scholarly journals The unified transform method for the Sasa–Satsuma equation on the half-line

2013 ◽  
Vol 469 (2159) ◽  
pp. 20130068 ◽  
Author(s):  
Jian Xu ◽  
Engui Fan
Keyword(s):  
Boundary Value ◽  
Initial Boundary ◽  
Half Line ◽  
Transform Method ◽  
Global Relation ◽  
Unified Transform Method ◽  
Initial Boundary Value ◽  
Dirichlet To Neumann Map

We implement the unified transform method to the initial-boundary value (IBV) problem of the Sasa–Satsuma equation on the half line. In addition to presenting the basic Riemann–Hilbert formalism, which linearizes this IBV problem, we also analyse the associated general Dirichlet to Neumann map using the so-called global relation.

ASYMPTOTICS FOR NONLINEAR NONLOCAL EQUATIONS ON A HALF-LINE

2006 ◽  
Vol 08 (02) ◽  
pp. 189-217 ◽  
Author(s):  
ROSA E. CARDIEL ◽  
ELENA I. KAIKINA ◽  
PAVEL I. NAUMKIN
Keyword(s):  
Differential Operator ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Inverse Laplace Transform ◽  
Half Line ◽  
Pseudo Differential Operator ◽  
Representation Of Solutions ◽  
Initial Boundary Value

We study the initial-boundary value problem for a general class of nonlinear pseudo-differential equations on a half-line [Formula: see text] where the number M depends on the order of the pseudo-differential operator [Formula: see text] on a half-line. The nonlinear term [Formula: see text] is such that [Formula: see text] as u, v → 0, with ρ, σ > 0. Pseudo-differential operator [Formula: see text] is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) and to find the main term of the asymptotic representation of solutions taking into account the influence of inhomogeneous boundary data and a source on the asymptotic properties of solutions.

scholarly journals Benjamin-Ono Equation on a Half-Line

2010 ◽  
Vol 2010 ◽  
pp. 1-38 ◽  
Author(s):  
Nakao Hayashi ◽  
Elena I. Kaikina
Keyword(s):  
Boundary Value Problem ◽  
Large Time ◽  
Boundary Value ◽  
Nonlinear Partial Differential Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Half Line ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Boundary Value ◽  
Time Existence

We consider the initial-boundary value problem for Benjamin-Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.

Hyperbolic inverse boundary-value problem and time-continuation of the non-stationary Dirichlet-to-Neumann map

2002 ◽  
Vol 132 (4) ◽  
pp. 931-949 ◽  
Author(s):  
Yaroslav Kurylev ◽  
Matti Lassas
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Geometric Condition ◽  
List Type ◽  
Response Operator ◽  
Gauge Transformations ◽  
Initial Boundary Value ◽  
Dirichlet To Neumann Map

Let M be a compact Riemannian manifold with non-empty boundary M. In this paper we consider an inverse problem for the second-order hyperbolic initial-boundary-value problem utt + but + a(x, D)u = 0 in M R+, u|MR+ = f, u|t=0 = ut|t=0 = 0. Our goal is to determine (M, g), b and a(x, D) from the knowledge of the non-stationary Dirichlet-to-Neumann map (the hyperbolic response operator) RT, with sufficiently large T 0. The response operator RT is the map , where is the normal derivative of the solution of the initial-boundary-value problem.More specifically, we show the following. (i)It is possible to determine Rt for any t 0 if we know RT for sufficiently large T and some geometric condition upon the geodesic behaviour on (M, g) is satisfied.(ii)It is then possible to determine (M, g) and b uniquely and the elliptic operator a(x, D) modulo generalized gauge transformations.

scholarly journals Well-posed two-point initial-boundary value problems with arbitrary boundary conditions

2011 ◽  
Vol 152 (3) ◽  
pp. 473-496 ◽  
Author(s):  
DAVID A. SMITH
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Evolution Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Transform Method ◽  
Arbitrary Boundary ◽  
Initial Boundary Value ◽  
Well Posed

AbstractWe study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the boundary conditions that specify well-posed problems using Fokas' transform method. We also give a sufficient condition guaranteeing that the solution can be represented using a series.The relevant condition, the analyticity at infinity of certain meromorphic functions within particular sectors, is significantly more concrete and easier to test than the previous criterion, based on the existence of admissible functions.

scholarly journals Fokas method for linear boundary value problems involving mixed spatial derivatives

2020 ◽  
Vol 476 (2239) ◽  
pp. 20200076
Author(s):  
A. Batal ◽  
A. S. Fokas ◽  
T. Özsarı
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Parabolic Pdes ◽  
Linear Change ◽  
Transform Method ◽  
Change Of Variables ◽  
Mixed Derivatives ◽  
Fokas Method ◽  
Initial Boundary Value

We obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well defined provided that certain analyticity issues are appropriately addressed.

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