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Article

Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials

1
Department of Mathematics, Post Graduate College, Ghazipur 233001, India
2
Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi 221005, India
3
Centre for Advanced Biomaterials and Tissue Engineering, Indian Institute of Technology (BHU) Varanasi, Varanasi 221005, India
4
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
5
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(3), 224; https://doi.org/10.3390/math7030224
Received: 15 January 2019 / Revised: 22 February 2019 / Accepted: 22 February 2019 / Published: 27 February 2019
The aim of this paper is to solve a class of non-linear fractional variational problems (NLFVPs) using the Ritz method and to perform a comparative study on the choice of different polynomials in the method. The Ritz method has allowed many researchers to solve different forms of fractional variational problems in recent years. The NLFVP is solved by applying the Ritz method using different orthogonal polynomials. Further, the approximate solution is obtained by solving a system of nonlinear algebraic equations. Error and convergence analysis of the discussed method is also provided. Numerical simulations are performed on illustrative examples to test the accuracy and applicability of the method. For comparison purposes, different polynomials such as 1) Shifted Legendre polynomials, 2) Shifted Chebyshev polynomials of the first kind, 3) Shifted Chebyshev polynomials of the third kind, 4) Shifted Chebyshev polynomials of the fourth kind, and 5) Gegenbauer polynomials are considered to perform the numerical investigations in the test examples. Further, the obtained results are presented in the form of tables and figures. The numerical results are also compared with some known methods from the literature. View Full-Text
Keywords: non-linear fractional variational problems; orthogonal polynomials; Rayleigh-Ritz method; error analysis; convergence analysis non-linear fractional variational problems; orthogonal polynomials; Rayleigh-Ritz method; error analysis; convergence analysis
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MDPI and ACS Style

Singh, H.; Pandey, R.K.; Srivastava, H.M. Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials. Mathematics 2019, 7, 224. https://doi.org/10.3390/math7030224

AMA Style

Singh H, Pandey RK, Srivastava HM. Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials. Mathematics. 2019; 7(3):224. https://doi.org/10.3390/math7030224

Chicago/Turabian Style

Singh, Harendra; Pandey, Rajesh K.; Srivastava, Hari M. 2019. "Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials" Mathematics 7, no. 3: 224. https://doi.org/10.3390/math7030224

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