi, S.A. (2013) A Numerical Technique for Solving a Class of Fractional Variational Problems. Journal of Computational and Applied Mathematics, 237, 633-643.

http://dx.doi.org/10.1016/j.cam.2012.08.005

[7] Atanackovic, T.M., Konjik, S., Pilipovi , S. and Simic, S. (2009) Variational Problems with Fractional Derivatives: Invariance Conditions and Nöther’s Theorem. Nonlinear Analysis: Theory, Methods & Applications, 71, 1504-1517.

http://dx.doi.org/10.1016/j.na.2008.12.043

[8] Gelfand, I.M. and Fomin, S.V. (1963) Calculus of Variations. Revised English Edition Translated and Edited by R. A. Silverman, Prentice-Hall, New Jersey.

[9] Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S. (2011) Efficient Chebyshev Spectral Methods for Solving Multi-Term Fractional Orders Differential Equations. Applied Mathematical Modelling, 35, 5662-5672.

http://dx.doi.org/10.1016/j.apm.2011.05.011

[10] Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S. (2015) An Efficient Legendre Spectral Tau Matrix Formulation for Solving Fractional Sub-Diffusion and Reaction Sub-Diffusion Equations. Journal of Computational and Nonlinear Dynamics, 10, Article ID: 021019.

http://dx.doi.org/10.1115/1.4027944

[11] Bhrawy, A.H., Doha, E.H., Ezz-Eldien, S.S. and Abdelkawy, M.A. (2015) A Numerical Technique Based on the Shifted Legendre Polynomials for Solving the Time-Fractional Coupled KdV Equation. Calcolo, in press.

http://dx.doi.org/10.1007/s10092-014-0132-x

[12] Bhrawy, A.H. and Abdelkawy, M.A. (2015) A Fully Spectral Collocation Approximation for Multi-Dimensional Fractional Schrödinger Equations. Journal of Computational Physics, 294, 462-483.

http://dx.doi.org/10.1016/j.jcp.2015.03.063

[13] Bhrawy, A.H. and Zaky, M.A. (2015) A Method Based on the Jacobi Tau Approximation for Solving Multi-Term Time-Space Fractional Partial Differential Equations. Journal of Comptuational Physics, 281, 876-895.

http://dx.doi.org/10.1016/j.jcp.2014.10.060

[14] Bell, W.W. (1968) Special Functions for Scientists and Engineers. Butler and Tanner Ltd., Frome.

[15] Khader, M.M. (2011) On the Numerical Solutions for the Fractional Diffusion Equation. Communications in Nonlinear Science and Numerical Simulation, 16, 2535-2542.

http://dx.doi.org/10.1016/j.cnsns.2010.09.007

[16] Bhrawy, A.H., Doha, E.H., Tenreiro Machado, J.A. and Ezz-Eldien, S.S. (2015) An Efficient Numerical Scheme for Solving Multi-Dimensional Fractional Optimal Control Problems with a Quadratic Performance Index. Asian Journal of Control, in press.

http://dx.doi.org/10.1002/asjc.1109

[17] Bhrawy, A.H., Doha, E.H., Baleanu, D., Ezz-Eldien, S.S. and Abdelkawy, M.A. (2015) An Accurate Numerical Technique for Solving Fractional Optimal Control Problems. Proceedings of the Romanian Academy Series A, 16, 47-54.

[18] Khader, M.M. and Hendy, A.S. (2012) The Approximate and Exact Solutions of the Fractional-Order Delay Differential Equations Using Legendre Pseudo-Spectral Method. International Journal of Pure and Applied Mathematics, 74, 287-297.

[19] Li, C.P., Zeng, F.H. and Liu, F.W. (2012) Spectral Approximations to the Fractional Integral and Derivative. Fractional Calculus and Applied Analysis, 15, 383-406.

http://dx.doi.org/10.2478/s13540-012-0028-x

[20] Khader, M.M. (2013) Numerical Treatment for Solving the Perturbed Fractional PDEs Using Hybrid Techniques. Journal of Computational Physics, 250, 565-573.

http://dx.doi.org/10.1016/j.jcp.2013.05.032

[21] Funaro, D. (1992) Polynomial Approximation of Differential Equations. Springer Verlag, New York.

[22] Khader, M.M. (2013) Numerical Treatment for Solving Fractional Riccati Differential Equation. Journal of the Egyptian Mathematical Society, 21, 32-37.

http://dx.doi.org/10.1016/j.joems.2012.09.005

[23] Khader, M.M. (2014) On the Numerical Solution and Convergence Study for System of Non-Linear Fractional Diffusion Equations. Canadian Journal of Physics, 92, 1658-1666.

http://dx.doi.org/10.1139/cjp-2013-0464

[24] Khader, M.M., Sweilam, N.H. and Mahdy, A.M.S. (2013) Numerical Study for the Fractional Differential Equations Generated by Optimization Problem Using Chebyshev Collocation Method and FDM. Applied Mathematics and Information Science, 7, 2011-2018.

http://dx.doi.org/10.12785/amis/070541

[25] Khader, M.M., El Danaf, T.S. and Hendy, A.S. (2013) A Computational Matrix Method for Solving Systems of High Order Fractional Differential Equations. Applied Mathematical Modelling, 37, 4035-4050.

http://dx.doi.org/10.1016/j.apm.2012.08.009

[26] Sweilam, N.H. and Khader, M.M. (2010) A Chebyshev Pseudo-Spectral Method for Solving Fractional Order Integro-Differential Equations. ANZIAM Journal, 51, 464-475.

http://dx.doi.org/10.1017/S1446181110000830

[27] Sweilam, N.H., Khader, M.M. and Mahdy, A.M.S. (2012) Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays. Journal of Applied Mathematics, 2012, Article ID: 764894.

[28] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, New York.

[29] Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wily and Sons Inc., New York.

[1] Agrawal, O.P. (2002) Formulation of Euler-Lagrange Equations for Fractional Variational Problems. Journal of Mathematical Analysis and Application, 272, 368-379.

http://dx.doi.org/10.1016/S0022-247X(02)00180-4

[2] Agrawal, O.P. (2008) A General Finite Element Formulation for Fractional Variational Problems. Journal of Mathematical Analysis and Application, 337, 1-12.

http://dx.doi.org/10.1016/j.jmaa.2007.03.105

[3] Dehghan, M. and Tatari, M. (2006) The Use of Adomian Decomposition Method for Solving Problems in Calculus of Variations. Mathematical Problems in Engineering, 2006, 12 p.

http://dx.doi.org/10.1155/MPE/2006/65379

[4] Elsgolts, L. (1977) Differential Equations and the Calculus of Variations. Translated from the Russian by G. Yankovsky, Mir, Moscow.

[5] Agrawal, O.P. (2001) A New Lagrangian and a New Lagrange Equation of Motion for Fractionally Damped Systems. Journal of Applied Mechanics, 68, 339-341.

http://dx.doi.org/10.1115/1.1352017

[6] Lotfi, A. and Yousefi, S.A. (2013) A Numerical Technique for Solving a Class of Fractional Variational Problems. Journal of Computational and Applied Mathematics, 237, 633-643.

http://dx.doi.org/10.1016/j.cam.2012.08.005

[7] Atanackovic, T.M., Konjik, S., Pilipovi , S. and Simic, S. (2009) Variational Problems with Fractional Derivatives: Invariance Conditions and Nöther’s Theorem. Nonlinear Analysis: Theory, Methods & Applications, 71, 1504-1517.

http://dx.doi.org/10.1016/j.na.2008.12.043

[8] Gelfand, I.M. and Fomin, S.V. (1963) Calculus of Variations. Revised English Edition Translated and Edited by R. A. Silverman, Prentice-Hall, New Jersey.

[9] Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S. (2011) Efficient Chebyshev Spectral Methods for Solving Multi-Term Fractional Orders Differential Equations. Applied Mathematical Modelling, 35, 5662-5672.

http://dx.doi.org/10.1016/j.apm.2011.05.011

[10] Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S. (2015) An Efficient Legendre Spectral Tau Matrix Formulation for Solving Fractional Sub-Diffusion and Reaction Sub-Diffusion Equations. Journal of Computational and Nonlinear Dynamics, 10, Article ID: 021019.

http://dx.doi.org/10.1115/1.4027944

[11] Bhrawy, A.H., Doha, E.H., Ezz-Eldien, S.S. and Abdelkawy, M.A. (2015) A Numerical Technique Based on the Shifted Legendre Polynomials for Solving the Time-Fractional Coupled KdV Equation. Calcolo, in press.

http://dx.doi.org/10.1007/s10092-014-0132-x

[12] Bhrawy, A.H. and Abdelkawy, M.A. (2015) A Fully Spectral Collocation Approximation for Multi-Dimensional Fractional Schrödinger Equations. Journal of Computational Physics, 294, 462-483.

http://dx.doi.org/10.1016/j.jcp.2015.03.063

[13] Bhrawy, A.H. and Zaky, M.A. (2015) A Method Based on the Jacobi Tau Approximation for Solving Multi-Term Time-Space Fractional Partial Differential Equations. Journal of Comptuational Physics, 281, 876-895.

http://dx.doi.org/10.1016/j.jcp.2014.10.060

[14] Bell, W.W. (1968) Special Functions for Scientists and Engineers. Butler and Tanner Ltd., Frome.

[15] Khader, M.M. (2011) On the Numerical Solutions for the Fractional Diffusion Equation. Communications in Nonlinear Science and Numerical Simulation, 16, 2535-2542.

http://dx.doi.org/10.1016/j.cnsns.2010.09.007

[16] Bhrawy, A.H., Doha, E.H., Tenreiro Machado, J.A. and Ezz-Eldien, S.S. (2015) An Efficient Numerical Scheme for Solving Multi-Dimensional Fractional Optimal Control Problems with a Quadratic Performance Index. Asian Journal of Control, in press.

http://dx.doi.org/10.1002/asjc.1109

[17] Bhrawy, A.H., Doha, E.H., Baleanu, D., Ezz-Eldien, S.S. and Abdelkawy, M.A. (2015) An Accurate Numerical Technique for Solving Fractional Optimal Control Problems. Proceedings of the Romanian Academy Series A, 16, 47-54.

[18] Khader, M.M. and Hendy, A.S. (2012) The Approximate and Exact Solutions of the Fractional-Order Delay Differential Equations Using Legendre Pseudo-Spectral Method. International Journal of Pure and Applied Mathematics, 74, 287-297.

[19] Li, C.P., Zeng, F.H. and Liu, F.W. (2012) Spectral Approximations to the Fractional Integral and Derivative. Fractional Calculus and Applied Analysis, 15, 383-406.

http://dx.doi.org/10.2478/s13540-012-0028-x

[20] Khader, M.M. (2013) Numerical Treatment for Solving the Perturbed Fractional PDEs Using Hybrid Techniques. Journal of Computational Physics, 250, 565-573.

http://dx.doi.org/10.1016/j.jcp.2013.05.032

[21] Funaro, D. (1992) Polynomial Approximation of Differential Equations. Springer Verlag, New York.

[22] Khader, M.M. (2013) Numerical Treatment for Solving Fractional Riccati Differential Equation. Journal of the Egyptian Mathematical Society, 21, 32-37.

http://dx.doi.org/10.1016/j.joems.2012.09.005

[23] Khader, M.M. (2014) On the Numerical Solution and Convergence Study for System of Non-Linear Fractional Diffusion Equations. Canadian Journal of Physics, 92, 1658-1666.

http://dx.doi.org/10.1139/cjp-2013-0464

[24] Khader, M.M., Sweilam, N.H. and Mahdy, A.M.S. (2013) Numerical Study for the Fractional Differential Equations Generated by Optimization Problem Using Chebyshev Collocation Method and FDM. Applied Mathematics and Information Science, 7, 2011-2018.

http://dx.doi.org/10.12785/amis/070541

[25] Khader, M.M., El Danaf, T.S. and Hendy, A.S. (2013) A Computational Matrix Method for Solving Systems of High Order Fractional Differential Equations. Applied Mathematical Modelling, 37, 4035-4050.

http://dx.doi.org/10.1016/j.apm.2012.08.009

[26] Sweilam, N.H. and Khader, M.M. (2010) A Chebyshev Pseudo-Spectral Method for Solving Fractional Order Integro-Differential Equations. ANZIAM Journal, 51, 464-475.

http://dx.doi.org/10.1017/S1446181110000830

[27] Sweilam, N.H., Khader, M.M. and Mahdy, A.M.S. (2012) Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays. Journal of Applied Mathematics, 2012, Article ID: 764894.

[28] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, New York.

[29] Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wily and Sons Inc., New York.