14 research outputs found
Invariant analysis and explicit solutions of the time fractional nonlinear perturbed Burgers equation
The Lie group analysis method is performed for the nonlinear perturbed Burgers equation and the time fractional nonlinear perturbed Burgers equation. All of the point symmetries of the equations are constructed. In view of the point symmetries, the vector fields of the equations are constructed. Subsequently, the symmetry reductions are investigated. In particular, some novel exact and explicit solutions are obtained
Wave breaking for the generalized Fornberg-Whitham equation
This paper aims to show that the Cauchy problem of the Burgers equation with
a weakly dispersive perturbation involving the Bessel potential (generalization
of the Fornberg-Whitham equation) can exhibit wave breaking for initial data
with large slope. We also comment on the dispersive properties of the equation
On the eventual periodicity of fractional order dispersive wave equations using RBFS and transform
In this research work, let’s focus on the construction of numerical scheme based on radial basis functions finite difference (RBF-FD) method combined with the Laplace transform for the solution of fractional order dispersive wave equations. The numerical scheme is then applied to examine the eventual periodicity of the proposed model subject to the periodic boundary conditions. The implementation of proposed technique for high order fractional and integer type nonlinear partial differential equations (PDEs) is beneficial because this method is local in nature, therefore it yields and resulted in sparse differentiation matrices instead of full and dense matrices. Only small dimensions of linear systems of equations are to be solved for every center in the domain and hence this procedure is more reliable and efficient to solve large scale physical and engineering problems in complex domain.
Laplace transform is utilized for obtaining the equivalent time-independent equation in Laplace space and also valuable to handle time-fractional derivatives in the Caputo sense.
Application of Laplace transform avoids the time steeping procedure which commonly encounters the time instability issues. The solution to the transformed model is then obtained by computing the inversion of Laplace transform with an appropriate contour in a complex space, which is approximated by trapezoidal rule with high accuracy. Also since the Laplace transform operator is linear, it cannot be used to transform non-linear terms therefore let’s use a linearization approach and an appropriate iterative scheme. The proposed approach is tasted for some nonlinear fractional order KdV and Burgers equations. The capacity, high order accuracy and efficiency of our approach are demonstrated using examples and resultsRBFs Method
A new numerical application of the generalized Rosenau-RLW equation
. This study implemented a collocation nite element method based on septic
B-splines as a tool to obtain the numerical solutions of the nonlinear generalized RosenauRLW equation. One of the advantages of this method is that when the bases are chosen
at a high degree, better numerical solutions are obtained. E ectiveness of the method
is demonstrated by solving the equation with various initial and boundary conditions.
Further, in order to detect the performance of the method, L2 and L1 error norms and
two lowest invariants IM and IE were computed. The obtained numerical results were
compared with some of those in the literature for similar parameters. This comparison
clearly shows that the obtained results are better than and in good conformity with some
of the earlier results. Stability analysis demonstrates that the proposed algorithm, based
on a Crank Nicolson approximation in time, is unconditionally stable
Global well posedness and inviscid limit for the Korteweg-de Vries-Burgers equation
Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation
\begin{eqnarray*} u_t+u_{xxx}+\epsilon |\partial_x|^{2\alpha}u+(u^2)_x=0, \
u(0)=\phi, \end{eqnarray*} where and is a
real-valued function, we show that it is globally well-posed in $H^s\
(s>s_\alpha)H^s (s>-3/4)\epsilon \in (0,1)T>0C([0,T]; H^s)\epsilon$ tends to
0.Comment: 35 pages, 0 figur
Global well-posedness and limit behavior for a higher-order Benjamin-Ono equation
In this paper, we prove that the Cauchy problem associated to the following
higher-order Benjamin-Ono equation is globally
well-posed in the energy space . Moreover, we study the limit
behavior when the small positive parameter tends to zero and show
that, under a condition on the coefficients , , and , the solution
to this equation converges to the corresponding solution of the
Benjamin-Ono equation
Zero-filter limit for the Camassa-Holm equation in Sobolev spaces
The aim of this paper is to answer the question left in \cite{GL} (Math. Z.
(2015) 281). We prove that the zero-filter limit of the Camassa-Holm equation
is the Burgers equation in the same topology of Sobolev spaces as the initial
data