A Generalization of the Hopf-Cole Transformation
Paulius Miskinis
TL;DR
This paper extends the Hopf-Cole transformation to generalized forms, exploring their relation to Burgers and diffusion equations, presenting explicit solutions, and analyzing traveling waves and nonlocal interactions.
Contribution
It introduces a generalized Hopf-Cole transformation and analyzes nonlocal extensions of Burgers and diffusion equations, providing new analytical solutions and insights.
Findings
Explicit analytical solution for the generalized transformation
Existence of traveling wave solutions confirmed
Interaction effects of nonlocal perturbations analyzed
Abstract
A generalization of the Hopf-Cole transformation and its relation to the Burgers equation of integer order and the diffusion equation with quadratic nonlinearity are discussed. The explicit form of a particular analytical solution is presented. The existence of the travelling wave solution and the interaction of nonlocal perturbation are considered. The nonlocal generalizations of the one-dimensional diffusion equation with quadratic nonlinearity and of the Burgers equation are analyzed.
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