A Problem of the Heat Conduction Equation with a Moving Boundary in Three-Dimensional Space, E3

  • Lian-ke Dong
  • Ben-lian Zhou


It is well known that a parabolic partial differential equation for the case of a moving boundary can describe some moving-boundary problems in different fields of science and technology. In this paper, we give an analytical method to obtain the analytical solution of the heat conduction equation for a system with a boundary moving in three-dimensional space:
$$\begin{array}{*{20}{c}} {\frac{{\partial T\left( {p;t} \right)}}{\partial } = \alpha {{\vartriangle }_{3}}T\left( {p;t} \right),p \in \dot{D},t > 0} \\ {T\left( {p;0} \right) = f\left( p \right),p \in \dot{D}} \\ {T\left[ {R\left( p \right)\theta ,z;t} \right] = {{T}_{c}},} \\ \end{array}$$
where T(p;t) is the temperature function, Ḋ C E3 is an open domain and R(t) is a time dependent function describing the moving boundary and having a continuous first derivative.

In this paper, we suggest a compressing transformation with respect to space-coordinates, by use of which problem (I) transforms to a homogeneous parabolic-type equation with variable coefficients for a system with fixed boundaries and a moving heat source; hence the existence and uniqueness of the solution of problem (I) will be proved. By a further transformation for the homogeneous parabolic-type equation with variable coefficients, an analytical solution is given.


Thermal Conductivity Heat Conduction Heat Source Variable Coefficient Research Academia 
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Copyright information

© Purdue Research Foundation 1985

Authors and Affiliations

  • Lian-ke Dong
    • 1
  • Ben-lian Zhou
    • 1
  1. 1.Institute of Metal ResearchAcademia SinicaShenyangChina

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