Initial-boundary value problems of the coupled modified Korteweg–de Vries equation on the half-line via the Fokas method

2017 ◽  
Vol 50 (39) ◽  
pp. 395204 ◽  
Author(s):  
Shou-Fu Tian
Keyword(s):  
Boundary Value Problems ◽  
Vries Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Half Line ◽  
Initial Boundary Value Problems ◽  
De Vries ◽  
Fokas Method ◽  
Initial Boundary Value

scholarly journals Blowing-up solutions of the time-fractional dispersive equations

2021 ◽  
Vol 10 (1) ◽  
pp. 952-971
Author(s):  
Ahmed Alsaedi ◽  
Bashir Ahmad ◽  
Mokhtar Kirane ◽  
Berikbol T. Torebek
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Nonlinear Capacity ◽  
Blowing Up ◽  
De Vries ◽  
Initial Boundary Value

Abstract 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.

scholarly journals Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite intervals

2009 ◽  
Vol 465 (2111) ◽  
pp. 3341-3360 ◽  
Author(s):  
Guillaume Michel Dujardin
Keyword(s):  
Schrödinger Equation ◽  
Asymptotic Behaviour ◽  
Boundary Value Problems ◽  
Boundary Data ◽  
Schrodinger Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Half Line ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems with constant coefficients on the half-line and on finite intervals. We assume that the boundary data are periodic in time and we investigate whether the solution becomes time-periodic after sufficiently long time. Using Fokas’ transformation method, we show that, for the linear Schrödinger equation, the linear heat equation and the linearized KdV equation on the half-line, the solutions indeed become periodic for large time. However, for the same linear Schrödinger equation on a finite interval, we show that the solution, in general, is not asymptotically periodic; actually, the asymptotic behaviour of the solution depends on the commensurability of the time period T of the boundary data with the square of the length of the interval over π .

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