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

## Abstract

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.

## Keywords

Hyperbolic heat conduction Heat source Step pulse function Exponential pulse function Laser interaction## List of symbols

- \(c_{\text{p}}\)
Specific heat at constant pressure

- \(H\)
Heaviside function

- \(I\left( t \right)\)
Laser incident intensity

- \(I_{r}\)
Arbitrary reference laser intensity

- \(L\)
Dimensionless length of the slab

- \(l\)
Length of the slab

- \(M\)
Number of complete pulses

- \(m\)
Coefficient controlling Off period

- \(N\)
Counter of pulses

- \(n\)
Fourier series counter

- \(Q\)
Heat source function

- \(q\)
Heat flux vector

- \(R\)
Surface reflectance

*T*Temperature

- \(T_{0} ,T_{m}\)
Reference temperatures

*T*_{p}Dimensionless step duration

*t*Time

*w*Speed of heat propagation

*x*Cartesian coordinate

*X*Dimensionless Cartesian coordinate

## Greek symbols

*α*Thermal diffusivity

*ρ*Density

- \(\lambda\)
Thermal conductivity

- \(\tau_{q}\)
Thermal relaxation time

- \(\theta\)
Dimensionless temperature

- \(\delta\)
Dirac function

- \(\mu\)
Penetration coefficient

- \(\beta\)
Dimensionless penetration coefficient

- \(\tau\)
Dimensionless time

- \(\psi\)
Dimensionless heat source function

- \(\psi_{0}\)
Constant coefficient

- \(\eta \left( \tau \right)\)
Dimensionless time-dependent laser function

## Notes

### Funding

Funding was provided by Iran University of Science and Technology.

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