The foam drainage equation with time- and space-fractional derivatives solved by the Adomian method

In this paper, by introducing the fractional derivative in the sense of Caputo, we apply the Adomian decomposition method for the foam drainage equation with time- and space-fractional derivative. As a result, numerical solutions are obtained in a form of rapidly convergent series with easily computable components

The well-posedness and solutions of Boussinesq-type equations

We develop well-posedness theory and analytical and numerical solution techniques for Boussinesq-type equations. Firstly, we consider the Cauchy problem for a generalized Boussinesq equation. We show that under suitable conditions, a global solution for this problem exists. In addition, we derive sufficient conditions for solution blow-up in finite time.Secondly, a generalized Jacobi/exponential expansion method for finding exact solutions of non-linear partial differential equations is discussed. We use the proposed expansion method to construct many new, previously undiscovered exact solutions for the Boussinesq and modified Korteweg-de Vries equations. We also apply it to the shallow water long wave approximate equations. New solutions are deduced for this system of partial differential equations.Finally, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem, which can be solved using many accurate numerical methods. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior

Nonlinear wave patterns in the complex KdV and nonlinear Schrodinger equations

This thesis is on the theory of nonlinear waves in physics. To begin with, we develop from first principles the theory of the complex Korteweg-de Vries (KdV) equation as an equation for the complex velocity of a weakly nonlinear wave in a shallow, ideal fluid. We show that this is completely consistent with the well-known theory of the real KdV equation as a special case, but has the advantage of directly giving complete information about the motion of all particles within the fluid. We show that the complex KdV equation also has conserved quantities which are completely consistent with the physical interpretation of the real KdV equation. When a periodic wave solution to the real KdV equation is expanded in the quasi-monochromatic approximation, it is known that the amplitude of the wave envelope is described by the nonlinear Schrodinger (NLS) equation. However, in the complex KdV equation, we show that the fundamental modes of the velocity are described by the split NLS equations, themselves a special case of the Ablowitz-Kaup-Newell-Segur system. This is a directly physical interpretation of the split NLS equations, which were primarily introduced as only a mathematical construct emerging from the Zakharov-Shabat equations. We also discuss an empirically obtained symmetry of the rational solutions to the KdV equations, which seems to have been unnoticed until now. Solutions which can be written in terms of Wronskian determinants are well-known; however, we show that these are actually part of a more general family of rational solutions. We show that a linear combination of the Wronskians of orders $n$ and $n+2$ generates a new, multi-peak rational solution to the KdV equation. We next move on to the integrable extensions of the NLS equation. These incorporate higher order nonlinear and dispersive terms in such a way that the system keeps the same conserved quantities, and is thus completely integrable. We obtain the general solution of the doubly-periodic solutions of the class I extension of the NLS equation, and discuss several special cases. These are the most general one-parameter first order solutions of the (class I) extended NLS equation. Building on this, we also discuss second order solutions to the extended NLS equation. We obtain the general 2-breather solutions, and discuss several special cases; among them, semirational breathers, the degenerate breather solution, the second-order rogue wave, and the rogue wave triplet solution. We also discuss the breather to soliton conversion, which is a solution which does not exist in the basic NLS equation where only the lowest order dispersive and nonlinear terms are present. Finally, we discuss a few possibilities for future research based on the work done in this thesis

1980 summer study program in geophysical fluid dynamics : coherent features in geophysical flows

Four principal lecturers shored the task of presenting the subject "Coherent Features in Geophysical Flows" to the participants of the twenty-second geophysical fluid dynamics summer program. Glenn Flierl introduced the topic and the Kortweg-de Vries equation via a model of finite amplitude motions on the beta plane. He extended the analysis to more complex flows in the ocean and the atmosphere and in the process treated motions of very large amplitude. Larry Redekopp's three lectures summarized an extensive body of the mathematical literature on coherent features. Andrew Ingersoll focussed on the many fascinating features in Jupiter's atmosphere. Joseph Keller supplemented an interesting summary of laboratory observations with suggestive models for treating the flows.Office of Naval Research under Contract N00014-79-C-067

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described