Numerical computation of fractional Fredholm integro-differential equation of order 2<i>β</i> arising in natural sciences

Alaroud, Mohammad; Al‐Smadi, Mohammed; Ahmad, Rokiah Rozita; Din, Ummul Khair Salma

doi:10.1088/1742-6596/1212/1/012022

Cited by 8 publications

(6 citation statements)

References 6 publications

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“…To determine w 1 ðxÞ, consider m = 1 in the series expansion (30) and substitute it into the first Laplace residual function as follows:…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Following the same argument, the 3 rd -transform function can be constructed by writing W 3 ðx, sÞ, of the fractional expansion (30) into the third Laplace residual function L Res 3 W ðx, sÞ of ( 29) such that…”

Section: Numerical Examplesmentioning

confidence: 99%

“…However, several mathematical techniques have been employed to investigate their solutions. For example, homotopy analysis technique, variational iteration technique, Adomian decomposition technique, residual power series technique, and other techniques are some of these methods, and for more details, see [21,[26][27][28][29][30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An Attractive Approach Associated with Transform Functions for Solving Certain Fractional Swift-Hohenberg Equation

Alaroud

Tahat

Al‐Omari

et al. 2021

Journal of Function Spaces

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Many phenomena in physics and engineering can be built by linear and nonlinear fractional partial differential equations which are considered an accurate instrument to interpret these phenomena. In the current manuscript, the approximate analytical solutions for linear and nonlinear time-fractional Swift-Hohenberg equations are created and studied by means of a recent superb technique, named the Laplace residual power series (LRPS) technique under the time-Caputo fractional derivatives. The proposed technique is a combination of the generalized Taylor’s formula and the Laplace transform operator, which depends mainly on the concept of limit at infinity to find the unknown functions for the fractional series expansions in the Laplace space with fewer computations and more accuracy comparing with the classical RPS that depends on the Caputo fractional derivative for each step in obtaining the coefficient expansion. To test the simplicity, performance, and applicability of the present method, three numerical problems of the time-fractional Swift-Hohenberg initial value problems are considered. The impact of the fractional order β on the behavior of the approximate solutions at fixed bifurcation parameter is shown graphically and numerically. Obtained results emphasized that the LRPS technique is an easy, efficient, and speed approach for the exact description of the linear and nonlinear time-fractional models that arise in natural sciences.

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“…To determine w 1 ðxÞ, consider m = 1 in the series expansion (30) and substitute it into the first Laplace residual function as follows:…”

Section: Numerical Examplesmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An Attractive Approach Associated with Transform Functions for Solving Certain Fractional Swift-Hohenberg Equation

Alaroud

Tahat

Al‐Omari

et al. 2021

Journal of Function Spaces

Self Cite

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show abstract

“…The objective of this work is to apply an advanced algorithm, called the fractional residual power series (FRPS) algorithm, for solving the time-FNWSE. The FRPS is a novel numeric-analytic technique for dealing with both linear and nonlinear issues, which enables us to obtain analytical and approximate solutions in convergent fractional power series (FPS) by combining Taylor's fractional series formula and residual error functions without requiring any constrained assumptions [19][20][21][22][23]. The outline of this work is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order

et al. 2019

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The Newell–Whitehead–Segel equation is one of the most nonlinear amplitude equations that plays a significant role in the modeling of various physical phenomena arising in fluid mechanics, solid-state physics, optics, plasma physics, dispersion, and convection system. In this analysis, a recent numeric-analytic technique, called the fractional residual power series (FRPS) approach, was successfully employed in obtaining effective approximate solutions to the Newell–Whitehead–Segel equation of the fractional sense. The proposed algorithm relies on a generalized classical power series under the Caputo sense and the concept of an error function that systematically produces an analytical solution in a convergent fractional power series form with accurately computable structures, without the need for any unphysical restrictive assumptions. Meanwhile, two illustrative applications are included to show the efficiency, reliability, and performance of the proposed technique. Plotted and numerical results indicated the compatibility between the exact and approximate solution obtained by the proposed technique. Furthermore, the solution behavior indicates that increasing the fractional parameter changes the nature of the solution with a smooth sense symmetrical to the integer-order state.

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“…Second, the present method provides the solutions in Taylor expansions; therefore, the exact solutions will be available when the solutions are polynomials. [20][21][22][23][24] This technique is a direct way to ensure the rate of convergence for series solution, as it depends on minimizing the residual error related. Third, the solutions along with their derivatives can be applied for each arbitrary point in the given interval.…”

Section: Introductionmentioning

confidence: 99%

Toward computational algorithm for time-fractional Fokker–Planck models

Freihet

Hasan

Alaroud

et al. 2019

Advances in Mechanical Engineering

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This article describes an efficient algorithm based on residual power series to approximate the solution of a class of partial differential equations of time-fractional Fokker–Planck model. The fractional derivative is assumed in the Caputo sense. The proposed algorithm gives the solution in a form of rapidly convergent fractional power series with easily computable coefficients. It does not require linearization, discretization, or small perturbation. To test simplicity, potentiality, and practical usefulness of the proposed algorithm, illustrative examples are provided. The approximate solutions of time-fractional Fokker–Planck equations are obtained by the residual power series method are compared with those obtained by other existing methods. The present results and graphics reveal the ability of residual power series method to deal with a wide range of partial fractional differential equations emerging in the modeling of physical phenomena of science and engineering.

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Numerical computation of fractional Fredholm integro-differential equation of order 2β arising in natural sciences

Cited by 8 publications

References 6 publications

An Attractive Approach Associated with Transform Functions for Solving Certain Fractional Swift-Hohenberg Equation

An Attractive Approach Associated with Transform Functions for Solving Certain Fractional Swift-Hohenberg Equation

Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order

Toward computational algorithm for time-fractional Fokker–Planck models

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