scholarly journals The Effect of Fractional Time Derivative on Two-Dimension Porous Materials Due to Pulse Heat Flux

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 207
Author(s):  
Tareq Saeed ◽  
Ibrahim A. Abbas
Keyword(s):  
Time Derivative ◽  
Wave Model ◽  
Physical Field ◽  
Thermoelastic Wave ◽  
Physical Fields ◽  
Transport Diffusion ◽  
Fractional Time Derivative ◽  
Without Energy Dissipation ◽  
Physical Quantities ◽  
Made In

In the present article, the generalized thermoelastic wave model with and without energy dissipation under fractional time derivative is used to study the physical field in porous two-dimensional media. By applying the Fourier-Laplace transforms and eigenvalues scheme, the physical quantities are presented analytically. The surface is shocked by heating (pulsed heat flow problem) and application of free traction on its outer surface (mechanical conditions) by the process of temperature transport (diffusion) to observe the full analytical solutions of the main physical fields. The magnesium (Mg) material is used to make the simulations and obtain numerical outcomes. The basic physical field quantities are graphed and discussed. Comparisons are made in the results obtained under the strong (SC), the weak (WC) and the normal (NC) conductivities.

scholarly journals Fractional-Order Thermoelastic Wave Assessment in a Two-Dimensional Fiber-Reinforced Anisotropic Material

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1609
Author(s):  
Samah Horrigue ◽  
Ibrahim A. Abbas
Keyword(s):  
Fractional Order ◽  
Time Derivative ◽  
Relaxation Times ◽  
Thermal Relaxation ◽  
Generalized Thermoelasticity ◽  
Fiber Reinforced ◽  
Thermoelastic Wave ◽  
Practical Applications ◽  
Fractional Time Derivative ◽  
Fiber Reinforced Material

The present work is aimed at studying the effect of fractional order and thermal relaxation time on an unbounded fiber-reinforced medium. In the context of generalized thermoelasticity theory, the fractional time derivative and the thermal relaxation times are employed to study the thermophysical quantities. The techniques of Fourier and Laplace transformations are used to present the problem exact solutions in the transformed domain by the eigenvalue approach. The inversions of the Fourier-Laplace transforms hold analytical and numerically. The numerical outcomes for the fiber-reinforced material are presented and graphically depicted. A comparison of the results for different theories under the fractional time derivative is presented. The properties of the fiber-reinforced material with the fractional derivative act to reduce the magnitudes of the variables considered, which can be significant in some practical applications and can be easily considered and accurately evaluated.

Fractional abstract Cauchy problem on complex interpolation scales

2020 ◽  
Vol 23 (4) ◽  
pp. 1125-1140
Author(s):  
Andriy Lopushansky ◽  
Oleh Lopushansky ◽  
Anna Szpila
Keyword(s):  
Cauchy Problem ◽  
Time Derivative ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Sectorial Operator ◽  
Complex Interpolation ◽  
Bounded Domains ◽  
Initial Boundary Value Problems ◽  
Fractional Time Derivative ◽  
Initial Boundary Value

AbstractAn fractional abstract Cauchy problem generated by a sectorial operator is investigated. An inequality of coercivity type for its solution with respect to a complex interpolation scale generated by a sectorial operator with the same parameters is established. An application to differential parabolic initial-boundary value problems in bounded domains with a fractional time derivative is shown.

scholarly journals New Iterative Method for Fractional Gas Dynamics and Coupled Burger’s Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Mohamed S. Al-luhaibi
Keyword(s):  
Iterative Method ◽  
Analytical Solutions ◽  
Gas Dynamics ◽  
Time Derivative ◽  
Explicit Solutions ◽  
Fractional Partial Differential Equations ◽  
Approximate Analytical Solutions ◽  
Fractional Time Derivative ◽  
Burger's Equations ◽  
New Iterative Method

This paper presents the approximate analytical solutions to solve the nonlinear gas dynamics and coupled Burger’s equations with fractional time derivative. By using initial values, the explicit solutions of the equations are solved by using a reliable algorithm. Numerical results show that the new iterative method is easy to implement and accurate when applied to time-fractional partial differential equations.

Application of Fractional Derivatives for Simulating Diffusion Into Porous Matrix in Mathematical Modeling of the Contaminant Transport in a Confined Fractured Porous Aquifer

2006 ◽  
Author(s):  
Sergei Fomin ◽  
Vladimir Chugunov ◽  
Toshiyuki Hashida
Keyword(s):  
Solute Transport ◽  
Contaminant Transport ◽  
Fractional Derivatives ◽  
Time Derivative ◽  
Porous Matrix ◽  
Confined Aquifer ◽  
Closed Form Solutions ◽  
Advection Dispersion ◽  
Porous Aquifer ◽  
Fractional Time Derivative

Solute transport in the fractured porous confined aquifer is modeled by the advection-dispersion equation with fractional time derivative of order γ, which may vary from 0 to 1. Accounting for diffusion in the surrounding rock mass leads to the introduction of an additional fractional time derivative of order 1/2 in the equation for solute transport. The closed-form solutions for concentrations in the aquifer and surrounding rocks are obtained for the arbitrary time-dependent source of contamination located in the inlet of the aquifer. Based on these solutions, different regimes of contamination of the aquifers with different physical properties are modeled and analyzed.

scholarly journals Fractional Time Derivative Seismic Wave Equation Modeling for Natural Gas Hydrate

Energies ◽  
2020 ◽  
Vol 13 (22) ◽  
pp. 5901
Author(s):  
Yanfei Wang ◽  
Yaxin Ning ◽  
Yibo Wang
Keyword(s):  
Wave Equation ◽  
Natural Gas ◽  
Seismic Wave ◽  
Gas Hydrate ◽  
Wave Equations ◽  
Time Derivative ◽  
Seismic Attenuation ◽  
Equation Modeling ◽  
Natural Gas Hydrate ◽  
Fractional Time Derivative

Simulation of the seismic wave propagation in natural gas hydrate (NGH) is of great importance. To finely portray the propagation of seismic wave in NGH, attenuation properties of the earth’s medium which causes reduced amplitude and dispersion need to be considered. The traditional viscoacoustic wave equations described by integer-order derivatives can only nearly describe the seismic attenuation. Differently, the fractional time derivative seismic wave-equation, which was rigorously derived from the Kjartansson’s constant-Q model, could be used to accurately describe the attenuation behavior in realistic media. We propose a new fractional finite-difference method, which is more accurate and faster with the short memory length. Numerical experiments are performed to show the feasibility of the proposed simulation scheme for NGH, which will be useful for next stage of seismic imaging of NGH.

scholarly journals Marine Wind and Wave Height Trends at Different ERA-Interim Forecast Ranges

2015 ◽  
Vol 28 (2) ◽  
pp. 819-837 ◽  
Author(s):  
Ole Johan Aarnes ◽  
Saleh Abdalla ◽  
Jean-Raymond Bidlot ◽  
Øyvind Breivik
Keyword(s):  
Wind Speed ◽  
Wave Height ◽  
North Atlantic ◽  
Wave Model ◽  
Altimeter Data ◽  
The North ◽  
The Impact ◽  
Global Reanalysis ◽  
Made In

Abstract Trends in marine wind speed and significant wave height are investigated using the global reanalysis ERA-Interim over the period 1979–2012, based on monthly-mean and monthly-maximum data. Besides the traditional reanalysis, the authors include trends obtained at different forecast range, available up to 10 days ahead. Any model biases that are corrected differently over time are likely to introduce spurious trends of variable magnitude. However, at increased forecast range the model tends to relax, being less affected by assimilation. Still, there is a trade-off between removing the impact of data assimilation at longer forecast range and getting a lower level of uncertainty in the predictions at shorter forecast range. Because of the sheer amount of assimilations made in ERA-Interim, directly and indirectly affecting the data, it is difficult, if not impossible, to distinguish effects imposed by all updates. Here, special emphasis is put on the introduction of wave altimeter data in August 1991, the only type of data directly affecting the wave field. From this, it is shown that areas of higher model bias introduce quite different trends depending on forecast range, most apparent in the North Atlantic and eastern tropical Pacific. Results are compared with 23 in situ measurements, Envisat altimeter winds, and two stand-alone ECMWF operational wave model (EC-WAM) runs with and without wave altimeter assimilation. Here, the 48-h forecast is suggested to be a better candidate for trend estimates of wave height, mainly due to the step change imposed by altimeter observations. Even though wind speed seems less affected by undesirable step changes, the authors believe that the 24–48-h forecast more effectively filters out any unwanted effects.

On the Frequency and Amplitude Dependence of the Payne Effect: Theory and Experiments

2003 ◽  
Vol 76 (2) ◽  
pp. 533-547 ◽  
Author(s):  
A. Lion ◽  
C. Kardelky ◽  
P. Haupt
Keyword(s):  
Time Scale ◽  
Time Derivative ◽  
Amplitude Dependence ◽  
Deformation History ◽  
Payne Effect ◽  
Physical Time ◽  
Starting Point ◽  
Effect Theory ◽  
Fractional Time Derivative ◽  
Analytical Expressions

Abstract In this paper, we develop a physical approach to represent the Payne effect which is observed in filler-reinforced elastomers. The starting point for the constitutive model is the well-known theory of linear viscoelasticity, where the stress is a linear functional of the deformation history. Since the corresponding relations are unable to describe any kind of amplitude dependence, we introduce a nonlinearity into the model. To this end, we replace the physical time, t, by a modified time scale, z, which is a functional of the deformation history. This approach was originally introduced by Valanis in the context of rate-independent plasticity and applied in a modified form; for example, by Haupt and Sedlan to represent process-dependent relaxation properties of rubber. The modified time, z, is a monotonic function of the physical time, t, and can be interpreted as an intrinsic time scale of the material. The rate of this time scale is non-negative and depends on the process history. We propose a constitutive relation for the variable z(t), which is driven by a fractional time derivative of the deformation, calculate analytical expressions for the storage and dissipation moduli and show that such phenomena as the frequency and the amplitude dependence, observed in experiments, are well represented.

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