14,910 research outputs found
A multiple exp-function method for nonlinear differential equations and its application
A multiple exp-function method to exact multiple wave solutions of nonlinear
partial differential equations is proposed. The method is oriented towards ease
of use and capability of computer algebra systems, and provides a direct and
systematical solution procedure which generalizes Hirota's perturbation scheme.
With help of Maple, an application of the approach to the dimensional
potential-Yu-Toda-Sasa-Fukuyama equation yields exact explicit 1-wave and
2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton
type solutions. Two cases with specific values of the involved parameters are
plotted for each of 2-wave and 3-wave solutions.Comment: 12 pages, 16 figure
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
Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs
Algorithms are presented for the tanh- and sech-methods, which lead to
closed-form solutions of nonlinear ordinary and partial differential equations
(ODEs and PDEs). New algorithms are given to find exact polynomial solutions of
ODEs and PDEs in terms of Jacobi's elliptic functions.
For systems with parameters, the algorithms determine the conditions on the
parameters so that the differential equations admit polynomial solutions in
tanh, sech, combinations thereof, Jacobi's sn or cn functions. Examples
illustrate key steps of the algorithms.
The new algorithms are implemented in Mathematica. The package
DDESpecialSolutions.m can be used to automatically compute new special
solutions of nonlinear PDEs. Use of the package, implementation issues, scope,
limitations, and future extensions of the software are addressed.
A survey is given of related algorithms and symbolic software to compute
exact solutions of nonlinear differential equations.Comment: 39 pages. Software available from Willy Hereman's home page at
http://www.mines.edu/fs_home/whereman
Potential Nonclassical Symmetries and Solutions of Fast Diffusion Equation
The fast diffusion equation is investigated from the
symmetry point of view in development of the paper by Gandarias [Phys. Lett. A
286 (2001) 153-160]. After studying equivalence of nonclassical symmetries with
respect to a transformation group, we completely classify the nonclassical
symmetries of the corresponding potential equation. As a result, new wide
classes of potential nonclassical symmetries of the fast diffusion equation are
obtained. The set of known exact non-Lie solutions are supplemented with the
similar ones. It is shown that all known non-Lie solutions of the fast
diffusion equation are exhausted by ones which can be constructed in a regular
way with the above potential nonclassical symmetries. Connection between
classes of nonclassical and potential nonclassical symmetries of the fast
diffusion equation is found.Comment: 13 pages, section 3 is essentially revise
Exactly solvable variable parametric Burgers type models
Exactly solvable variable parametric Burgers type equations in one-dimension
are introduced, and two different approaches for solving the corresponding
initial value problems are given. The first one is using the relationship
between the variable parametric models and their standard counterparts. The
second approach is a direct linearization of the variable parametric Burgers
model to a variable parametric parabolic model via a generalized Cole-Hopf
transform. Eventually, the problem of finding analytic and exact solutions of
the variable parametric models reduces to that of solving a corresponding
second order linear ODE with time dependent coefficients. This makes our
results applicable to a wide class of exactly solvable Burgers type equations
related with the classical Sturm-Liouville problems for the orthogonal
polynomials
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