期刊
FRACTAL AND FRACTIONAL
卷 7, 期 9, 页码 -出版社
MDPI
DOI: 10.3390/fractalfract7090652
关键词
time-fractional Kawahara equation; generalized Gegenbauer polynomials; Chebyshev polynomials; collocation method; connection formulas; convergence analysis
In this study, an innovative approach using a spectral collocation algorithm is proposed to obtain numerical solutions of the nonlinear time-fractional generalized Kawahara equation. The method involves introducing a new set of orthogonal polynomials as fundamental functions and transforming the equation into a set of nonlinear algebraic equations. The effectiveness and reliability of the approach are validated through rigorous analysis and numerical experiments.
In this study, we present an innovative approach involving a spectral collocation algorithm to effectively obtain numerical solutions of the nonlinear time-fractional generalized Kawahara equation (NTFGKE). We introduce a new set of orthogonal polynomials (OPs) referred to as Eighth-kind Chebyshev polynomials (CPs). These polynomials are special kinds of generalized Gegenbauer polynomials. To achieve the proposed numerical approximations, we first derive some new theoretical results for eighth-kind CPs, and after that, we employ the spectral collocation technique and incorporate the shifted eighth-kind CPs as fundamental functions. This method facilitates the transformation of the equation and its inherent conditions into a set of nonlinear algebraic equations. By harnessing Newton's method, we obtain the necessary semi-analytical solutions. Rigorous analysis is dedicated to evaluating convergence and errors. The effectiveness and reliability of our approach are validated through a series of numerical experiments accompanied by comparative assessments. By undertaking these steps, we seek to communicate our findings comprehensively while ensuring the method's applicability and precision are demonstrated.
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