This article employs certain polynomials that generalize standard Fermat polynomials, called convolved Fermat polynomials, to numerically solve the fractional Burgers’ equation. New theoretical results of these polynomials are developed and utilized along with the collocation method to find approximate solutions of the fractional Burgers’ equation. The basic idea behind the proposed numerical algorithm is based on establishing the operational matrices of derivatives of both integer and fractional derivatives of the convolved Fermat polynomials that help to convert the equation governed by its underlying conditions into an algebraic system of equations that can be treated numerically. A comprehensive study is performed to analyze the error of the proposed convolved Fermat expansion. Some numerical examples are presented to test our proposed numerical algorithm, and some comparisons are made. The results indicate that the proposed algorithm is applicable and accurate.

In this paper, we present a collocation algorithm for numerically treating the time-fractional Kuramoto–Sivashinsky equation (TFKSE). Certain orthogonal polynomials, which are expressed as combinations of Chebyshev polynomials, and their shifted polynomials are introduced. Some new theoretical formulas regarding these polynomials have been developed, including their operational matrices of both integer and fractional derivatives. The derived formulas will be the foundation for designing the proposed numerical algorithm, which relies on converting the governing problem with its underlying conditions into a nonlinear algebraic system, which can be solved using Newton’s iteration technique. A rigorous error analysis for the proposed combined Chebyshev expansion is presented. Some numerical examples are given to ensure the applicability and efficiency of the presented algorithm. These results demonstrate that the proposed algorithm attains superior accuracy with fewer expansion terms.

This study develops an effective numerical method for addressing the time-fractional gas dynamics equation formulated with the Caputo time-fractional derivative. Novel basis functions are utilized, formulated as particular generalized Fibonacci polynomials contingent on a free parameter. This family generalizes the second kind of Chebyshev family. For the proposed polynomials, we establish basic analytical properties, including closed-form series expansion, inverse relation, moment and linearization formulas, and operational matrices for both integer-order and Caputo fractional derivatives. Using these tools, the fractional model, together with its underlying conditions, can be transformed into a finite system of nonlinear algebraic equations via a collocation strategy. Using Newton’s iterative method, the resulting system can be treated. A full convergence analysis of the double generalized Chebyshev expansion is provided. We demonstrate the accuracy and reliability of the presented method through several numerical simulations. Comparisons with existing numerical methods show that this approach achieves higher accuracy and faster execution.

In order to establish and assess the dynamics of kink solitons in the integrable Kairat-X equation, which explains the differential geometry of curves and equivalence aspects, the present investigation put forward two variants of a unique transformation-based analytical technique. These modifications were referred to as the generalized ($ r+\frac{G'}{G} $)-expansion method and the simple ($ \frac{G'}{G} $)-expansion approach. The proposed methods spilled over the aimed Kairat-X equation into a nonlinear ordinary differential equation by means of a variable transformation. Immediately following that, it was presumed that the resultant nonlinear ordinary differential equation had a closed form solution, which turned it into a system of algebraic equations. The resultant set of algebraic equations was solved to find new families of soliton solutions which took the forms of hyperbolic, rational and periodic functions. An assortment of contour, 2D and 3D graphs were used to visually show the dynamics of certain generated soliton solutions. This indicated that these soliton solutions likely took the structures of kink solitons prominently. Moreover, our proposed methods demonstrated their use by constructing a multiplicity of soliton solutions, offering significant understanding into the evolution of the focused model, and suggesting possible applications in dealing with related nonlinear phenomena.

I investigated soliton phenomena in a prominent nonlinear fractional partial differential equation (FPDE) namely the conformable coupled Drinfeld-Sokolov-Wilson system (CCDSWS) using a novel variant of the novel extended direct algebraic method (EDAM), namely $ r $+mEDAM. The conformable fractional derivatives are used to generalize the model due to the memory and hereditary features that are inherent in the fractional dynamics. The model was initially transformed into a more manageable system of integer-order nonlinear ordinary differential equations (NODEs) through the implementation of complex transformation. The obtained system of NODEs is further transformed into a system of algebraic equations, which yields, by solving new plethora of soliton solutions for CCDSWS in the form of generalized trigonometrical, exponential hyperbolical, and rational functions. Moreover, we employed 2D, 3D, and contour graphics to show the behavior of acquired solitons, making it abundantly evident that the obtained solitons take the shape of kink, anti-kink, bright, dark, bright-dark, and bell-shaped kink solitons within the framework of CCDSWS. The results confirmed the efficiency of the presented approach in finding solitonic solutions, which in its turn expands knowledge of nonlinear FPDEs. The aimed to theoretical and application perspectives in fractional solitons applicable in areas such fluid mechanics, plasma physic, optical communications, etc.

A spectral Galerkin algorithm is developed for the numerical treatment of modified anomalous time-fractional sub-diffusion equations (MATFSDEs). The algorithm is constructed using Fibonacci coefficient polynomials, from which two space–time trial families are formed. These families are selected so that the homogeneous initial and boundary conditions are automatically incorporated after a suitable transformation of the original problem. The Galerkin formulation then reduces the model to a finite algebraic matrix system whose entries can be explicitly evaluated. To support the construction, inversion, moment, and linearization identities for the selected polynomials are obtained and then used to express the required matrices in closed form. The convergence of the expansion is investigated, and explicit error bounds are derived. Numerical tests are reported to demonstrate the accuracy, applicability, and competitiveness of the proposed scheme in comparison with some existing methods in the literature.