• Journal of the Korean Mathematical Society
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• v.52 no.5
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• pp.1069-1096
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• 2015

In this article we introduce a numerical method, named Gegenbauer wavelets method, which is derived from conventional Gegenbauer polynomials, for solving fractional initial and boundary value problems. The operational matrices are derived and utilized to reduce the linear fractional differential equation to a system of algebraic equations. We perform the convergence analysis for the Gegenbauer wavelets method. We also combine Gegenbauer wavelets operational matrix method with quasilinearization technique for solving fractional nonlinear differential equation. Quasilinearization technique is used to discretize the nonlinear fractional ordinary differential equation and then the Gegenbauer wavelet method is applied to discretized fractional ordinary differential equations. In each iteration of quasilinearization technique, solution is updated by the Gegenbauer wavelet method. Numerical examples are provided to illustrate the efficiency and accuracy of the methods.

• The Pure and Applied Mathematics
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• v.21 no.3
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• pp.207-218
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• 2014

The principal object of this paper is to study a class of matrix polynomials associated with Humbert polynomials. These polynomials generalize the well known class of Gegenbauer, Legendre, Pincherl, Horadam, Horadam-Pethe and Kinney polynomials. We shall give some basic relations involving the Humbert matrix polynomials and then take up several generating functions, hypergeometric representations and expansions in series of matrix polynomials.

• Korean Journal of Mathematics
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• v.32 no.1
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• pp.183-193
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• 2024

In this paper, a new class of bi-univalent functions that are described by Gegenbauer polynomials is presented. We obtain the estimates of the Taylor-Maclaurin coefficients |m₂| and |m₃| for each function in this class of bi-univalent functions. In addition, the Fekete-Szegö problems function new are also studied.

• Kyungpook Mathematical Journal
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• v.62 no.2
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• pp.257-269
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• 2022

In the present paper, a subclass of analytic and bi-univalent functions is defined using a symmetric q-derivative operator by means of Gegenbauer polynomials. Coefficients bounds for functions belonging to this subclass are obtained. Furthermore, the Fekete-Szegö problem for this subclass is solved. A number of known or new results are shown to follow upon specializing the parameters involved in our main results.

• Journal of Power Electronics
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• v.19 no.3
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• pp.771-783
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• 2019

Due to nonlinear-synthetic uncertainty including the total unknown nonlinear load torque, the total parameter variation and the fixed load torque, a synchronous reluctance motor (SynRM) driving a continuously variable transmission (CVT) system causes a lot of nonlinear effects. Linear control methods make it hard to achieve good control performance. To increase the control performance and reduce the influence of nonlinear time-synthetic uncertainty, an admixed recurrent Gegenbauer orthogonal polynomials neural network (ARGOPNN) with a modified particle swarm optimization (MPSO) control system is proposed to achieve better control performance. The ARGOPNN with a MPSO control system is composed of an observer controller, a recurrent Gegenbauer orthogonal polynomial neural network (RGOPNN) controller and a remunerated controller. To insure the stability of the control system, the RGOPNN controller with an adaptive law and the remunerated controller with a reckoned law are derived according to the Lyapunov stability theorem. In addition, the two learning rates of the weights in the RGOPNN are regulating by using the MPSO algorithm to enhance convergence. Finally, three types of experimental results with comparative studies are presented to confirm the usefulness of the proposed ARGOPNN with a MPSO control system.

• Journal of Advanced Navigation Technology
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• v.11 no.2
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• pp.196-202
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• 2007

In this paper, The TE(transverse electric) scattering problems by a resistive strip grating over a grounded dielectric plane with edge boundary condition are analyzed by applying the FGMM(Fourier-Galerkin Moment Method) known as a numerical procedure. For a TE scattering problem, the induced surface current density is expected to the zero value at both edges of the resistive strip, then the induced surface current density on the resistive strip is expanded in a series of the multiplication of Gegenbauer(Ultraspherical) polynomials with the first order and functions of appropriate edge boundary condition. To verify the validity of the proposed method, the numerical results of normalized reflected power for the uniform resistivity R = 100 ohms/square and R = 0 as a conductive strip case show in good agreement with those in the existing papers.

• International Journal of Naval Architecture and Ocean Engineering
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• v.11 no.2
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• pp.952-969
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• 2019

This study develops new analytical solutions to water wave diffraction by vertical truncated cylinders in the context of linear potential theory. Three typical truncated surface-piercing cylinders, a submerged bottom-standing cylinder and a submerged floating cylinder are examined. The analytical solutions utilize the multi-term Galerkin method, which is able to model the cube-root singularity of fluid velocity near the edges of the truncated cylinders by expanding the fluid velocity into a set of basis function involving the Gegenbauer polynomials. The convergence of the present analytical solution is rapid, and a few truncated numbers in the series of the basis function can yield results of six-figure accuracy for wave forces and moments. The present solutions are in good agreement with those by a higher-order BEM (boundary element method) model. Comparisons between present results and experimental results in literature and results by Froude-Krylov theory are conducted. The variation of wave forces and moments with different parameters are presented. This study not only gives a new analytical approach to wave diffraction by truncated cylinders but also provides a reliable benchmark for numerical investigations of wave diffraction by structures.