Boundary Control of Temperature Distribution in a Rectangular Functionally Graded Material Plate

2010 ◽  
Vol 133 (2) ◽  
Author(s):  
Hossein Rastgoftar ◽  
Mohammad Eghtesad ◽  
Alireza Khayatian
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Flux ◽  
Steady State ◽  
Temperature Distribution ◽  
Functionally Graded ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Fg Plates ◽  
Graded Material

In this paper, an analytical method and a partial differential equation based solution to control temperature distribution for functionally graded (FG) plates is introduced. For the rectangular FG plate under consideration, it is assumed that the material properties such as thermal conductivity, density, and specific heat capacity vary in the width direction, and the governing heat conduction equation of the plate is a second-order partial differential equation. Using Lyapunov’s theorem, it is shown that by applying controlled heat flux through the boundary of the domain, the temperature distribution of the plate will approach a desired steady-state distribution. Numerical simulation is provided to verify the effectiveness of the proposed method such that by applying the boundary transient heat flux, in-domain distributed temperature converges to its desired steady-state temperature.

Three-Dimensional Elastic Solution of a Transversely Isotropic Functionally Graded Rectangular Plate

2012 ◽  
Vol 591-593 ◽  
pp. 2655-2660 ◽  
Author(s):  
Guo Jun Nie ◽  
Zhao Yang Feng ◽  
Jun Tao Shi ◽  
Ying Ya Lu ◽  
Zheng Zhong
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Rectangular Plate ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Functionally Graded ◽  
Elastic Solution ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Present Solution

Three-dimensional elastic solution of a simply supported, transversely isotropic functionally graded rectangular plate is presented in this paper. Suppose that all elastic coefficients of the material have the same power-law dependence on the thickness coordinate. By introducing two new displacement functions, three equations of equilibrium in terms of displacements are reduced to two uncoupled partial differential equations. Exact solution for a second-order partial differential equation expressed by one of displacement functions is obtained and analytical solution for another fourth-order partial differential equation expressed by another displacement function is found by employing the Frobenius method. The validity of the present solution is first investigated. And the effect of the gradation of material properties on the mechanical behavior of the plate is studied through numerical examples.

scholarly journals Modelling heat transfer in heterogeneous media using fractional calculus

2013 ◽  
Vol 371 (1990) ◽  
pp. 20120146 ◽  
Author(s):  
Dominik Sierociuk ◽  
Andrzej Dzieliński ◽  
Grzegorz Sarwas ◽  
Ivo Petras ◽  
Igor Podlubny ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Flux ◽  
Transfer Function ◽  
Transfer Process ◽  
Heterogeneous Media ◽  
Heat Transfer Process ◽  
Order Partial Differential Equation ◽  
Partial Differential

This paper presents the results of modelling the heat transfer process in heterogeneous media with the assumption that part of the heat flux is dispersed in the air around the beam. The heat transfer process in a solid material (beam) can be described by an integer order partial differential equation. However, in heterogeneous media, it can be described by a sub- or hyperdiffusion equation which results in a fractional order partial differential equation. Taking into consideration that part of the heat flux is dispersed into the neighbouring environment we additionally modify the main relation between heat flux and the temperature, and we obtain in this case the heat transfer equation in a new form. This leads to the transfer function that describes the dependency between the heat flux at the beginning of the beam and the temperature at a given distance. This article also presents the experimental results of modelling real plant in the frequency domain based on the obtained transfer function.

Modeling Heat Transfer in Heterogeneous Media Using Fractional Calculus

2011 ◽  
Author(s):  
Dominik Sierociuk ◽  
Andrzej Dzielin´ski ◽  
Grzegorz Sarwas ◽  
Ivo Petras ◽  
Igor Podlubny ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Flux ◽  
Transfer Function ◽  
Transfer Process ◽  
Heterogeneous Media ◽  
Heat Transfer Process ◽  
Order Partial Differential Equation ◽  
Partial Differential

The paper presents the results of modeling the heat transfer process in heterogeneous media with the assumption that part of the heat flux is dispersed in the air around the beam. The heat transfer process in solid material (beam) can be described by integer order partial differential equation. However, in heterogeneous media it can be described by sub- or hyperdiffusion equation which results in fractional order partial differential equation. Taking into consideration that the part of the heat flux is dispersed into the neighbouring environment we additionally modify the main relation between heat flux and the temperature, and we obtain in this case the heat transfer equation in the new form. This leads to the transfer function which describes the dependency between the heat flux at the beginning of the beam and the temperature at the given distance. The article also presents the experimental results of modeling real plant in the frequency domain basing on the obtained transfer function.

ESTIMATES FOR APPROXIMATE SOLUTIONS TO A FUNCTIONAL DIFFERENTIAL EQUATION MODEL OF CELL DIVISION

2021 ◽  
pp. 1-20
Author(s):  
STEPHEN TAYLOR ◽  
XUESHAN YANG
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cell Division ◽  
Nonnegative Function ◽  
Approximate Solutions ◽  
Equation Model ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
First Order ◽  
Functional Differential

Abstract The functional partial differential equation (FPDE) for cell division, $$ \begin{align*} &\frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t))\\ &\quad = -(b(x,t)+\mu(x,t))n(x,t)+b(\alpha x,t)\alpha n(\alpha x,t)+b(\beta x,t)\beta n(\beta x,t), \end{align*} $$ is not amenable to analytical solution techniques, despite being closely related to the first-order partial differential equation (PDE) $$ \begin{align*} \frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)+F(x,t), \end{align*} $$ which, with known $F(x,t)$ , can be solved by the method of characteristics. The difficulty is due to the advanced functional terms $n(\alpha x,t)$ and $n(\beta x,t)$ , where $\beta \ge 2 \ge \alpha \ge 1$ , which arise because cells of size x are created when cells of size $\alpha x$ and $\beta x$ divide. The nonnegative function, $n(x,t)$ , denotes the density of cells at time t with respect to cell size x. The functions $g(x,t)$ , $b(x,t)$ and $\mu (x,t)$ are, respectively, the growth rate, splitting rate and death rate of cells of size x. The total number of cells, $\int _{0}^{\infty }n(x,t)\,dx$ , coincides with the $L^1$ norm of n. The goal of this paper is to find estimates in $L^1$ (and, with some restrictions, $L^p$ for $p>1$ ) for a sequence of approximate solutions to the FPDE that are generated by solving the first-order PDE. Our goal is to provide a framework for the analysis and computation of such FPDEs, and we give examples of such computations at the end of the paper.

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