241 research outputs found
Seven common errors in finding exact solutions of nonlinear differential equations
We analyze the common errors of the recent papers in which the solitary wave
solutions of nonlinear differential equations are presented. Seven common
errors are formulated and classified. These errors are illustrated by using
multiple examples of the common errors from the recent publications. We show
that many popular methods in finding of the exact solutions are equivalent each
other. We demonstrate that some authors look for the solitary wave solutions of
nonlinear ordinary differential equations and do not take into account the well
- known general solutions of these equations. We illustrate several cases when
authors present some functions for describing solutions but do not use
arbitrary constants. As this fact takes place the redundant solutions of
differential equations are found. A few examples of incorrect solutions by some
authors are presented. Several other errors in finding the exact solutions of
nonlinear differential equations are also discussed.Comment: 42 page
Reduction operators and exact solutions of generalized Burgers equations
Reduction operators of generalized Burgers equations are studied. A
connection between these equations and potential fast diffusion equations with
power nonlinearity -1 via reduction operators is established. Exact solutions
of generalized Burgers equations are constructed using this connection and
known solutions of the constant-coefficient potential fast diffusion equation.Comment: 7 page
Stability of equilibrium solutions of a double power reaction diffusion equation with a Dirac interaction
In this paper we provide detailed information about the instability of
equilibrium solutions of a nonlinear family of localized reaction-difussion
equations in dimensione one. Beyond we provide explicit formulas to the
equilibrium solutions, via perturbation method and we calculate the exact
number of positive eigenvalues of the linear operator associated to the
stability problem, which allow us to compute the dimension of the unstable
manifold.Comment: 16 pages, 3 figure
Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations
We emphasize two connections, one well known and another less known, between
the dissipative nonlinear second order differential equations and the Abel
equations which in its first kind form have only cubic and quadratic terms.
Then, employing an old integrability criterion due to Chiellini, we introduce
the corresponding integrable dissipative equations. For illustration, we
present the cases of some integrable dissipative Fisher, nonlinear pendulum,
and Burgers-Huxley type equations which are obtained in this way and can be of
interest in applications. We also show how to obtain Abel solutions directly
from the factorization of second-order nonlinear equationsComment: 6 pages, 7 figures, published versio
A finite volume-complete flux scheme for the singularly perturbed generalized Burgers-Huxley equation
In this paper the finite volume-complete flux scheme is proposed to numerically solve the generalized Burgers-Huxley equation. The scheme is applied in an iterative manner. Numerical computations are performed for traveling wave-type problems as a validation of the method. Convection-dominated problems are used to assess the method on boundary layers. The results are in good agreement with reference results
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