Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel

Atta, A. G.; Youssri, Y. H.

doi:10.1007/s40314-022-02096-7

Cited by 44 publications

(15 citation statements)

References 27 publications

(25 reference statements)

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“…Solving FDEs is one of the important numerical analysis problems. In recent years, various methods have been proposed to solve them [24,25]. In [26], a method based on applying the Petrov-Galerkin procedure to discretize the differential problem into a system of equations with unknown expansion coefficients was proposed.…”

Section: Introductionmentioning

confidence: 99%

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

Erfanifar,

Hajarian,

Sayevand

2023

Math Methods in App Sciences

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In this work, first, a family of fourth‐order methods is proposed to solve nonlinear equations. The methods satisfy the Kung‐Traub optimality conjecture. By developing the methods into memory methods, their efficiency indices are increased. Then, the methods are extended to the multi‐step methods for finding the solutions to systems of problems. The formula for the order of convergence of the multi‐step iterative methods is , where is the step number of the methods. It is clear that computing the Jacobian matrix derivative evaluation and its inversion are expensive; therefore, we compute them only once in every cycle of the methods. The important feature of these multi‐step methods is their high‐efficiency index. Numerical examples that confirm the theoretical results are performed. In applications, some nonlinear problems related to the numerical approximation of fractional differential equations (FDEs) are constructed and solved by the proposed methods.

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Section: Introductionmentioning

confidence: 99%

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

Erfanifar,

Hajarian,

Sayevand

2023

Math Methods in App Sciences

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“…Lane–Emden, Bratu equations and singular perturbed problems are considered, and their numerical solutions are obtained by assembling two spectral Legendre's derivative algorithms in the article by Abdelhakem and Youssri [30]. A nonlinear time fractional partial integro‐differential equation with a weakly singular kernel is solved using advanced shifted first kind Chebyshev collocation approach by Atta and Youssri [31]. Not only motivated by the aforementioned works but also due to the limited study of a good approximation method based on the orthogonal polynomials to solve the stochastic Fisher–KPP equation, we have employed two‐dimensional SLP approximation to obtain the approximate solutions of stochastic Fisher–KPP equations of the form () in the interval

\left[0,1\right]

.…”

Section: Introductionmentioning

confidence: 99%

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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“…The Tau method has the advantage of the two selected families not being identical (see [5]). Because of its ability to treat any type of differential equation, the collocation method is widely used to solve a variety of differential equations (see, for example, [6][7][8]).…”

Section: Introductionmentioning

confidence: 99%

Numerical Contrivance for Kawahara-Type Differential Equations Based on Fifth-Kind Chebyshev Polynomials

et al. 2023

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This article proposes a numerical algorithm utilizing the spectral Tau method for numerically handling the Kawahara partial differential equation. The double basis of the fifth-kind Chebyshev polynomials and their shifted ones are used as basis functions. Some theoretical results of the fifth-kind Chebyshev polynomials and their shifted ones are used in deriving our proposed numerical algorithm. The nonlinear term in the equation is linearized using a new product formula of the fifth-kind Chebyshev polynomials with their first derivative polynomials. Some illustrative examples are presented to ensure the applicability and efficiency of the proposed algorithm. Furthermore, our proposed algorithm is compared with other methods in the literature. The presented numerical method results ensure the accuracy and applicability of the presented algorithm.

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Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel

Cited by 44 publications

References 27 publications

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

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

Numerical Contrivance for Kawahara-Type Differential Equations Based on Fifth-Kind Chebyshev Polynomials

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