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Register a new userConvection-Diffusion-Reaction equation with similarity solutions
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We consider similarity solutions of the generalized convection-diffusion-reaction equation with both space- and time-dependent convection, diffusion and reaction terms. By introducing the similarity variable, the reaction-diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable convection-diffusion-reaction systems. Some representative examples of exactly solvable systems are presented. We also describe how an equivalent convection-diffusion-reaction system can be constructed which admits the same similarity solution of another convection-diffusion-reaction system.
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We consider solvability of the generalized reaction-diffusion equation with both space- and time-dependent diffusion and reaction terms by means of the similarity method. By introducing the similarity variable, the reaction-diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable reaction-diffusion systems. Several representative examples of exactly solvable reaction-diffusion equations are presented.
We consider quasi-stationary (travelling wave type) solutions to a general nonlinear reaction-convection-diffusion equation with arbitrary, autonomous coefficients. The second order nonlinear equation describing one dimensional travelling waves can be reduced to a first kind first order Abel type differential equation By using two integrability conditions for the Abel equation (the Chiellini lemma and the Lemke transformation), several classes of exact travelling wave solutions of the general reaction--convection-diffusion equation are obtained, corresponding to different functional relations imposed between the diffusion, convection and reaction functions. In particular, we obtain travelling wave solutions for two non-linear second order partial differential equations, representing generalizations of the standard diffusion equation and of the classical Fisher--Kolmogorov equation, to which they reduce for some limiting values of the model parameters. The models correspond to some specific, power law type choices of the reaction and convection functions, respectively. The travelling wave solutions of these two classes of differential equations are investigated in detail by using both numerical and semi-analytical methods.
We consider the solvability of the Fokker-Planck equation with both time-dependent drift and diffusion coefficients by means of the similarity method. By the introduction of the similarity variable, the Fokker-Planck equation is reduced to an ordinary differential equation. Adopting the natural requirement that the probability current density vanishes at the boundary, the resulting ordinary differential equation turns out to be integrable, and the probability density function can be given in closed form. New examples of exactly solvable Fokker-Planck equations are presented.
This work studies exact solvability of a class of fractional reaction-diffusion equation with the Riemann-Liouville fractional derivatives on the half-line in terms of the similarity solutions. We derived the conditions for the equation to possess scaling symmetry even with the fractional derivatives. Relations among the scaling exponents are determined, and the appropriate similarity variable introduced. With the similarity variable we reduced the stochastic partial differential equation to a fractional ordinary differential equation. Exactly solvable systems are then identified by matching the resulted ordinary differential equation with the known exactly solvable fractional ones. Several examples involving the three-parameter Mittag-Leffler function (Kilbas-Saigo function) are presented. The models discussed here turn out to correspond to superdiffusive systems.
We establish an integral variational principle for the spreading speed of the one dimensional reaction diffusion equation with Stefan boundary conditions, for arbitrary reaction terms. This principle allows to obtain in a simple way the dependence of the speed on the Stefan constant. As an application a generalized Zeldovich-Frank-Kamenetskii lower bound for the speed, valid for monostable and combustion reaction terms, is given.
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