Next Article in Journal
Study of a High Order Family: Local Convergence and Dynamics
Next Article in Special Issue
On the Solvability of a Mixed Problem for a High-Order Partial Differential Equation with Fractional Derivatives with Respect to Time, with Laplace Operators with Spatial Variables and Nonlocal Boundary Conditions in Sobolev Classes
Previous Article in Journal
On Ulam Stability and Multiplicity Results to a Nonlinear Coupled System with Integral Boundary Conditions
Previous Article in Special Issue
Approximate Controllability of Sub-Diffusion Equation with Impulsive Condition
Article Menu

Export Article

Open AccessArticle
Mathematics 2019, 7(3), 224; https://doi.org/10.3390/math7030224

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.
Received: 15 January 2019 / Revised: 22 February 2019 / Accepted: 22 February 2019 / Published: 27 February 2019
Full-Text   |   PDF [5440 KB, uploaded 6 March 2019]   |  

Abstract

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
Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
SciFeed

Share & Cite This Article

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.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Mathematics EISSN 2227-7390 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top