Behavior of a one-dimensional nonlinear flux-limited diffusion equation and finite speed of propagation
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It is well-known that the classical diffusion equation has an unrealistic property that it gives infinite speed of propagation. That is to say that if the initial data is non- negative and positive somewhere then the solution (temperature etc.) at any later time is positive everywhere. Here we modify the classical diffusion equation and obtain a nonlinear equation which possibly exhibits finite speed of propagation. We do so by modifying the Fick's (or Fourier) law to be a bounded function of gradient. The equation is analyzed using analytical, perturbation, and numerical methods. The equation is also analyzed using similarity solutions to find a family of solutions exhibiting slowed diffusion. Using perturbation methods we determine the analytic solution at zeroth and first order. Numerical solution is also obtained by using finite difference method, using forward difference for time and central difference for space. Our results show that the diffusion in our case is much slower than the classical one.