scholarly journals A self-adaptive LGSM to recover initial condition or heat source of one-dimensional heat conduction equation by using only minimal boundary thermal data

2011 ◽  
Vol 54 (7-8) ◽  
pp. 1305-1312 ◽  
Author(s):  
Chein-Shan Liu
Keyword(s):  
Heat Conduction ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Initial Condition ◽  
One Dimensional ◽  
Thermal Data ◽  
Self Adaptive

scholarly journals Determining An Unknown Boundary Condition by An Iteration Method

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 409
Author(s):  
Dejian Huang ◽  
Yanqing Li ◽  
Donghe Pei
Keyword(s):  
Boundary Condition ◽  
Exact Solution ◽  
Heat Conduction ◽  
Iteration Method ◽  
Heat Conduction Equation ◽  
Heat Conduction Problem ◽  
Variational Iteration Method ◽  
Conduction Equation ◽  
Unknown Boundary ◽  
Initial Condition

This paper investigates the boundary value in the heat conduction problem by a variational iteration method. Applying the iteration method, a sequence of convergent functions is constructed, the limit approximates the exact solution of the heat conduction equation in a few iterations using only the initial condition. This method does not require discretization of the variables. Numerical results show that this method is quite simple and straightforward for models that are currently under research.

Analytical Solution for Three-Dimensional Hyperbolic Heat Conduction Equation with Time-Dependent and Distributed Heat Source

2016 ◽  
Vol 33 (1) ◽  
pp. 65-75 ◽  
Author(s):  
M. R. Talaee ◽  
V. Sarafrazi
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Time Dependent ◽  
Conduction Equation ◽  
Hyperbolic Heat Conduction

AbstractThis paper is devoted to the analytical solution of three-dimensional hyperbolic heat conduction equation in a finite solid medium with rectangular cross-section under time dependent and non-uniform internal heat source. The closed form solution of both Fourier and non-Fourier profiles are introduced with Eigen function expansion method. The solution is applied for simple simulation of absorption of a continues laser in biological tissue. The results show the depth of laser absorption in tissue and considerable difference between the Fourier and Non-Fourier temperature profiles. In addition the solution can be applied as a verification branch for other numerical solutions.

Analytical solution to thermal stresses of high speed PM rotor considering thermal load and heat convection

2020 ◽  
Vol 64 (1-4) ◽  
pp. 533-540
Author(s):  
Wenjie Cheng ◽  
Zhikai Deng ◽  
Guangdong Cao ◽  
Ling Xiao ◽  
Huimin Qi ◽  
...  
Keyword(s):  
Temperature Field ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Stress Field ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
High Speed ◽  
Heat Convection ◽  
Conduction Equation ◽  
Air Cooling

Aiming at the high speed permanent magnet (PM) rotor with the heat source, this work investigates the analytic solution to the transient temperature field and thermal stress field of the rotor, considering the influence of the forced air cooling of rotor surface on the stress field. Firstly, dimensionless formulation of the transient heat conduction equation including interior heat source is derived, where the axially non-uniform heat convection coefficient and the temperature of main flow region are equivalent to their mean values. Secondly, the Fourier integral transform method is used to solve the dimensionless heat conduction equation. Then, the obtained temperature field is loaded into the analytical solution of strength, in which three types of stress sources such as interference fit, centrifugal force and temperature gradient are included. Finally, examples are carried out to verify the analytical solutions and relative results are discussed.

Analytical Solution of Hyperbolic Heat Conduction Equation for a Finite Slab With Arbitrary Boundaries, Initial Condition, and Stationary Heat Source

2008 ◽  
Author(s):  
Hossein Shokouhmand ◽  
Seyed Reza Mahmoudi ◽  
Kaveh Habibi
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Transient Temperature ◽  
Hyperbolic Heat Conduction ◽  
High Heat Flux ◽  
Adequate Model ◽  
Finite Slab

This paper presents an analytical solution of the hyperbolic heat conduction equation for a finite slab that sides are subjected to arbitrary heat source, boundary, and initial conditions. In the mathematical model used in this study, the heating on both sides treated as an apparent heat source while sides of the slab assumed to be insulated. Distribution of the apparent heat source for a problem with arbitrary heating on two boundaries is solved. The solution obtained by separation of variable method using appropriate Fourier series. Being a Sturm-Liouville problem in x-direction, suitable orthogonal functions can be allocated to hyperbolic heat conduction equation depending on the type of boundary conditions. Despite ease of proposed method, very few works has been done to solve hyperbolic heat conduction problems using this method by authors. The main feature of the method is straightforward formulation. In the analysis of heat conduction involving extremely short times, the parabolic heat conduction equation breaks down. By increasing the applications of the fast heat sources such as laser pulse for annealing of semiconductors and high heat flux applications, the need for adequate model of heat conduction has arisen. The hyperbolic heat conduction equation eliminates the paradox of an infinite speed of propagation of thermal disturbances which contradicts with Einstein’s theory of relativity. Moreover, it describes the highly transient temperature distribution in a finite medium more accurately.

Truncation Error Estimates for Numerical and Analog Solutions of the Heat-Conduction Equation

1961 ◽  
Vol 83 (3) ◽  
pp. 382-383 ◽  
Author(s):  
N. H. Freed ◽  
C. J. Rallis
Keyword(s):  
Heat Conduction ◽  
Heat Flow ◽  
Finite Difference ◽  
Error Estimates ◽  
Heat Conduction Equation ◽  
Truncation Error ◽  
Mesh Size ◽  
Practical Method ◽  
Conduction Equation ◽  
One Dimensional

A practical method is presented for obtaining a meaningful estimate of the truncation error associated with fully finite-difference forms of the heat-conduction equation. The analysis is applied in this instance to the Liebmann analog equations. It may also be used with other manual and analog methods of computation, where the error due to mesh size is relatively large. An example is given deriving error estimates for a case of one-dimensional heat flow.

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