Analysis of Two-Dimensional Heat Diffusion Using FEA and the Fundamental Collocation Method

2016 ◽  
Author(s):  
Amir Khalilollahi ◽  
Enayat Mahajerin ◽  
Gary Burgess
Keyword(s):  
Laplace Transform ◽  
Collocation Method ◽  
Initial Conditions ◽  
Time Derivative ◽  
Heat Diffusion ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Two Dimensional ◽  
Heat Diffusion Equation ◽  
The Laplace Transform

Finite Element Analysis (FEA) and the Laplace Transform-Based Fundamental Collocation Method (FCM) are used to solve the heat diffusion equation in two-dimensional regions having arbitrary shapes and subjected to arbitrary initial and mixed type boundary conditions. In the FEA method, the time derivative is replaced with a finite difference approximation. The resulting time dependent global equations are solved incrementally starting with the initial conditions. The FCM approach is applied in the Laplace transform domain to obtain temperatures in the s-domain, T(x,y,s). An inversion technique is used to retrieve the time domain solution, T(x,y,t). To compare applicability and accuracy of these methods, both techniques are applied to transient heat flow problems for which exact solutions are known.

scholarly journals Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives

2017 ◽  
Vol 21 (6 Part A) ◽  
pp. 2299-2305 ◽  
Author(s):  
Ilknur Koca ◽  
Abdon Atangana
Keyword(s):  
Diffusion Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Heat Diffusion ◽  
Relaxation Kernel ◽  
Extended Version ◽  
Heat Diffusion Equation ◽  
The Laplace Transform ◽  
Non Local

Recently Hristov using the concept of a relaxation kernel with no singularity developed a new model of elastic heat diffusion equation based on the Caputo-Fabrizio fractional derivative as an extended version of Cattaneo model of heat diffusion equation. In the present article, we solve exactly the Cattaneo-Hristov model and extend it by the concept of a derivative with non-local and non-singular kernel by using the new Atangana-Baleanu derivative. The Cattaneo-Hristov model with the extended derivative is solved analytically with the Laplace transform, and numerically using the Crank-Nicholson scheme.

scholarly journals Heat-balance integral to fractional (half-time) heat diffusion sub-model

2010 ◽  
Vol 14 (2) ◽  
pp. 291-316 ◽  
Author(s):  
Jordan Hristov
Keyword(s):  
Diffusion Equation ◽  
Heat Balance ◽  
Integral Method ◽  
Time Derivative ◽  
Heat Diffusion ◽  
Half Time ◽  
Heat Diffusion Equation ◽  
Parabolic Profile ◽  
Heat Balance Integral Method ◽  
The Heat Balance

The fractional (half-time) sub-model of the heat diffusion equation, known as Dirac-like evolution diffusion equation has been solved by the heat-balance integral method and a parabolic profile with unspecified exponent. The fractional heat-balance integral method has been tested with two classic examples: fixed temperature and fixed flux at the boundary. The heat-balance technique allows easily the convolution integral of the fractional half-time derivative to be solved as a convolution of the time-independent approximating function. The fractional sub-model provides an artificial boundary condition at the boundary that closes the set of the equations required to express all parameters of the approximating profile as function of the thermal layer depth. This allows the exponent of the parabolic profile to be defined by a straightforward manner. The elegant solution performed by the fractional heat-balance integral method has been analyzed and the main efforts have been oriented towards the evaluation of fractional (half-time) derivatives by use of approximate profile across the penetration layer.

Initial conditions and the Laplace transform

1965 ◽  
Vol 11 (11) ◽  
pp. 385
Author(s):  
J.H. Brodie ◽  
C. Jones ◽  
S.E. Tweedy ◽  
E. Besag
Keyword(s):  
Laplace Transform ◽  
Initial Conditions ◽  
The Laplace Transform

scholarly journals New Numerical Treatment for a Family of Two-Dimensional Fractional Fredholm Integro-Differential Equations

Algorithms ◽  
2020 ◽  
Vol 13 (2) ◽  
pp. 37
Author(s):  
Amer Darweesh ◽  
Marwan Alquran ◽  
Khawla Aghzawi
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Haar Wavelet ◽  
Two Dimensional ◽  
Numerical Treatment ◽  
Wavelet Method ◽  
New Approach ◽  
Robust Algorithm ◽  
Haar Wavelet Method ◽  
The Laplace Transform

In this paper, we present a robust algorithm to solve numerically a family of two-dimensional fractional integro differential equations. The Haar wavelet method is upgraded to include in its construction the Laplace transform step. This modification has proven to reduce the accumulative errors that will be obtained in case of using the regular Haar wavelet technique. Different examples are discussed to serve two goals, the methodology and the accuracy of our new approach.

Asymptotic solution of the Vlasov and Poisson equations for an inhomogeneous plasma

1991 ◽  
Vol 45 (1) ◽  
pp. 59-70 ◽  
Author(s):  
Riccardo Croci
Keyword(s):  
Magnetic Field ◽  
Laplace Transform ◽  
Integral Equations ◽  
Initial Conditions ◽  
Inhomogeneous Plasma ◽  
Algebraic Equations ◽  
Poisson Equations ◽  
The Laplace Transform ◽  
Eigenvalue Parameter ◽  
The Fourier Transform

The purpose of this paper is to derive the asymptotic solutions to a class of inhomogeneous integral equations that reduce to algebraic equations when a parameter ε goes to zero (the kernel becoming proportional to a Dirac δ function). This class includes the integral equations obtained from the system of Vlasov and Poisson equations for the Fourier transform in space and the Laplace transform in time of the electrostatic potential, when the equilibrium magnetic field is uniform and the equilibrium plasma density depends on εx, with the co-ordinate z being the direction of the magnetic field. In this case the inhomogeneous term is given by the initial conditions and possibly by sources, and the Laplace-transform variable ω is the eigenvalue parameter.

Initial conditions and the Laplace transform

1965 ◽  
Vol 11 (10) ◽  
pp. 349
Author(s):  
G.K. Steel ◽  
D.J. Storey ◽  
H.M. Power
Keyword(s):  
Laplace Transform ◽  
Initial Conditions ◽  
The Laplace Transform

Digital simulation of photoacoustic impulse responses

1986 ◽  
Vol 64 (9) ◽  
pp. 1049-1052 ◽  
Author(s):  
Richard M. Miller
Keyword(s):  
Finite Difference ◽  
Impulse Response ◽  
Depth Distribution ◽  
Digital Simulation ◽  
Heat Diffusion ◽  
Finite Difference Approximation ◽  
Diffusion Equations ◽  
Difference Approximation ◽  
Impulse Responses ◽  
Solid Samples

Impulse-response photoacoustic spectroscopy provides information on the depth distribution of chromophores in solid samples. To gain an understanding of the way in which sample properties affect the impulse response, a digital model has been generated. This model is based on discretization of time and space coupled with a finite-difference approximation of the governing heat-diffusion equations. The simulations are compared with experimental results.

Two-Dimensional Transient Solutions for Crossflow Heat Exchangers With Neither Gas Mixed

1987 ◽  
Vol 109 (2) ◽  
pp. 281-286 ◽  
Author(s):  
G. Spiga ◽  
M. Spiga
Keyword(s):  
Heat Exchangers ◽  
Initial Conditions ◽  
Transient Behavior ◽  
Computational Time ◽  
Two Dimensional ◽  
Transform Method ◽  
Wide Range ◽  
The Laplace Transform ◽  
Inlet Conditions ◽  
Conductance Ratio

The two-dimensional transient behavior of gas-to-gas crossflow heat exchangers is investigated, solving by analytical methods the thermal balance equations in order to determine the transient distribution of temperatures in the core wall and in both the unmixed gases. Assuming large wall capacitance, the general solutions are deduced by the Laplace transform method and are presented as integrals of modified Bessel functions on space and time, for a transient response with any arbitrary initial and inlet conditions, in terms of the number of transfer units, capacity rate and conductance ratio. Specializing the entrance temperature and assuming constant initial conditions, the most meaningful transient conditions (such as step, ramp, and exponential responses) have been simulated and the relevant solutions, expressed by means of either integrals or series, have been accurately computed with extremely low computational time. The temperature responses are then presented in graphic form for a wide range of the number of transfer units.

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