I have a solution to the Burgers' equation that should ideally produce a triangular wave-like graph but, unfortunately, I have been unable to plot this successfully. So far, I have only succeeded in producing a Gaussian-type curve, representing the distribution of shock-wave energies, that diffuses-out with time, in Mathematica. I think that the latter is a solution to the Heat equation and the former triangular wave should plot the solution to the Diffusion equation. Can anyone first tell me how to plot the triangular wave (using Mathematica), and can you describe the vanishingly small values that each displays, such as of the order of 0.0000001 (metres) for the maximum height of the Gaussian, at least? I am trying to model neurological shock-wave energies, which should be vastly larger, even if only just a few Joules, and I am actually unsure about what both curves signify – apart from the fact that each spreads out and approaches zero as time tends to infinity, as one would expect for diffusion processes. When I integrated the solution to the Burgers’ equation (obtained by solving the Heat-Diffusion equation via the Hopf-Cole Transformation) last year, I obtained an ‘energy’ value of around 25,000 Joules, as expected, and it also matched the values of other calculations obtained via other equations and methods. The diffusion coefficient that I am using has a magnitude of 0.000000001 square metres per second. Have I misunderstood the Burgers’ and the Heat and Diffusion equation solutions?
You will never have a triangular wave or even a Gaussian-type wave propagating as such by Burgers' equation. It will produced a shock. In the triangular case some kind of "triangular" shock. If a diffusion equation acts on this perturbation it will tends to smooth it immediately loosing it corners but without moving but if a transport term is added.
Thank you, Jorge and Ambika. I am aware of the shock-discontinuity, and the rarefaction wave aspects, and I have already discussed them in my paper. I was expecting a triangular wave (including the Heat-Equation based symmetric Gaussian that I already succeeded in plotting), since there is a diffusion component in my models as well as convection - and both convection and diffusion (Advection) are major features of the Burgers' equation (and the more complicated Navier Stokes' equation), all of which are obtained using the Heat-Diffusion equation via the Hopf-Cole Transformation. I do have a programme to run the triangular wave, but it won't work, and I wonder if my diffusion coefficient of 0.000000001 metres squared per second (hence the shock-discontinuity and the Rankine-Hugoniot condition etc.) is too small for the programme to run!? I have already modelled the shock- and the resulting pressure-wave and associated energies using more conventional methods associated with solution of the Burgers' equation, but wanted to verify these more modest results by the former Gaussian and Triangular methods that I have already just mentioned. The Triangular wave, which is a solution to the Burgers' equation, in particular takes into consideration the highly discontinuous nature of the latter, and illustrates the fact that the wave first steepens, only to then 'overtake' itself and to then finally diffuse out to zero over time - like a 'breaker-type-wave' that rolls in over the beach that does pretty much the same thing. I am also very much aware that the subject of shock-waves is no ordinary area of Mathematics and Physics, and that the usual models of wave-mechanics do not hold - hence PDEs and the Burgers' equation etc..
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