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

Application of fractional order theory of thermoelasticity to a 2D problem for a half-space

In this work, we consider a two-dimensional problem for a visco-elastic half-space. Laplace and exponential Fourier transforms are used to solve the problem. The solution in the transformed domain is obtained by a direct approach. The inverse Fourier transforms are obtained by using the inversion formula of the transform, while the inverse Laplace transforms are obtained using a numerical method. The temperature, displacement and stress distributions are computed and represented graphically.

2D problem for a half-space in the generalized theory of thermo-viscoelasticity

In this work, we apply the fractional order theory of thermoelasticity to a 1D problem for a half-space overlaid by a thick layer of a different material. The upper surface of the layer is taken to be traction free and is subjected to a constant thermal shock. There are no body forces or heat sources affecting the medium. Laplace transform techniques are used to eliminate the time variable t. The solution in the transformed domain is obtained by using a direct approach. The inverse Laplace transforms are obtained by using a numerical method based on Fourier expansion techniques. The predictions of the fractional order theory are discussed and compared with those for the generalized theory of thermoelasticity. We also study the effect of the fractional derivative parameters of the two media on the behavior of the solution. Numerical results are computed and represented graphically for the temperature, displacement and stress distributions.

Fractional model of thermoelasticity for a half‐space overlaid by a thick layer

Here, this work is concerned with some different one-dimensional (1D) problems of distribution of the thermal stresses and temperature in fractional generalized thermoelastic material, as follows: (i) 1D problem for a half-space of elastic material in the presence of heat sources has been solved by using Laplace transform and state space techniques; and (ii) 1D problem of distribution of thermal stresses and temperature in infinite medium with a spherical cavity subjected to a sudden change in the temperature of its internal boundary, which is assumed to be traction free, has been solved by Laplace transform and direct approach techniques. In the preceding problems, the numerical results for dimensionless variable fields are given and illustrated graphically. According to the numerical results, some comparisons have been shown in figures to estimate the effect of the fractional derivative parameter α about the new theory on all variable fields and discussion has been established for a copper-like ...

1D applications on fractional generalized thermoelasticity associated with two relaxation times

IN THIS WORK WE CONSIDER A PROBLEM IN THE CONTEXT OF THE GENERALIZED THEORY OF THERMOELASTICITY FOR A HALF SPACE. THE MATERIAL OF THE HALF SPACE IS FUNCTIONALLY GRADED IN WHICH LAMA©ÂS MODULII ARE FUNCTIONS OF THE VERTICAL DISTANCE FROM THE SURFACE OF THE MEDIUM. THE SURFACE IS TRACTION FREE AND SUBJECTED TO A TIME DEPENDENT THERMAL SHOCK. THE PROBLEM WAS SOLVED BY USING THE LAPLACE TRANSFORM METHOD TOGETHER WITH THE PERTURBATION TECHNIQUE. THE OBTAINED RESULTS ARE DISCUSSED AND COMPARED WITH THE SOLUTION WHEN LAMA©ÂS MODULII ARE CONSTANTS. NUMERICAL RESULTS ARE COMPUTED AND REPRESENTED GRAPHICALLY FOR THE TEMPERATURE, DISPLACEMENT AND STRESS DISTRIBUTIONS.

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A. M. Abd El-Latief

Papers

Application of fractional order theory of thermoelasticity to a 2D problem for a half-space

2D problem for a half-space in the generalized theory of thermo-viscoelasticity

Fractional model of thermoelasticity for a half‐space overlaid by a thick layer

1D applications on fractional generalized thermoelasticity associated with two relaxation times

Modeling of Variable Lamé's Modulii for a FGM Generalized Thermoelastic Half Space