Time fractional dual-phase-lag heat conduction equation

Xu, Huanying; Jiang, Xiaoyun

doi:10.1088/1674-1056/24/3/034401

Cited by 55 publications

(14 citation statements)

References 49 publications

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“…For the moment, we are not aware of references concerning the well-posedness and regularity of the multi-dimensional fractional Dual-Phase-Lag model (7); however, the suitable proof-technique is available in papers concerning fractional diffusion. The 1D solvability was treated in [5], where analytical solutions expressed by H-functions are obtained by using the Laplace and Fourier transforms method.…”

Section: Existing Results Of the Phase-lag Type Modelsmentioning

confidence: 99%

“…Considering the fractional version of this relation, we replace the two first order derivatives with respect to time by the fractional operators, and the Phase-Lags τ q and τ T by τ α q and τ β T , which are introduced to maintain the dimensions in order. The time fractional Dual-Phase-Lag model then reads as (see [5])…”

Section: Modelingmentioning

confidence: 99%

“…Here, the external (antenna) source is considered in a separated form (space and time). By combining this relation with (5), we arrive at the final fractional form of Dual-Phase-Lag heat equation for (x, t) ∈ Q T ,…”

Section: Derivation Of the Source Termmentioning

confidence: 99%

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Some Inverse Source Problems of Determining a Space Dependent Source in Fractional-Dual-Phase-Lag Type Equations

Maes

Slodička

2020

Mathematics

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The dual-phase-lag heat transfer models attract a lot of interest of researchers in the last few decades. These are used in problems arising from non-classical thermal models, which are based on a non-Fourier type law. We study uniqueness of solutions to some inverse source problems for fractional partial differential equations of the Dual-Phase-Lag type. The source term is supposed to be of the form h(t)f(x) with a known function h(t). The unknown space dependent source f(x) is determined from the final time observation. New uniqueness results are formulated in Theorem 1 (for a general fractional Jeffrey-type model). Here, the variational approach was used. Theorem 2 derives uniqueness results under weaker assumptions on h(t) (monotonically increasing character of h(t) was removed) in a case of dominant parabolic behavior. The proof technique was based on spectral analysis. Section 4 shows that an analogy of Theorem 2 for dominant hyperbolic behavior (fractional Cattaneo–Vernotte equation) is not possible.

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Section: Existing Results Of the Phase-lag Type Modelsmentioning

confidence: 99%

Section: Modelingmentioning

confidence: 99%

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Some Inverse Source Problems of Determining a Space Dependent Source in Fractional-Dual-Phase-Lag Type Equations

Maes

Slodička

2020

Mathematics

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“…Hence, there are many difficulties to use them in model simulation and actual production. In the past few years, many researchers have studied the numerical methods to solve fractional partial differential equations [16–19]. Sun and Gu [20] employed a mesh‐less method for solving 3D time fractional diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

Fast evaluation for magnetohydrodynamic flow and heat transfer of fractional Oldroyd‐B fluids between parallel plates

2021

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In this work, we establish a model to describe the unsteady magnetohydrodynamic (MHD) flow of fractional Oldroyd‐B fluid between parallel plates on heat transfer, by introducing the fractional derivative into the constitutive equation. The flow is driven by magnetic, electroosmotic and pressure gradient forcings. In the numerical implementation, the spectral collocation method combined with fast algorithm based on sum‐of‐exponentials (SOE) approximation is performed to solve the model. Effects of several parameters on the velocity profiles, temperature field and Nusselt number are investigated graphically.

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“…[17][18][19][20][21] Sun, 22 proposed a finite difference scheme to solve the time-fractional DPL model which subjects to the temperature jump boundary condition, in which the L1 scheme was employed to approximate the Caputo type fractional derivative. Jiang et al analyzed the analytical solution 23 using the time-fractional DPL model to describe the experimental results for processed meat. A finite element Legendre wavelet-Galerkin method was proposed to simulate heat transfer in skin tissue during heat treatment.…”

Section: Introductionmentioning

confidence: 99%

Efficient and accurate spectral method for the time‐fractional dual‐phase‐lag heat transfer model and its parameter estimation

Zheng

Jiang

Zhang

2019

Math Methods in App Sciences

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In the current paper, a heat transfer model is suggested based on a time‐fractional dual‐phase‐lag (DPL) model. We discuss the model in two parts, the direct problem and the inverse problem. Firstly, for solving it, the finite difference/Legendre spectral method is constructed. In the temporal direction, we employ the weighted and shifted Grünwald approximation, which can achieve second order convergence. In the spatial direction, the Legendre spectral method is used, it can obtain spectral accuracy. The stability and convergence are theoretically analyzed. For the inverse problem, the Bayesian method is used to construct an algorithm to estimate the four parameters for the model, namely, the time‐fractional order α, the time‐fractional order β, the delay time τT, and the relaxation time τq. Next, numerical experiments are provided to demonstrate the effectiveness of our scheme, with the values of τq and τT for processed meat employed. We also make a comparison with another method. The data obtained for the direct problem are used in the parameter estimation. The paper provides an accurate and efficient numerical method for the time‐fractional DPL model.

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Time fractional dual-phase-lag heat conduction equation

Cited by 55 publications

References 49 publications

Some Inverse Source Problems of Determining a Space Dependent Source in Fractional-Dual-Phase-Lag Type Equations

Some Inverse Source Problems of Determining a Space Dependent Source in Fractional-Dual-Phase-Lag Type Equations

Fast evaluation for magnetohydrodynamic flow and heat transfer of fractional Oldroyd‐B fluids between parallel plates

Efficient and accurate spectral method for the time‐fractional dual‐phase‐lag heat transfer model and its parameter estimation

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