Monic Chebyshev pseudospectral differentiation matrices for higher-order IVPs and BVPs: applications to certain types of real-life problems

Abdelhakem, M.; Ahmed, Abd El-Latif S.; Băleanu, Dumitru; El-Kady, M.

doi:10.1007/s40314-022-01940-0

Cited by 21 publications

(7 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Talaei and Asgar [41] and Chen et al [42] solved numerically this problem. In [41], the authors suggested an operational approach based on the Chelyshkov-collocation spectral method (CCSM) for the numerical solution of (34), while the authors in [42] applied the Haar wavelets method (HWM) for the numerical treatment of (34). The L 2 and L ∞ -errors of our presented method for different values of N with υ 1 = 0.25 and b = 2 are shown in Table 1.…”

Section: Illustrative Problems and Comparisonsmentioning

confidence: 99%

See 1 more Smart Citation

Numerical Treatment of Multi-Term Fractional Differential Equations via New Kind of Generalized Chebyshev Polynomials

Abd-Elhameed

2023

Fractal Fract

View full text Add to dashboard Cite

The main aim of this paper is to introduce a new class of orthogonal polynomials that generalizes the class of Chebyshev polynomials of the first kind. Some basic properties of the generalized Chebyshev polynomials and their shifted ones are established. Additionally, some new formulas concerned with these generalized polynomials are established. These generalized orthogonal polynomials are employed to treat the multi-term linear fractional differential equations (FDEs) that include some specific problems that arise in many applications. The basic idea behind the derivation of our proposed algorithm is built on utilizing a new power form representation of the shifted generalized Chebyshev polynomials along with the application of the spectral Galerkin method to transform the FDE governed by its initial conditions into a system of linear equations that can be efficiently solved via a suitable numerical solver. Some illustrative examples accompanied by comparisons with some other methods are presented to show that the presented algorithm is useful and effective.

show abstract

Section: Illustrative Problems and Comparisonsmentioning

confidence: 99%

“…In addition, they employed this class of polynomials to treat certain types of even-order BVPs. The authors in [33,34] handled some BVPs and IVP using the Chebyshev polynomials' first derivative.…”

Section: Introductionmentioning

confidence: 99%

Numerical Treatment of Multi-Term Fractional Differential Equations via New Kind of Generalized Chebyshev Polynomials

Abd-Elhameed

2023

Fractal Fract

View full text Add to dashboard Cite

show abstract

“…These polynomials are used as basis functions in various papers when different spectral polynomials are applied. For example, the first kind of CPs are employed in [39] to treat some real-life problems. The same polynomials were utilized to solve some BVPs in [40].…”

Section: Introductionmentioning

confidence: 99%

A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices

Ahmed,

Hafez,

Abd-Elhameed

2024

Phys. Scr.

View full text Add to dashboard Cite

This paper uses a new method to numerically solve the nonlinear time-fractional generalized Kawahara equations (NTFGKE) with uniform initial boundary conditions (IBCs). A class of modified shifted eighth-kind Chebyshev polynomials (MSEKCPs) is defined to satisfy the given IBCs. The proposed method is based on using operational matrices (OMs) for the ordinary derivatives (ODs) and fractional derivatives (FDs) of MSEKCPs. These OMs are employed together with the spectral collocation method (SCM). Our presented algorithm enables the extraction of efficient and accurate numerical solutions. The convergence of the suggested method and the error analysis have been developed. Three numerical examples are presented to demonstrate the applicability and accuracy of our algorithm. Some comparisons of the presented numerical results with other existing ones are offered to validate the efficiency and superiority of our approach. Tables and graphs demonstrate that the proposed approach produces approximate solutions with high accuracy.

show abstract

“…The fundamental idea behind these methods is choosing suitable linear combinations of different special functions, often orthogonal polynomials. The spectral method uses different types of orthogonal polynomials, which are called basis functions, such as Chebyshev polynomials [9,10] or their derivatives [11], Legendre polynomials [12] or their derivatives [13][14][15], and Ultraspherical polynomials [16]. Spectral methods can solve ordinary differential equations by representing the unknown function in these equations as a finite series of well-known polynomials.…”

Section: Introductionmentioning

confidence: 99%

Solving some types of ordinary differential equations by using Chebyshev derivatives direct residual spectral method

Gamal,

El-Kady,

Abdelhakem

2024

Advances in Basic and Applied Sciences

View full text Add to dashboard Cite

Herein, novel basis orthogonal polynomials have developed. These developed polynomials have been used to find the approximation solutions for some types of linear and non-linear ordinary differential equations by direct numerical method. This numerical method depends on the Chebyshev polynomials' derivatives. We shall present these solutions in the form of a finite sum of the Chebyshev polynomials' derivatives and unknown coefficients involving these polynomials. By substituting into the differential equation, the given differential equation will be converted into a system of algebraic equations. The obtained algebraic system can be solved easily to get the values of the spectral expansion constant. In addition, an algorithm for the approximated process has been designed to be easily used in the coding process. Consequently, some ordinary differential equations have been solved via the introduced Chebyshev polynomials' derivatives. Finally, the approximated solutions have been compared with exact and other methods solutions to illustrate the efficiency and accuracy of the used method.

show abstract

Monic Chebyshev pseudospectral differentiation matrices for higher-order IVPs and BVPs: applications to certain types of real-life problems

Cited by 21 publications

References 36 publications

Numerical Treatment of Multi-Term Fractional Differential Equations via New Kind of Generalized Chebyshev Polynomials

Numerical Treatment of Multi-Term Fractional Differential Equations via New Kind of Generalized Chebyshev Polynomials

A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices

Solving some types of ordinary differential equations by using Chebyshev derivatives direct residual spectral method

Contact Info

Product

Resources

About