Revisiting the Group Classification of the General Nonlinear Heat Equation <i>u<sub>t</sub></i> = (<i>K</i>(<i>u</i>)<i>u<sub>x</sub></i>)<i><sub>x</sub></i>

Revisiting the Group Classification of the General Nonlinear Heat Equation <i>u<sub>t</sub></i> = (<i>K</i>(<i>u</i>)<i>u<sub>x</sub></i>)<i><sub>x</sub></i>

Group classification is a powerful tool for identifying and selecting the free elements—functions or parameters—in partial differential equations (PDEs) that maximize the symmetry properties of the model. In this paper, we revisit the group classification of the general nonlinear heat (or diffusion)...

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Bibliographic Details
Main Author: Winter Sinkala
Format: Article
Language:English
Published: MDPI AG 2025-03-01
Series:Mathematics
Subjects:
group classification
nonlinear heat equation
Lie symmetries
diffusion models
Online Access:https://www.mdpi.com/2227-7390/13/6/911
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Summary:Group classification is a powerful tool for identifying and selecting the free elements—functions or parameters—in partial differential equations (PDEs) that maximize the symmetry properties of the model. In this paper, we revisit the group classification of the general nonlinear heat (or diffusion) equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>u</mi><mi>t</mi></msub><mo>=</mo><msub><mfenced separators="" open="(" close=")"><mi>K</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><msub><mi>u</mi><mi>x</mi></msub></mfenced><mi>x</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a non-constant function of the dependent variable. We present the group classification framework, derive the determining equations for the coefficients of the infinitesimal generators of the admitted symmetry groups, and systematically solve for admissible forms of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Our analysis recovers the classical results of Ovsyannikov and Bluman and provides additional clarity and detailed derivations. The classification yields multiple cases, each corresponding to a specific form of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and reveals the dimension of the associated Lie symmetry algebra.
ISSN:2227-7390

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