Numerical analysis of the fractional evolution model for heat flow in materials with memory

Nikan, O.; Jafari, Hossein; Golbabai, A.

doi:10.1016/j.aej.2020.04.026

Cited by 53 publications

(20 citation statements)

References 30 publications

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“…By replacing relations (15) and (16) in Eq. (19) and substituting collocation nodes for time steps k = n, n + 1 yields in the following time semi-discretization equation…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…Many authors derived that fractional differential equations (FDEs) are more suitable than integer order ones, because fractional derivatives describe the memory and hereditary properties of diverse materials and processes [12,13,17,8]. Recently, FDEs have gained much interest in many research areas such as engineering, physics, chemistry, economics, and other branches of science [20,25,7,17,8,16]. Consequently, the RLWE related to the derivatives of fractional order generalizes of the RLWE (1) to interpret the water waves.…”

mentioning

confidence: 99%

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Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves

Nikan¹,

Molavi-Arabshai²,

Jafari³

2021

DCDS-S

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This paper aimed at obtaining the traveling-wave solution of the nonlinear time fractional regularized long-wave equation. In this approach, firstly, the time fractional derivative is accomplished using a finite difference with convergence order O(δt 2−α) for 0 < α < 1 and the nonlinear term is linearized by a linearization technique. Then, the spatial terms are approximated with the help of the radial basis function (RBF). Numerical stability of the method is analyzed by applying the Von-Neumann linear stability analysis. Three invariant quantities corresponding to mass, momentum and energy are evaluated for further validation. Numerical results demonstrate the accuracy and validity of the proposed method.

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“…By replacing relations (15) and (16) in Eq. (19) and substituting collocation nodes for time steps k = n, n + 1 yields in the following time semi-discretization equation…”

Section: Spatial Discretizationmentioning

confidence: 99%

mentioning

confidence: 99%

Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves

Nikan¹,

Molavi-Arabshai²,

Jafari³

2021

DCDS-S

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show abstract

“…Although there are relations between these different types of fractional operators, they may differ in the physical interpretation of the definitions. For more information and application for fractional calculus, we refer the reader to other studies 3–24 . Therefore, in order to benefit from many advantages of the fractional‐order operator, especially the memory effect, we analyze the abovementioned phytoplankton growth and biological models theoretically and numerically with the help of Caputo fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

Analysis of fractional‐order nonlinear dynamic systems under Caputo differential operator

Yusuf

Acay

2021

Math Methods in App Sciences

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The current study presents a detailed analysis of two crucial real‐world problems under the Caputo fractional derivative in order to deliver some desired results for the ecosystem. In view of the fact that memory effect plays a vital role in the application, we utilize an advantageous non‐local fractional operator to investigate and analyze a mathematical model of the planktonic ecosystem and biological system for the ecosystem on Planet GLIA‐2. On the other hand, theoretical and numerical results are given for the model created for phytoplankton, which is of great importance in preventing global warming, and the biological model. Existence and uniqueness are discussed for the solutions of both models with the help of the fixed‐point theorem under the Caputo operator. Additionally, the first‐order convergent numerical technique which is accurate, conditionally stable, and convergent in obtaining the solution of fractional‐order nonlinear systems of ordinary differential equations is utilized to simulate the two governing models. Numerical simulations including different values of arbitrary order ρ indicate the righteousness of the conditions for phytoplankton, Jancor, Murrot, and Vekton populations to develop.

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“…Within the past decades, the field of fractional calculus that is the study of integrals and derivatives of arbitrary order has attained significant popularity and credit. Since the fractional derivatives are better than customary integer-order kind due to the inherent nonlocality and their revealed applications, they mainly are successful in abundant and prevalent fields in science, finance, engineering, and physics [1][2][3][4][5][6][7][8]. Indeed, one of the most significant applications is to interpret the superdiffusion and subdiffusion phenomena [9][10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

“…The residual sections of the article are as follows. In Section 2, we exert the compact finite difference (CFD) to discrete in the temporal direction, present the approximation of the space fractional derivative by using the spectral method based on the Chebyshev polynomials of the second kind and then formulate the fully discrete for Equation (1). In Section 3, we prove that the semidiscrete scheme has approximate the exact ones with ( 2 ) and derive the stable analysis.…”

Section: Introductionmentioning

confidence: 99%

A novel numerical manner for two‐dimensional space fractional diffusion equation arising in transport phenomena

Tuấn¹,

Aghdam

Jafari

et al. 2020

Numerical Methods Partial

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Fractional diffusion equations include a consistent and efficient explanation of transport phenomena that manifest abnormal diffusion, that cannot be often represented by second‐order diffusion equations. In this article, a two‐dimensional space fractional diffusion equation (SFDE‐2D) with nonhomogeneous and homogeneous boundary conditions is considered in Caputo derivative sense. An instant and nevertheless accurate scheme is obtained by the finite‐difference discretization to get the semidiscrete in temporal derivative with convergence order Ofalse(δτ2false). Moreover, space fractional derivative can be approximated based on the Chebyshev polynomials of second kind which are powerful methods for basing the operational matrix. The convergence and stability of the proposed scheme are discussed theoretically in detail. Finally, two numerical problems with an exact solution are given that numerical results show the effectiveness of the new techniques. These schemes can be simply extended to three spatial dimensions, which will be the subject of our subsequent research.

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Numerical analysis of the fractional evolution model for heat flow in materials with memory

Cited by 53 publications

References 30 publications

Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves

Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves

Analysis of fractional‐order nonlinear dynamic systems under Caputo differential operator

A novel numerical manner for two‐dimensional space fractional diffusion equation arising in transport phenomena

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