Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Similarity solutions of the nonlinear diffusion equation


Author: R. E. Grundy
Journal: Quart. Appl. Math. 37 (1979), 259-280
MSC: Primary 35K60
DOI: https://doi.org/10.1090/qam/548987
MathSciNet review: 548987
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Abstract: The paper considers similarity solutions of the nonlinear diffusion equation of the form $ {t^\alpha }f\left( \eta \right)$ where $ \eta = r{t^{ - \delta }}$ or $ \exp \left( {\alpha t} \right)f\left( \eta \right)$ where $ \eta = r\exp \left( { - \delta t} \right)$. The novel feature of the paper is that the second-order differential equation for $ f$ is reduced to a system of first-order equations and a phase plane analysis of one member of the system can be made. In this way we may discuss the existence and uniqueness of all the solutions for $ f\left( n \right)$. Restricting the discussion to plane geometry, we list all the continuous solutions to the basic problem on $ 0 \le \eta \le \infty $ with $ f\left( 0 \right) = U \ge 0$ and $ f\left( \infty \right) = 0$. Solutions of previous authors are identified as special cases.


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DOI: https://doi.org/10.1090/qam/548987
Article copyright: © Copyright 1979 American Mathematical Society

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