scholarly journals Comparison of KP and BBM-KP Models

2007 ◽  
Vol 2007 ◽  
pp. 1-20 ◽  
Author(s):  
Gideon P. Daspan ◽  
Michael M. Tom
Keyword(s):  
Initial Value Problem ◽  
Time Scale ◽  
Relative Difference ◽  
Initial Value ◽  
Order Of Accuracy ◽  
Kp Equations ◽  
Wave Profiles

It is shown that the solutions of the pure initial-value problem for the KP and regularized KP equations are the same, within the order of accuracy attributable to either, on the time scale0≤t≤ε−3/2, during which nonlinear and dispersive effects may accumulate to make an order-one relative difference to the wave profiles.

scholarly journals Dispersive Estimates for Full Dispersion KP Equations

2021 ◽  
Vol 23 (1) ◽  
Author(s):  
Didier Pilod ◽  
Jean-Claude Saut ◽  
Sigmund Selberg ◽  
Achenef Tesfahun
Keyword(s):  
Stationary Phase ◽  
Initial Value Problem ◽  
Bessel Functions ◽  
Linear Part ◽  
Independent Interest ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Initial Value ◽  
Kp Equations ◽  
Well Posed

AbstractWe prove several dispersive estimates for the linear part of the Full Dispersion Kadomtsev–Petviashvili introduced by David Lannes to overcome some shortcomings of the classical Kadomtsev–Petviashvili equations. The proof of these estimates combines the stationary phase method with sharp asymptotics on asymmetric Bessel functions, which may be of independent interest. As a consequence, we prove that the initial value problem associated to the Full Dispersion Kadomtsev–Petviashvili is locally well-posed in $$H^s(\mathbb R^2)$$ H s ( R 2 ) , for $$s>\frac{7}{4}$$ s > 7 4 , in the capillary-gravity setting.

scholarly journals Methods for solving the initial value problem with a two-sided estimate of the local error

2021 ◽  
pp. 88-92
Author(s):  
Yaroslav Pelekh ◽  
Andrii Kunynets ◽  
Halyna Beregova ◽  
Tatiana Magerovska
Keyword(s):  
Differential Equation ◽  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Local Error ◽  
Initial Value ◽  
Order Of Accuracy ◽  
Embedded Methods ◽  
The Right

Numerical methods for solving the initial value problem for ordinary differential equations are proposed. Embedded methods of order of accuracy 2(1), 3(2) and 4(3) are constructed. To estimate the local error, two-sided calculation formulas were used, which give estimates of the main terms of the error without additional calculations of the right-hand side of the differential equation, which favorably distinguishes them from traditional two-sided methods of the Runge- Kutta type.

Problems with different time scales

Acta Numerica ◽  
1992 ◽  
Vol 1 ◽  
pp. 101-139 ◽  
Author(s):  
Heinz-Otto Kreiss
Keyword(s):  
Time Scales ◽  
Complex Number ◽  
Initial Value Problem ◽  
Time Scale ◽  
Homogeneous Equation ◽  
Simple Problem ◽  
Initial Value ◽  
Small Constant ◽  
Image Position ◽  
Different Time Scales

In this section we discuss a very simple problem. Consider the scalar initial value problemHere ε > 0 is a small constant and a = a1 + ia2, a1, a2 real, is a complex number with |a| = 1. We can write down the solution of (1.1) explicity. It iswhereis the forced solution andis a solution of the homogeneous equationyS varies on the time scale ‘1’ while yF varies on the much faster scale 1/ε. We say that yS, yF vary on the slow and fast scale, respectively. We use also the phrase: yS and yF are the slow and the fast part of the solution, respectively.

scholarly journals Stability of a Second Order of Accuracy Difference Scheme for Hyperbolic Equation in a Hilbert Space

2007 ◽  
Vol 2007 ◽  
pp. 1-25 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Mehmet Emir Koksal
Keyword(s):  
Hilbert Space ◽  
Difference Scheme ◽  
Hyperbolic Equation ◽  
Initial Value Problem ◽  
Second Order ◽  
Stability Estimates ◽  
Initial Value ◽  
Order Of Accuracy ◽  
Positive Definite Operators ◽  
Accuracy Difference

The initial-value problem for hyperbolic equation d2u(t)/dt2+A(t)u(t)=f(t)(0≤t≤T), u(0)=ϕ,u′(0)=ψ in a Hilbert space H with the self-adjoint positive definite operators A(t) is considered. The second order of accuracy difference scheme for the approximately solving this initial-value problem is presented. The stability estimates for the solution of this difference scheme are established.

scholarly journals The Convolution on Time Scales

2007 ◽  
Vol 2007 ◽  
pp. 1-24 ◽  
Author(s):  
Martin Bohner ◽  
Gusein Sh. Guseinov
Keyword(s):  
Power Series ◽  
Transport Equation ◽  
Time Scales ◽  
Difference Equations ◽  
Initial Value Problem ◽  
Time Scale ◽  
Convolution Theorem ◽  
Main Theme ◽  
Initial Value ◽  
Dynamic Version

The main theme in this paper is an initial value problem containing a dynamic version of the transport equation. Via this problem, the delay (or shift) of a function defined on a time scale is introduced, and the delay in turn is used to introduce the convolution of two functions defined on the time scale. In this paper, we give some elementary properties of the delay and of the convolution and we also prove the convolution theorem. Our investigation contains a study of the initial value problem under consideration as well as some results about power series on time scales. As an extensive example, we consider theq-difference equations case.

scholarly journals Multistep Methods of the Hybrid Type and Their Application to Solve the Second Kind Volterra Integral Equation

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1087
Author(s):  
Vagif Ibrahimov ◽  
Mehriban Imanova
Keyword(s):  
Initial Value Problem ◽  
Multistep Methods ◽  
Sufficient Conditions ◽  
Algebraic Equations ◽  
Exact Methods ◽  
Initial Value ◽  
Order Of Accuracy ◽  
Advantages And Disadvantages ◽  
Construction Methods ◽  
Second Derivative Method

There are some classes of methods for solving integral equations of the variable boundaries. It is known that each method has its own advantages and disadvantages. By taking into account the disadvantages of known methods, here was constructed a new method free from them. For this, we have used multistep methods of advanced and hybrid types for the construction methods, with the best properties of the intersection of them. We also show some connection of the methods constructed here with the methods which are using solving of the initial-value problem for ODEs of the first order. Some of the constructed methods have been applied to solve model problems. A formula is proposed to determine the maximal values of the order of accuracy for the stable and unstable methods, constructed here. Note that to construct the new methods, here we propose to use the system of algebraic equations which allows us to construct methods with the best properties by using the minimal volume of the computational works at each step. For the construction of more exact methods, here we have proposed to use the multistep second derivative method, which has comparisons with the known methods. We have constructed some formulas to determine the maximal order of accuracy, and also determined the necessary and sufficient conditions for the convergence of the methods constructed here. One can proved by multistep methods, which are usually applied to solve the initial-value problem for ODE, demonstrating the applications of these methods to solve Volterra integro-differential equations. For the illustration of the results, we have constructed some concrete methods, and one of them has been applied to solve a model equation.

scholarly journals New properties of the time-scale fractional operators with application to dynamic equations

2021 ◽  
Vol 25 (1) ◽  
pp. 123-136
Author(s):  
Cherif Benaissa ◽  
Ladrani Zohra
Keyword(s):  
Differential Equations ◽  
Time Scales ◽  
Initial Value Problem ◽  
Time Scale ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Sufficient Conditions ◽  
Dynamic Equations ◽  
Fractional Operators ◽  
Initial Value

We introduce new properties of Riemann-Liouville fractional integral and derivative on time scales. As well as sufficient conditions for existence and uniqueness of solution to an initial value problem for a class differential equations on time scales.

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