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.
Highlights
It is necessary to determine the maxima and minima of certain functionals in study problems in analysis, mechanics, and geometry
We extend the Rayleigh-Ritz method together with operational matrices of different orthogonal polynomials such as Shifted Legendre polynomials, Shifted Chebyshev polynomials of the first kind, Shifted Chebyshev polynomials of the third kind, Shifted Chebyshev polynomials of the fourth kind, and Gegenbauer polynomials to solve a special class of non-linear fractional variational problems (NLFVPs)
We have the following non-linear fractional variational problem [19]:
Summary
It is necessary to determine the maxima and minima of certain functionals in study problems in analysis, mechanics, and geometry. Due to the importance of the fractional derivatives established through real-life applications, several authors have considered problems in calculus of variations by replacing the integer-order derivative with fractional orders in objective functionals, and this is known as fractional calculus of variations. Some of these studies are of a fractionally damped system [15], energy control for a fractional linear control system [16], a fractional model of a vibrating string [17], Mathematics 2019, 7, 224; doi:10.3390/math7030224 www.mdpi.com/journal/mathematics
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