In this article we propose a new fractional derivative without singular
 kernel. We consider the potential application for modeling the steady
 heat-conduction problem. The analytical solution of the fractional-order heat
 flow is also obtained by means of the Laplace transform.

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A NEW FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL Application to the Modelling of the Steady Heat Flow

Following over 170+ pages and additional appendixes are formed based on content of Course: Fundamentals of Heat Transfer. Mainly this summarizes relevant parts on Book of Fundamentals of Heat and Mass Transfer (Incropera), but also other references introducing the same concepts are included. Student’s point of view has been consideredwith following highlights: (1) Relevant topics are presented in a nutshell to provide fast digestion of principles of heat transfer. (2) Appendixes include terminology dictionary. (3) Totally 22 illustrating examples are connecting theory to practical applications and quantifying heat transfer to understandable forms as: temperatures, heat transfer rates, heat fluxes, resistances and etc. (4) Most important Learning outcomes are presented for each topic separately. The Book, Fundamentals of Heat and Mass Transfer (Incropera), is certainly recommended for those going beyond basic knowledge of heat transfer. Lecture Notes consists of four primary content-wise objectives: (1) Give understanding to physical mechanisms of heat transfer, (2)Present basic concepts and terminology relevant for conduction, convection and radiation (3) Introduce thermal performance analysis methods for steady state and transient conduction systems. (4) Provide fast-to-digest phenomenological understanding required for basic design of thermal models

Fundamentals of heat transfer

Effect of variable thermal conductivity on a half-space under the fractional order theory of thermoelasticity

In this study, we consider an infinitely long solid circular cylinder, whose lateral surface is traction free and subject to a known surrounding temperature. The problem is in the context of the fractional order thermoelasticity theory. The medium is assumed to be initially quiescent. The solution is obtained by a direct approach without the customary use of potential functions. Laplace transform technique is used to obtain the general solution for any set of boundary conditions. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical results are given and represented graphically.

Fractional Order Thermoelastic Problem for an Infinitely Long Solid Circular Cylinder

Fractional thermoelastic models have been formulated from classical thermoelasticity or extended thermoelasticity, i.e. Lord-Shulman or Green-Naghdi theory. It seems different authors have different physical pictures on such topic, for instance, there exist several approaches from Lord-Shulman model to fractional order ones. This work is aimed to simplify the theoretical frameworks by clarifying connections between existed models. To this end, a new fractional thermoelasticity is established by directly extending classical thermoelasticity with the aids of new forms of fractional derivatives, i.e. Caputo-Fabrizio, Atangana–Baleanu and Tempered-Caputo definitions. All definitions are introduced into Fourier's law by a unified way with relaxation time incorporated. Theoretically, the present model may be simplified into existed fractional theory. Numerically, it has the capability of describing thermoelastic behaviors from fractional Lord-Shulman models. For numerical studies, Laplace transform method is adopted, and a two-layered structure subjected to thermal heating is considered, from which the effects of different fractional derivatives and relaxation time are firstly uncovered. And then, the influences of material constants of two layers, especially with interfacial conditions, are discussed in detail. Finally, some concluding remarks are made.

Fractional thermoelasticity revisited with new definitions of fractional derivative

In this work, we derive a set of governing equations for a mathematical model of generalized thermoelasticity in poroelastic materials. This model predicts finite speeds of propagation of waves contrary to the model of coupled thermoelasticity where an infinite speed of propagation is inherent. Next, we prove the uniqueness of solution of these equations under suitable conditions. We also obtain a reciprocity theorem for these equations. A thermal shock problem for a half-space composed of a poroelastic material saturated with a liquid is then considered. The surface of the half-space is assumed to be traction free, permeable, and subjected to heating. The Laplace transform technique is used to solve the problem. Numerical results for the temperature in the elastic body and fluid, displacement of the elastic body, velocity of the fluid, and stresses for both components are obtained and represented graphically.

A Mathematical Model for Short-Time Filtration in Poroelastic Media with Thermal Relaxation and Two Temperatures

Two-dimensional axisymmetric problems are considered within the context of the fractional order thermoelasticity theory. The general solution is obtained in the Laplace transform domain by using a ...

The effect of fractional thermoelasticity on two-dimensional problems in spherical regions under axisymmetric distributions

Effect of fractional parameter on thermoelastic half-space subjected to a moving heat source

We obtain the fundamental solution of thermoelasticity with two relaxation times for an infinite space which is acted upon by a spherically symmetric instantaneous point source of heat. The operational method and the Laplace transform technique are used to derive the solution in the form of a series of functions. Numerical results are given and represented graphically.

Eman M. Hussein

Papers

A Mathematical Model for Short-Time Filtration in Poroelastic Media with Thermal Relaxation and Two Temperatures

Fractional Order Thermoelastic Problem for an Infinitely Long Solid Circular Cylinder

The effect of fractional thermoelasticity on two-dimensional problems in spherical regions under axisymmetric distributions

Effect of fractional parameter on thermoelastic half-space subjected to a moving heat source

Fundamental solution of thermoelasticity with two relaxation times for an infinite spherically symmetric space