Shifted Gegenbauer-Gauss Collocation Method for Solving Fractional Neutral Functional-Differential Equations with Proportional Delays

HAFEZ, R. M.; Youssri, Y. H.

doi:10.46793/kgjmat2206.981h

Cited by 27 publications

(11 citation statements)

References 22 publications

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“…Since the approximation has been carried out for first

\left(N&#x0002B;1\right)

terms, the remaining terms in the approximation till the last term forms an error and these error terms should approach to zero for larger values. The difference between the approximated terms and the remaining terms denote the residual error [34]. The error function is defined, and its convergence to zero is established in the following theorem.…”

Section: Convergence Analysismentioning

confidence: 99%

“…Various spectral methods and their improvements have been effectively studied to solve a variety of linear and nonlinear differential equations. Neutral fractional functional differential equations with propotional delays have been solved by Hafez and Youssri [29]. It is done by approximating the unknown function using shifted Gagenbauer polynomials and further using Gauss quadrature integration.…”

Section: Introductionmentioning

confidence: 99%

“…In order to justify the efficiency of the method, in the convergence analysis, we derive the upper bound of the error function

{e}_N\left(t,z\right)

. The convergence analysis of a truncated series of Legendre polynomials is discussed in depth in Hafez and Youssri [34]. Two examples are taken into account to substantiate the applicability of the proposed method, and the convergence of the solution is also verified.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A mathematical modeling and numerical study for stochastic Fisher–SI model driven by space uniform white noise

Uma

Jafari

Balachandar

et al. 2023

Math Methods in App Sciences

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In this study, we propose an approximate solution based on two‐dimensional shifted Legendre polynomials to solve nonlinear stochastic partial differential equations with variable coefficients. For this purpose, we have considered a Fisher‐Kolmogorov‐Petrovsky‐Piskunov (Fisher–KPP) equation with space uniform white noise for the same. New stochastic operational matrix of integration based on shifted Legendre polynomials is generated. This operational matrix reduces the problem under study into solving a system of algebraic equations. The convergence analysis is discussed, and the error bound in L2$$ {L}^2 $$ norm is derived and obtained as ‖‖eNfalse(t,xfalse)2≤6Kfalse(16N4false)2false(1−6Sfalse)$$ {\left\Vert {e}_N\left(t,x\right)\right\Vert}^2\le \frac{6K}{{\left(16{N}^4\right)}^2\left(1-6S\right)} $$. The theoretical analysis confirms that as the degree of the approximation polynomial is increased, the solution is on par with the exact solution. The numerical examples confirm the accuracy and the applicability of the proposed method. A comparative study is carried out with explicit 1.5 order Runge–Kutta method. The time complexity of the proposed technique is also studied.

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“…Since the approximation has been carried out for first

\left(N&#x0002B;1\right)

Section: Convergence Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In order to justify the efficiency of the method, in the convergence analysis, we derive the upper bound of the error function

{e}_N\left(t,z\right)

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A mathematical modeling and numerical study for stochastic Fisher–SI model driven by space uniform white noise

Uma

Jafari

Balachandar

et al. 2023

Math Methods in App Sciences

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show abstract

“…However, there are some numerical techniques proposed for solving many fractional problems, and the error analysis has mainly focused on smooth solutions. For instance, see [16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Novel spectral schemes to fractional problems with nonsmooth solutions

Atta

Abd-Elhameed

Moatimid

et al. 2023

Math Methods in App Sciences

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In this article, we present two numerical methods for treating the fractional initial‐value problem (FIVP) and time‐fractional partial differential problem (FPDP) that caused the error to decay exponentially rapidly. The derivations of the proposed schemes rely on the use of a spectral Galerkin method that reduces each of the FIVP and FPDP into an algebraic system of equations in the unknown expansion coefficients. The class of orthogonal polynomials, namely, Chebyshev polynomials of the fifth kind is utilized. In terms of new basis functions called regular shifted Chebyshev poly‐fractionomials of fifth kind, approximate solutions to the FIVP and FPDP are obtained. Moreover, convergence and error analysis of the two problems are investigated in depth. Some numerical examples are presented with some comparisons. In conclusion, our spectral methods are effective and convenient.

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“…Recently, spectral methods have been applied to solve different FDDEs. Hafez and Youssri (2022) used the Gegenbauer-Gauss SC method for fractional neutral functionaldifferential equations. Alsuyuti et al (2021) used the Legendre polynomials as basis function of SG approach for solving a general form of multi-order fractional pantograph equations.…”

Section: Introductionmentioning

confidence: 99%

Robust spectral treatment for time-fractional delay partial differential equations

2023

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Fractional delay differential equations (FDDEs) and time-fractional delay partial differential equations (TFDPDEs) are the focus of the present research. The FDDEs is converted into a system of algebraic equations utilizing a novel numerical approach based on the spectral Galerkin (SG) technique. The suggested numerical technique is likewise utilized for TFDPDEs. In terms of shifted Jacobi polynomials, suitable trial functions are developed to fulfill the initial-boundary conditions of the main problems. According to the authors, this is the first time utilizing the SG technique to solve TFDPDEs. The approximate solution of five numerical examples is provided and compared with those of other approaches and with the analytic solutions to test the superiority of the proposed method.

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Shifted Gegenbauer-Gauss Collocation Method for Solving Fractional Neutral Functional-Differential Equations with Proportional Delays

Cited by 27 publications

References 22 publications

A mathematical modeling and numerical study for stochastic Fisher–SI model driven by space uniform white noise

A mathematical modeling and numerical study for stochastic Fisher–SI model driven by space uniform white noise

Novel spectral schemes to fractional problems with nonsmooth solutions

Robust spectral treatment for time-fractional delay partial differential equations

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