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

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

Large-time Behavior of the Solutions to Rosenau Type Approximations to the Heat Equation

In this paper we study the large-time behavior of the solution to a general Rosenau type approximation to the heat equation, by showing that the solution to this approximation approaches the fundamental solution of the heat equation at a sub-optimal rate. The result is valid in particular for the central differences scheme approximation of the heat equation, a property which to our knowledge has never been observed before.Comment: 20 page

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

Numerical Methods and Analysis for Several Kinds of Partial Differential Equations

\\本篇论文我们探讨几类偏微分方程的数值方法，针对这些方程，我们提出一些数值格式，严格给出该格式的稳定性结果和收敛性误差估计，并给出一些例子验证我们的结论。 我们主要讨论的是分数阶型微分方程。近几年，此类方程在数学建模中的应用越来越广泛，由不同分数阶微分方程导出的模型被很多领 域被提出，如材料、机械和生物系统，并发现针对一些具有记忆，不均匀或遗传性质的材料，分数阶微分方程相对于整数阶更有优势。随 着分数阶微分方程在建模领域的发展，其数值格式的构造也引起了越来越多人的兴趣。 \\论文主要内容如下： 第一章，介绍分数阶微积分的历史和背景，并回顾一些现有的工作，基本定义和预备知识，这些...In this dissertation, we investigate numerical methods for several partial differential equations. A number of numerical schemes are proposed and analyzed for numerical solutions to these equations. Some stability and error estimates are vigorously derived and numerically tested. Our work is focused on fractional differential equations. The use of fractional differential equations in mathema...学位：理学博士院系专业：数学科学学院_计算数学学号：1902013015444