Sixth-Kind Chebyshev and Bernoulli Polynomial Numerical Methods for Solving Nonlinear Mixed Partial Integrodifferential Equations with Continuous Kernels

Al-Bugami, Abeer M.; Abdou, Mohamed A.; Mahdy, Amr M. S.

doi:10.1155/2023/6647649

Cited by 3 publications

(1 citation statement)

References 39 publications

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“…Every day, substantial efforts are undertaken to solve differential equations, which have numerous applications in various scientific domains. [1][2][3] Fractional-order differential equations (FDEs) have been successfully used to describe phenomena in various scientific and engineering fields, including fluid mechanics, medicine, heat conduction, electromagnetism, viscoelasticity, biology, wave propagation, optimal control, and more. [4][5][6][7][8][9][10][11][12][13][14] As a result of their importance, solutions to fractional ordinary or partial differential equations with physical meaning have received special attention.…”

Section: Introductionmentioning

confidence: 99%

Hybrid technique of conformal mapping and Chebyshev collocation method for solving time–space fractional order wave equation

El‐Sayed,

Boulaaras

2024

Int J Numerical Modelling

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This work presents a numerical approach for solving the time–space fractional‐order wave equation. The time‐fractional derivative is described in the conformal sense, whereas the space‐fractional derivative is given in the Caputo sense. The investigated technique is based on the third‐kind of shifted Chebyshev polynomials. The main problem is converted into a system of ordinary differential equations using conformal mapping, Caputo derivatives, and the properties of the third‐kind shifted Chebyshev polynomials. Then, the Chebyshev collocation method and the non‐standard finite difference method will be used to convert this system into a system of algebraic equations. Finally, this system can be solved numerically via Newton's iteration method. In the end, physics numerical examples and comparison results are provided to confirm the accuracy, applicability, and efficiency of the suggested approach.

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Section: Introductionmentioning

confidence: 99%

Hybrid technique of conformal mapping and Chebyshev collocation method for solving time–space fractional order wave equation

El‐Sayed,

Boulaaras

2024

Int J Numerical Modelling

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Optical soliton solutions of generalized Pochammer Chree equation

Tarla,

Ali,

Günerhan

2024

Opt Quant Electron

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This research investigates the utilization of a modified version of the Sardar sub-equation method to discover novel exact solutions for the generalized Pochammer Chree equation. The equation itself represents the propagation of longitudinal deformation waves in an elastic rod. By employing this modified method, we aim to identify previously unknown solutions for the equation under consideration, which can contribute to a deeper understanding of the behavior of deformation waves in elastic rods. The solutions obtained are represented by hyperbolic, trigonometric, exponential functions, dark, dark-bright, periodic, singular, and bright solutions. By selecting suitable values for the physical parameters, the dynamic behaviors of these solutions can be demonstrated. This allows for a comprehensive understanding of how the solutions evolve and behave over time. The effectiveness of these methods in capturing the dynamics of the solutions contributes to our understanding of complex physical phenomena. The study’s findings show how effective the selected approaches are in explaining nonlinear dynamic processes. The findings reveal that the chosen techniques are not only effective but also easily implementable, making them applicable to nonlinear model across various fields, particularly in studying the propagation of longitudinal deformation waves in an elastic rod. Furthermore, the results demonstrate that the given model possesses solutions with potentially diverse structures.

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The orthogonal polynomials method using Gegenbauer polynomials to solve mixed integral equations with a Carleman kernel

Alalyani,

Abdou,

Basseem

2024

MATH

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<abstract> <p>The orthogonal polynomials approach with Gegenbauer polynomials is an effective tool for analyzing mixed integral equations (MIEs) due to their orthogonality qualities. This article reviewed recent breakthroughs in the use of Gegenbauer polynomials to solve mixed integral problems. Previous authors studied the problem with a continuous kernel that combined both Volterra (V) and Fredholm (F) components; however, in this paper, we focused on a singular Carleman kernel. The kernel of FI was measured with respect to position in the space <inline-formula><tex-math id="M1">\begin{document}$ {L}_{2}[-\mathrm{1, 1}], $\end{document}</tex-math></inline-formula> while the kernel of Ⅵ was considered as a function of time in the space <inline-formula><tex-math id="M2">\begin{document}$ C[0, T], T < 1 $\end{document}</tex-math></inline-formula>. The existence of a unique solution was discussed in <inline-formula><tex-math id="M3">\begin{document}$ {L}_{2}\left[-\mathrm{1, 1}\right]\times C\left[0, T\right] $\end{document}</tex-math></inline-formula> space. The solution and its error stability were both investigated and commented on. Finally, numerical examples were reviewed, and their estimated errors were assessed using Maple (2022) software.</p> </abstract>

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Sixth-Kind Chebyshev and Bernoulli Polynomial Numerical Methods for Solving Nonlinear Mixed Partial Integrodifferential Equations with Continuous Kernels

Cited by 3 publications

References 39 publications

Hybrid technique of conformal mapping and Chebyshev collocation method for solving time–space fractional order wave equation

Hybrid technique of conformal mapping and Chebyshev collocation method for solving time–space fractional order wave equation

Optical soliton solutions of generalized Pochammer Chree equation

The orthogonal polynomials method using Gegenbauer polynomials to solve mixed integral equations with a Carleman kernel

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