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Extensions and solutions for nonlinear diffusion equations and random walks.

E K Lenzi1,2, M K Lenzi3, H V Ribeiro4

  • 1Departamento de Física, Universidade Estadual de Ponta Grossa, Av. Gen. Carlos Cavalcanti 4748, 84030-900, Ponta Grossa, PR, Brazil.

Proceedings. Mathematical, Physical, and Engineering Sciences
|December 12, 2019
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Summary
This summary is machine-generated.

This study links random walks to nonlinear diffusion equations, modifying Einstein's framework for new physical scenarios. It reveals how fractal and fractional derivatives can arise in these complex systems.

Keywords:
Tsallis entropyexact solutionsnonlinear diffusion equationrandom walk

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Mathematical Physics

Background:

  • Einstein's Brownian motion framework provides a foundation for diffusion.
  • Understanding nonlinear diffusion is crucial for various physical systems.
  • Random walks offer a microscopic perspective on diffusion processes.

Purpose of the Study:

  • To explore the connection between random walks and nonlinear diffusion equations.
  • To adapt Einstein's framework for diverse physical scenarios.
  • To analyze the emergence of fractal and fractional derivatives in nonlinear diffusion.

Main Methods:

  • Modifying Einstein's Brownian motion framework.
  • Deriving solutions for nonlinear diffusion equations from a random walk model.
  • Analyzing connections with generalized thermostatistics.

Main Results:

  • Successfully obtained solutions for nonlinear diffusion equations.
  • Demonstrated the emergence of fractal and fractional derivatives under specific conditions.
  • Established a link between random walks, nonlinear diffusion, and generalized thermostatistics.

Conclusions:

  • The study successfully connects random walks and nonlinear diffusion.
  • Modified frameworks can accommodate various physical situations.
  • Fractal and fractional dynamics are potential outcomes in nonlinear diffusion systems.