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Lie symmetry method for a nonlinear heat-diffusion equation

Julieta Bollati, Ernesto A. Borrego Rodriguez, Adriana C. Briozzo

Abstract

We investigate the nonlinear heat-diffusion equation \( C(u)\,\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\!\left( K(u)\,\frac{\partial u}{\partial x} \right) \), where \( C(u) \) and \( K(u) \) are coefficients that depend on \( u \). By applying the classical Lie symmetry method, we determine the admitted Lie point symmetries and compute the corresponding infinitesimal generators according to the functional relationship between \( C(u) \) and \( K(u) \). The admitted symmetries are used to reduce the partial differential equation to ordinary differential equations and to construct invariant solutions. Particular cases of physical interest are analyzed in detail, including Storm-type materials and power-law dependence of \( C(u) \) and \( K(u) \) on \( u \). For these cases, similarity solutions are obtained.

Lie symmetry method for a nonlinear heat-diffusion equation

Abstract

We investigate the nonlinear heat-diffusion equation \( C(u)\,\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\!\left( K(u)\,\frac{\partial u}{\partial x} \right) \), where \( C(u) \) and \( K(u) \) are coefficients that depend on . By applying the classical Lie symmetry method, we determine the admitted Lie point symmetries and compute the corresponding infinitesimal generators according to the functional relationship between \( C(u) \) and \( K(u) \). The admitted symmetries are used to reduce the partial differential equation to ordinary differential equations and to construct invariant solutions. Particular cases of physical interest are analyzed in detail, including Storm-type materials and power-law dependence of \( C(u) \) and \( K(u) \) on . For these cases, similarity solutions are obtained.
Paper Structure (10 sections, 141 equations, 4 tables)

This paper contains 10 sections, 141 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Lie point symmetries
  3. Invariant solutions $u = \Theta(x,t)$ of the PDE \ref{['eq']}
  4. Case 1
  5. Case 2
  6. Particular differential equations of the type \ref{['eq']}
  7. Power-type $C(u)$ and constant $K(u)$
  8. Storm's condition for $C(u)$ and $K(u)$
  9. Power type $C(u)$ and $K(u)$
  10. Conclusion

Theorems & Definitions (3)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3