Fractional Differential Equations

This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

Two general models of fractional heat conduction for non-homogeneous anisotropic elastic solids are introduced and the constitutive equations for thermoelasticity theory are obtained, uniqueness an...

On fractional thermoelasticity

A novel generalized thermoelasticity model based on memory-dependent derivative

Generalized thermo-viscoelasticity with memory-dependent derivatives

In this work, a new state-space approach in the context of fractional-order theory of thermoelasticity is introduced. This new method utilizes the diagonalization of a square matrix by the use of its eigenvalues and vectors. This new approach is applied to a 1-D problem for a half-space subjected to a thermal shock. Laplace transform technique is used throughout. The inversion of the transforms is performed using a numerical inversion technique based on the Fourier series expansion technique. The effects of the time and the fractional parameter on variable field distributions are discussed and represented graphically.

New state-space approach and its application in thermoelasticity

The fractional mathematical model of Maxwell’s equations in an electromagnetic field and the fractional generalized thermoelastic theory associated with two relaxation times are applied to a 1D problem for a thick plate. Laplace transform is used. The solution in Laplace transform domain has been obtained using a direct method and its inversion is calculated numerically using a method based on Fourier series expansion technique. Finally, the effects of the two fractional parameters (thermo and magneto) on variable fields distributions are made. Numerical results are represented graphically.

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Application of Generalized Fractional Thermoelasticity Theory with Two Relaxation Times to an Electromagnetothermoelastic Thick Plate

We introduce a mathematical model of unsteady thermoelectric MHD flow and heat transfer of two immiscible fractional second-grade fluids, with thermal fractional parameters and mechanical fractional parameters , . The Laplace transform with respect to time is used to obtain the solution in the transformed domain. The inversion of Laplace transform is obtained by using numerical method based on a Fourier-series expansion. The numerical results for temperature, velocity, and the stress distributions are represented graphically for different values of and . The graphs describe the fractional thermomechanical parameters effect on the case of two immiscible fluids and the case of a single fluid.

/pdf/thermomechanical-fractional-model-of-two-immiscible-temhd-ly6z538vm0.pdf

Thermomechanical Fractional Model of Two Immiscible TEMHD

In this work, we study the effects of the thermal and diffusion memories in the seas and oceans water using the methodology of fractional calculus for modeling a viscous fluid. In our model, we derive a model with two fractional parameters representing the thermal and diffusion effects for this type of fluid. We formulate a problem of mixed convection heat and mass transfer flow of a viscous fluid through a Limestone porous medium in a vertical channel whose walls are kept at variable temperature and concentration. The influences of heat‐mass memories in the presence of chemical reaction, heat generation/absorption and, the Soret effect are discussed and represented graphically on all variables fields. Also, the influences of Darcy, Prandtl, Schmidt, Grashof parameters are compared with the effect of fractional parameters.

The memory time effects for unsteady seas and oceans water flow through the limestone porous medium in the presence of chemical reaction and Soret effects

In this work, the generalized thermoelastic theory due to Lord-Shulman is used to solve 2D problem for a long circular cylinder formed of a realistic material. The thermal conductivity (TC) of the considered material is variable. The traction is zero on the lateral surface of the cylinder which is subjected to the boundary condition that varies peripherally. Laplace transform is used. The inverse of the Laplace transforms is obtained numerically. The temperature, displacement and stresses distributions were shown graphically. Comparison is made between the cases of variable and constant thermal conductivity. The effect of variability of the TC on the behavior of the stress is somewhat big but is small on the displacement and temperature. 

A. M. Abd El-Latief

Papers

New state-space approach and its application in thermoelasticity

Application of Generalized Fractional Thermoelasticity Theory with Two Relaxation Times to an Electromagnetothermoelastic Thick Plate

Thermomechanical Fractional Model of Two Immiscible TEMHD

The memory time effects for unsteady seas and oceans water flow through the limestone porous medium in the presence of chemical reaction and Soret effects

2D long generalized thermoelastic cylinder with realistic thermal conductivity