Numerical Hilbert space solution of fractional Sobolev equation in $$\left(1+1\right)$$-dimensional space

Arqub, Omar Abu; Alsulami, Hamed; Alhodaly, Mohammed Sh.

doi:10.1007/s40096-022-00495-9

Cited by 21 publications

(5 citation statements)

References 27 publications

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“…Gao et al [21] used the local discontinuous Galerkin finite element method to solve a particular class of 2D Sobolev equations, while another work [22] applied the weak Galerkin finite element method, providing an error estimate. Abu et al [23] presented a method for solving time-and spacefractional Sobolev equations with Caputo fractional derivatives in n-dimensional space, utilizing the reproducing kernel Hilbert space method, which is particularly effective for Caputo class derivatives. A meshless RBFs technique for solving 2D time-fractional Sobolev equations was presented by Hussain et al [24].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Numerical Solution of a Multi-Dimensional Two-Term Fractional Order PDE via a Hybrid Methodology: The Caputo–Lucas–Fibonacci Approach with Strang Splitting

Ahmad,

Alshammari,

Jan

et al. 2024

Fractal Fract

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The utilization of time-fractional PDEs in diverse fields within science and technology has attracted significant interest from researchers. This paper presents a relatively new numerical approach aimed at solving two-term time-fractional PDE models in two and three dimensions. We combined the Liouville–Caputo fractional derivative scheme with the Strang splitting algorithm for the temporal component and employed a meshless technique for spatial derivatives utilizing Lucas and Fibonacci polynomials. The rising demand for meshless methods stems from their inherent mesh-free nature and suitability for higher dimensions. Moreover, this approach demonstrates the effective approximation of solutions across both regular and irregular domains. Error norms were used to assess the accuracy of the methodology across both regular and irregular domains. A comparative analysis was conducted between the exact solution and alternative numerical methods found in the contemporary literature. The findings demonstrate that our proposed approach exhibited better performance while demanding fewer computational resources.

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Section: Introductionmentioning

confidence: 99%

An Efficient Numerical Solution of a Multi-Dimensional Two-Term Fractional Order PDE via a Hybrid Methodology: The Caputo–Lucas–Fibonacci Approach with Strang Splitting

Ahmad,

Alshammari,

Jan

et al. 2024

Fractal Fract

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“…Many models that describe many phenomena in the applied sciences can be modeled by Fractional Differential Equations (FDEs), see for example [1][2][3][4][5]. One of the fundamental equations of mathematical physics is the fractional diffusion equation; it generalizes the classical diffusion equation, treating super-diffusive flow processes; it becomes increasingly sought after in recent years.…”

Section: Introductionmentioning

confidence: 99%

Petrov-Galerkin Lucas Polynomials Procedure for the Time-Fractional Diffusion Equation

Youssri

Atta

2023

Contemp. Math.

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Herein, we build and implement a combination of Lucas polynomials basis that fulfills the spatial homogenous boundary conditions. This basis is then used to solve the time-fractional diffusion equation spectrally. The elements of all spectral matrices are explicitly obtained in terms of the Gauss hypergeometric function. The convergence and error analysis of the proposed Lucas expansion is studied. Numerical results indicate the high accuracy and applicability of the suggested algorithm.

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“…The application of fractional differential equations to many engineering and scientific disciplines is very important, as numerous fractional-order derivatives are used in the mathematical modeling in the fields of physics, chemistry, electrodynamics of complex media, and polymer rheology, see [1][2][3][4][5][6][7][8][9][10]. Currently, fractional differential equations are used extensively in every branch of science, for example, the electrical closed loops can be expressed as fractional equations by Kirchhoff 's law [11].…”

Section: Introductionmentioning

confidence: 99%

Controllability of Hilfer fractional Langevin evolution equations

Wang

2023

Front. Appl. Math. Stat.

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The existence of fractional evolution equations has attracted a growing interest in recent years. The mild solution of fractional evolution equations constructed by a probability density function was first introduced by El-Borai. Inspired by El-Borai, Zhou and Jiao gave a definition of mild solution for fractional evolution equations with Caputo fractional derivative. Exact controllability is one of the fundamental issues in control theory: under some admissible control input, a system can be steered from an arbitrary given initial state to an arbitrary desired final state. In this article, using the (α, β) resolvent operator and three different fixed point theorems, we discuss the control problem for a class of Hilfer fractional Langevin evolution equations. The exact controllability of Hilfer fractional Langevin systems is established. An example is also discussed to illustrate the results.

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Numerical Hilbert space solution of fractional Sobolev equation in $$\left(1+1\right)$$-dimensional space

Cited by 21 publications

References 27 publications

An Efficient Numerical Solution of a Multi-Dimensional Two-Term Fractional Order PDE via a Hybrid Methodology: The Caputo–Lucas–Fibonacci Approach with Strang Splitting

An Efficient Numerical Solution of a Multi-Dimensional Two-Term Fractional Order PDE via a Hybrid Methodology: The Caputo–Lucas–Fibonacci Approach with Strang Splitting

Petrov-Galerkin Lucas Polynomials Procedure for the Time-Fractional Diffusion Equation

Controllability of Hilfer fractional Langevin evolution equations

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