Analytical Solution of Hyperbolic Heat Conduction Equation in a Finite Medium Under Pulsatile Heat Source

  • Mohammad Reza Talaee
  • Ali Kabiri
  • Reza Khodarahmi
Research Paper


This paper presents a pure analytical solution of one-dimensional hyperbolic heat conduction equation in a homogeneous finite medium under series of time pulsed heat source which is exponentially distributed and acts symmetrically on both sides. The solution is obtained without any numerical procedures, using the Eigenvalue function. The problem is solved under two types of step and exponential time pulse series functions, which are used in simulation of laser interaction of tissues, and the closed-form solutions are introduced. The ability of the solution to estimate the effect of pulse duration and intensity is investigated. The results can be applied as a verification branch for other numerical solutions such as pulse laser interaction phenomenon.


Hyperbolic heat conduction Heat source Step pulse function Exponential pulse function Laser interaction 

List of symbols


Specific heat at constant pressure


Heaviside function

\(I\left( t \right)\)

Laser incident intensity


Arbitrary reference laser intensity


Dimensionless length of the slab


Length of the slab


Number of complete pulses


Coefficient controlling Off period


Counter of pulses


Fourier series counter


Heat source function


Heat flux vector


Surface reflectance



\(T_{0} ,T_{m}\)

Reference temperatures


Dimensionless step duration




Speed of heat propagation


Cartesian coordinate


Dimensionless Cartesian coordinate

Greek symbols


Thermal diffusivity




Thermal conductivity


Thermal relaxation time


Dimensionless temperature


Dirac function


Penetration coefficient


Dimensionless penetration coefficient


Dimensionless time


Dimensionless heat source function


Constant coefficient

\(\eta \left( \tau \right)\)

Dimensionless time-dependent laser function



Funding was provided by Iran University of Science and Technology.


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Copyright information

© Shiraz University 2017

Authors and Affiliations

  • Mohammad Reza Talaee
    • 1
  • Ali Kabiri
    • 1
  • Reza Khodarahmi
    • 1
  1. 1.School of Railway Engineering, Rolling StockIran University of Science and Technology (IUST)TehranIran

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