88 research outputs found
Blowing-up solutions of the time-fractional dispersive equations
This paper is devoted to the study of initial-boundary value problems for
time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers,
Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional
modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient
conditions for the blowing-up of solutions in finite time of aforementioned
equations are presented. We also discuss the maximum principle and influence of
gradient non-linearity on the global solvability of initial-boundary value
problems for the time-fractional Burgers equation. The main tool of our study
is the Pohozhaev nonlinear capacity method. We also provide some illustrative
examples.Comment: 24 page
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
Galerkin finite element solution for benjamin-bona-mahony-burgers equation with cubic b-splines
In this article, we study solitary-wave solutions of the nonlinear Benjamin–Bona–Mahony–
Burgers(BBM–Burgers) equation based on a lumped Galerkin technique using cubic Bspline finite elements for the spatial approximation. The existence and uniqueness of
solutions of the Galerkin version of the solutions have been established. An accuracy
analysis of the Galerkin finite element scheme for the spatial approximation has been well
studied. The proposed scheme is carried out for four test problems including dispersion
of single solitary wave, interaction of two, three solitary waves and development of an
undular bore. Then we propose a full discrete scheme for the resulting IVP. Von Neumann
theory is used to establish stability analysis of the full discrete numerical algorithm. To
display applicability and durableness of the new scheme, error norms L2, L∞ and three
invariants I1, I2 and I3 are computed and the acquired results are demonstrated both
numerically and graphically. The obtained results specify that our new scheme ensures
an apparent and an operative mathematical instrument for solving nonlinear evolution
equation
Solitary Wave Solutions of the Generalized Rosenau-KdV-RLW Equation
This paper investigates the solitary wave solutions of the generalized Rosenau–Korteweg-de Vries-regularized-long wave equation. This model is obtained by coupling the Rosenau–Korteweg-de Vries and Rosenau-regularized-long wave equations. The solution of the equation is approximated by a local meshless technique called radial basis function (RBF) and the finite-difference (FD) method. The association of the two techniques leads to a meshless algorithm that does not requires the linearization of the nonlinear terms. First, the partial differential equation is transformed into a system of ordinary differential equations (ODEs) using radial kernels. Then, the ODE system is solved by means of an ODE solver of higher-order. It is shown that the proposed method is stable. In order to illustrate the validity and the efficiency of the technique, five problems are tested and the results compared with those provided by other schemes.info:eu-repo/semantics/publishedVersio
Incompressible Lagrangian fluid flow with thermal coupling
In this monograph is presented a method for the solution of an incompressible viscous fluid flow with heat transfer and solidification usin a fully Lagrangian description on the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation.Postprint (published version
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