61,045 research outputs found
Exact Solutions of a Remarkable Fin Equation
A model "remarkable" fin equation is singled out from a class of nonlinear
(1+1)-dimensional fin equations. For this equation a number of exact solutions
are constructed by means of using both classical Lie algorithm and different
modern techniques (functional separation of variables, generalized conditional
symmetries, hidden symmetries etc).Comment: 6 page
Boundary knot method: A meshless, exponential convergence, integration-free, and boundary-only RBF technique
Based on the radial basis function (RBF), non-singular general solution and
dual reciprocity principle (DRM), this paper presents an inheretnly meshless,
exponential convergence, integration-free, boundary-only collocation techniques
for numerical solution of general partial differential equation systems. The
basic ideas behind this methodology are very mathematically simple and
generally effective. The RBFs are used in this study to approximate the
inhomogeneous terms of system equations in terms of the DRM, while non-singular
general solution leads to a boundary-only RBF formulation. The present method
is named as the boundary knot method (BKM) to differentiate it from the other
numerical techniques. In particular, due to the use of non-singular general
solutions rather than singular fundamental solutions, the BKM is different from
the method of fundamental solution in that the former does no need to introduce
the artificial boundary and results in the symmetric system equations under
certain conditions. It is also found that the BKM can solve nonlinear partial
differential equations one-step without iteration if only boundary knots are
used. The efficiency and utility of this new technique are validated through
some typical numerical examples. Some promising developments of the BKM are
also discussed.Comment: 36 pages, 2 figures, Welcome to contact me on this paper: Email:
[email protected] or [email protected]
An improvement of the product integration method for a weakly singular Hammerstein equation
We present a new method to solve nonlinear Hammerstein equations with weakly
singular kernels. The process to approximate the solution, followed usually,
consists in adapting the discretization scheme from the linear case in order to
obtain a nonlinear system in a finite dimensional space and solve it by any
linearization method. In this paper, we propose to first linearize, via Newton
method, the nonlinear operator equation and only then to discretize the
obtained linear equations by the product integration method. We prove that the
iterates, issued from our method, tends to the exact solution of the nonlinear
Hammerstein equation when the number of Newton iterations tends to infinity,
whatever the discretization parameter can be. This is not the case when the
discretization is done first: in this case, the accuracy of the approximation
is limited by the mesh size discretization. A Numerical example is given to
confirm the theorical result
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
Generalized conditional symmetries of evolution equations
We analyze the relationship of generalized conditional symmetries of
evolution equations to the formal compatibility and passivity of systems of
differential equations as well as to systems of vector fields in involution.
Earlier results on the connection between generalized conditional invariance
and generalized reduction of evolution equations are revisited. This leads to a
no-go theorem on determining equations for operators of generalized conditional
symmetry. It is also shown that up to certain equivalences there exists a
one-to-one correspondence between generalized conditional symmetries of an
evolution equation and parametric families of its solutions.Comment: 23 pages, extended versio
Do peaked solitary water waves indeed exist?
Many models of shallow water waves admit peaked solitary waves. However, it
is an open question whether or not the widely accepted peaked solitary waves
can be derived from the fully nonlinear wave equations. In this paper, a
unified wave model (UWM) based on the symmetry and the fully nonlinear wave
equations is put forward for progressive waves with permanent form in finite
water depth. Different from traditional wave models, the flows described by the
UWM are not necessarily irrotational at crest, so that it is more general. The
unified wave model admits not only the traditional progressive waves with
smooth crest, but also a new kind of solitary waves with peaked crest that
include the famous peaked solitary waves given by the Camassa-Holm equation.
Besides, it is proved that Kelvin's theorem still holds everywhere for the
newly found peaked solitary waves. Thus, the UWM unifies, for the first time,
both of the traditional smooth waves and the peaked solitary waves. In other
words, the peaked solitary waves are consistent with the traditional smooth
ones. So, in the frame of inviscid fluid, the peaked solitary waves are as
acceptable and reasonable as the traditional smooth ones. It is found that the
peaked solitary waves have some unusual and unique characteristics. First of
all, they have a peaked crest with a discontinuous vertical velocity at crest.
Especially, the phase speed of the peaked solitary waves has nothing to do with
wave height. In addition, the kinetic energy of the peaked solitary waves
either increases or almost keeps the same from free surface to bottom. All of
these unusual properties show the novelty of the peaked solitary waves,
although it is still an open question whether or not they are reasonable in
physics if the viscosity of fluid and surface tension are considered.Comment: 53 pages, 13 figures, 7 tables. Accepted by Communications in
Nonlinear Science and Numerical Simulatio
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