scholarly journals Fractional-Differential Model of Heat Conductivity Process in Ferroelectrics under the Intensive Heating Conditions

2019 ◽  
pp. 29-47
Author(s):  
L. I. Moroz ◽  
A. G. Maslovskaya
Keyword(s):  
Finite Difference ◽  
Heat Conductivity ◽  
Time Derivative ◽  
Model Verification ◽  
Total System ◽  
Ferroelectric Crystal ◽  
Differential Model ◽  
Linear Algebraic Equations ◽  
Fractional Differential ◽  
Fractional Time Derivative

Ferroelectrics, due a number of characteristics, behave as hereditary materials with fractal structure. To model mathematically the systems with so-called memory effects one can use the fractional time-derivatives. The pyro-electric properties of ferroelectrics arouse interest in developing the fractional-differential approach to simulating heat conductivity process.The present study deals with development and numerical implementation of fractal heat conductivity model for hereditary materials using the concepts of fractional-differential calculus applied to the simulation of intensive heating processes in ferroelectrics.The paper proposes a mathematical model governed through mixed initial-boundary value problem for partial differential equation containing a fractional time-derivative as well as nonlinear temperature dependence on the heat capacity. To solve the problem the computational algorithm was designed which is based on an analog of the Crank – Nicolson finite difference scheme combining with the Grunwald – Letnikov formula for fractional time-derivative approximation. The approximation of Neumann boundary condition is included into the finite difference problem statement using scheme of fictitious mesh points. The total system of linear algebraic equations is solved by sweep method.The designed application program allows one to perform the computer simulation of heat conductivity process in hereditary materials. The model verification was performed for numerical solving test problem with known analytical solution. The results of computational experiments are demonstrated for the example of estimating heat distribution in a typical ferroelectric crystal of TGS (triglycine sulfate) near the temperature of phase transition. The fractional derivative order was approximately evaluated to be ~0.7 at variation of this parameter. We applied the comparison of fractal model implementation results with experimental data related to the time when the ferroelectric crystal is heated to Curie temperature. These findings demonstrate that one needs to use the modified models at the analysis of the field effects arising in hereditary materials.

Dynamic Viscoelastic Rod Stability Modeling by Fractional Differential Operator

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
D. Ingman ◽  
J. Suzdalnitsky
Keyword(s):  
Damping Ratio ◽  
Time Derivative ◽  
Additional Term ◽  
Bifurcation Line ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
The Stability ◽  
Fractional Time Derivative ◽  
Stability Modeling

The need to take into account the oscillation of a system is a special feature in the linear problem of the stability of a cantilevered rod under a follower force. Involvement of viscoelastic materials leads to damping of the oscillation hence to overestimation of critical loads. This new problem is solved here by means of an additional term introduced into the constitutive equation and proportional to the fractional time derivative with complex order—besides the inertial one. The effects contributed by the damping ratio, the real part of the order and the corrective role of its imaginary part on the shape of the bifurcation line, on its maximum and on the disposition of the inflection and maximal deflection points on the centerline of the deformed rod during the secondary loss of stability, are discussed.

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 A parallel in time/spectral collocation combined with finite difference method for the time fractional differential equations

2021 ◽  
Vol 15 ◽  
pp. 174830262110084
Author(s):  
Xianjuan Li ◽  
Yanhui Su
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Spectral Method ◽  
Fractional Differential Equations ◽  
Difference Method ◽  
Spectral Collocation ◽  
Parallel Method ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

In this article, we consider the numerical solution for the time fractional differential equations (TFDEs). We propose a parallel in time method, combined with a spectral collocation scheme and the finite difference scheme for the TFDEs. The parallel in time method follows the same sprit as the domain decomposition that consists in breaking the domain of computation into subdomains and solving iteratively the sub-problems over each subdomain in a parallel way. Concretely, the iterative scheme falls in the category of the predictor-corrector scheme, where the predictor is solved by finite difference method in a sequential way, while the corrector is solved by computing the difference between spectral collocation and finite difference method in a parallel way. The solution of the iterative method converges to the solution of the spectral method with high accuracy. Some numerical tests are performed to confirm the efficiency of the method in three areas: (i) convergence behaviors with respect to the discretization parameters are tested; (ii) the overall CPU time in parallel machine is compared with that for solving the original problem by spectral method in a single processor; (iii) for the fixed precision, while the parallel elements grow larger, the iteration number of the parallel method always keep constant, which plays the key role in the efficiency of the time parallel method.

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.

Time Fractional Differential Equation Model With Random Derivative Order

2009 ◽  
Author(s):  
Hongguang Sun ◽  
Yangquan Chen ◽  
Wen Chen
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Diffusion Processes ◽  
Time Derivative ◽  
Equation Model ◽  
Differential Equation Model ◽  
New Model ◽  
Potential Applications ◽  
Fractional Differential ◽  
Derivative Order

This paper proposes a new type of fractional differential equation model, named time fractional differential equation model, in which noise term is included in the time derivative order. The new model is applied to anomalous relaxation and diffusion processes suffering noisy field. The analysis and numerical simulation results show that our model can well describes the feature of these processes. We also find that the scale parameter and the frequency of the noise play a crucial role in the behaviors of these systems. At the end, we recognize some potential applications of this new model.

scholarly journals Memory effects on the proliferative function in the cycle-specific of chemotherapy

2021 ◽  
Author(s):  
Najma Ahmed ◽  
Dumitru Vieru ◽  
Fiazud Din Zaman
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Classical Model ◽  
Cell Mass ◽  
Time Derivative ◽  
Fractional Model ◽  
Fractional Differential ◽  
Mathcad Software

A generalized mathematical model of the breast and ovarian cancer is developed by considering the fractional differential equations with Caputo time-fractional derivatives. The use of the fractional model shows that the time-evolution of the proliferating cell mass, the quiescent cell mass, and the proliferative function are significantly influenced by their history. Even if the classical model, based on the derivative of integer order has been studied in many papers, its analytical solutions are presented in order to make the comparison between the classical model and the fractional model. Using the finite difference method, numerical schemes to the Caputo derivative operator and Riemann-Liouville fractional integral operator are obtained. Numerical solutions to the fractional differential equations of the generalized mathematical model are determined for the chemotherapy scheme based on the function of "on-off" type. Numerical results, obtained with the Mathcad software, are discussed and presented in graphical illustrations. The presence of the fractional order of the time-derivative as a parameter of solutions gives important information regarding the proliferative function, therefore, could give the possible rules for more efficient chemotherapy.

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

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