Heat and Mass Transfer

, Volume 46, Issue 7, pp 707–716 | Cite as

Analytical solution to the unsteady one-dimensional conduction problem with two time-varying boundary conditions: Duhamel’s theorem and separation of variables



An analytical resolution of the time-dependent one-dimensional heat conduction problem with time-dependent boundary conditions using the method of separation of variables and Duhamel’s theorem is presented. The two boundary conditions used are a time-dependent heat flux at one end and a varying temperature at the other end of the one-dimensional domain. It is put forth because the author found that the prescribed resolution method using separation of variables and Duhamel’s theorem presented in heat conduction textbooks is not directly applicable to problems with more than one time-dependent boundary condition. The analytical method presented in this paper makes use of one of the property of the heat conduction equation: the apparent linearity of the solutions. For that reason, in order to solve a problem with two time-dependent boundary conditions, the author first separates the initial problem into two independent but complementary problems, each with only one time-dependent boundary condition. Doing that, both simpler problems can be solved independently using a prescribed method that is known to work and the final solution can be obtained by joining the two independent solutions from the simpler separated problems. Every step of the resolution method is presented in this paper, along with a numerical validation of the final solution of three test case problems.


Heat Conduction Equation Auxiliary Problem Heat Conduction Problem Resolution Method Transient Solution 

List of symbols

Dimensional variables


Heat capacity (J kg−1 K−1)


Thermal conductivity (W m−1 K−1)


Length of the solid rod (m)


Summation index


Summation index


Summation index


Heat flux (W m−2)


Time (s)


Temperature (K)


Coordinate (m)

Greek symbols


Thermal diffusivity of the solid (m2 s−1)


Eigenvalues from the method of separation of variables


Numerical time step (s)


Distance between nodes (m)


Modified temperature variable (K)


Density (kg m−3)


Extra time variable in Duhamel’s theorem (s)


Period (s)


Temperature variable in auxiliary problem (K)



Complementary problem 1


Complementary problem 2


Heat flux coefficient in the cosine summation


Eastern boundary of the control volume


Eastern node




Position at x = L


Summation index


Summation index


Node N


Summation index


Present node




Stationary solution


Heat flux coefficient in the sine summation


Transient solution


Western boundary of the control volume


Western node



Previous time step



The author is grateful to the Natural Science and Engineering Research Council of Canada and to Dalhousie University for their financial support.


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentDalhousie UniversityHalifaxCanada

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