A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam Polynomials

Mathematics, Journal Year: 2025, Volume and Issue: 13(2), P. 300 - 300

Published: Jan. 17, 2025

This paper proposes a numerical algorithm for the nonlinear fifth-order Korteweg–de Vries equations. class of equations is known its significance in modeling various complex wave phenomena physics and engineering. The approximate solutions are expressed terms certain shifted Horadam polynomials. A theoretical background these polynomials first introduced. derivatives their operational metrics established to tackle problem using typical collocation method transform equation governed by underlying conditions into system algebraic equations, thereby obtaining solutions. also includes rigorous convergence analysis proposed expansion. To validate method, we present several examples illustrating accuracy effectiveness.

Language: Английский

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An Efficient Petrov–Galerkin Scheme for the Euler–Bernoulli Beam Equation via Second-Kind Chebyshev Polynomials

Fractal and Fractional, Journal Year: 2025, Volume and Issue: 9(2), P. 78 - 78

Published: Jan. 24, 2025

The current article introduces a Petrov–Galerkin method (PGM) to address the fourth-order uniform Euler–Bernoulli pinned–pinned beam equation. Utilizing suitable combination of second-kind Chebyshev polynomials as basis in spatial variables, proposed elegantly and simultaneously satisfies clamped–clamped boundary conditions. To make PGM application easier, explicit formulas for inner product between these functions their derivatives with are derived. This leads simplified system algebraic equations recognizable pattern that facilitates effective inversion produce an approximate spectral solution. Presentations made regarding method’s convergence analysis computational cost matrix inversion. efficiency described precisely solving equation under different scenarios has been validated by numerical testing. Additionally, procedure this paper is more compared other existing techniques.

Language: Английский

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A Collocation Procedure for Treating the Time-Fractional FitzHugh–Nagumo Differential Equation Using Shifted Lucas Polynomials

Mathematics, Journal Year: 2024, Volume and Issue: 12(23), P. 3672 - 3672

Published: Nov. 23, 2024

This work employs newly shifted Lucas polynomials to approximate solutions the time-fractional Fitzhugh–Nagumo differential equation (TFFNDE) relevant neuroscience. Novel essential formulae for are crucial developing our suggested numerical approach. The analytic and inversion formulas introduced, after that, new that express these polynomials’ integer fractional derivatives derived facilitate construction of operational matrices derivatives. Employing with typical collocation method converts TFFNDE into a system algebraic equations can be addressed standard solvers. convergence analysis expansion is carefully investigated. Certain inequalities involving golden ratio established in this context. evaluated using several examples verify its applicability efficiency.

Language: Английский

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Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation

AIMS Mathematics, Journal Year: 2024, Volume and Issue: 9(12), P. 34567 - 34587

Published: Jan. 1, 2024

<p>Here, we provide a new method to solve the time-fractional diffusion equation (TFDE) following spectral tau approach. Our proposed numerical solution is expressed in terms of double Lucas expansion. The discretization technique based on several formulas about polynomials, such as those for explicit integer and fractional derivatives, products, certain definite integrals these polynomials. These aid transforming TFDE its conditions into matrix system that can be treated through suitable procedure. We conduct study convergence analysis In addition, few examples ensure approach applicable efficient.</p>

Language: Английский

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A collocation procedure for the numerical treatment of FitzHugh–Nagumo equation using a kind of Chebyshev polynomials

AIMS Mathematics, Journal Year: 2025, Volume and Issue: 10(1), P. 1201 - 1223

Published: Jan. 1, 2025

<p>This manuscript aims to provide numerical solutions for the FitzHugh–Nagumo (FH–N) problem. The suggested approximate are spectral and may be achieved using standard collocation technique. We introduce utilize specific polynomials of generalized Gegenbauer polynomials. These introduced have connections with Chebyshev polynomials' series representation, orthogonality property, derivative expressions among new formulas developed these transform obtain their counterparts shifted polynomials, which serve as basis functions solutions. convergence expansion is thoroughly examined. several tests comparisons confirm applicability accuracy our proposed algorithm.</p>

Language: Английский

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A hybrid method based on the classical/piecewise Chebyshev cardinal functions for multi-dimensional fractional Rayleigh–Stokes equations

Results in Applied Mathematics, Journal Year: 2025, Volume and Issue: 25, P. 100541 - 100541

Published: Jan. 30, 2025

Language: Английский

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Approximate Petrov–Galerkin Solution for the Time Fractional Diffusion Wave Equation

Mathematical Methods in the Applied Sciences, Journal Year: 2025, Volume and Issue: unknown

Published: April 20, 2025

ABSTRACT This paper discusses the Petrov–Galerkin method's application in solving time fractional diffusion wave equation (TFDWE). The method is based on using two modified sets of shifted fourth‐kind Chebyshev polynomials (FKCPs) as basis functions. explicit forms all spectral matrices were reported. These are essential to transforming TFDWE and its underlying homogeneous conditions into a matrix system. An appropriate algorithm can be used solve this system obtain desired approximate solutions. error analysis was studied depth. Four numerical examples provided that included comparisons with other existing methods literature.

Language: Английский

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On generalized Hermite polynomials

AIMS Mathematics, Journal Year: 2024, Volume and Issue: 9(11), P. 32463 - 32490

Published: Jan. 1, 2024

<p>This article is devoted to establishing new formulas concerning generalized Hermite polynomials (GHPs) that generalize the classical polynomials. Derivative expressions of these involve one parameter are found in terms other Some important formulas, such as linearization and connection between some polynomials, also given. Most coefficients represented hypergeometric functions can be reduced specific cases using standard formulas. Two applications developed this paper The first application concerned with introducing weighted definite integrals involving GHPs. In contrast, second operational matrix integer derivatives GHPs.</p>

Language: Английский

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Generalized third-kind Chebyshev tau approach for treating the time fractional cable problem

Electronic Research Archive, Journal Year: 2024, Volume and Issue: 32(11), P. 6200 - 6224

Published: Jan. 1, 2024

<p>This work introduces a computational method for solving the time-fractional cable equation (TFCE). We utilize tau numerical treatment of TFCE, using generalized Chebyshev polynomials third kind (GCPs3) as basis functions. The integer and fractional derivatives GCPs3 are essential formulas that serve to transform TFCE with its underlying conditions into matrix system. This system can be solved suitable algorithm obtain desired approximate solutions. error bound resulting from approximation by proposed is given. has been validated against existing methods presenting examples.</p>

Language: Английский

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A numerical approach for multi-dimensional ψ-Hilfer fractional nonlinear Galilei invariant advection–diffusion equations

Results in Physics, Journal Year: 2024, Volume and Issue: 68, P. 108067 - 108067

Published: Dec. 10, 2024

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