scholarly journals Initial-boundary-value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations

2008 ◽  
Vol 24 (6) ◽  
pp. 065011 ◽  
Author(s):  
Gino Biondini ◽  
Guenbo Hwang
Keyword(s):  
Boundary Value Problems ◽  
Evolution Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Schrödinger Equations ◽  
Initial Boundary Value Problems ◽  
Nonlinear Schrodinger Equations ◽  
Discrete Nonlinear Schrödinger Equations ◽  
Discrete Evolution Equations ◽  
Initial Boundary Value

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.

Initial-boundary value problems for linear PDEs with variable coefficients

2007 ◽  
Vol 143 (1) ◽  
pp. 221-242 ◽  
Author(s):  
P. A. TREHARNE ◽  
A. S. FOKAS
Keyword(s):  
Boundary Value Problems ◽  
Evolution Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Initial Boundary Value Problems ◽  
Linear Evolution ◽  
Higher Derivatives ◽  
Initial Boundary Value

AbstractA new approach for studying initial-boundary value problems for linear partial differential equations (PDEs) with variable coefficients was introduced recently by the second author, and was applied to PDEs involving second order derivatives. Here, we extend this approach further to solve an initial-boundary value problem for a third-order evolution PDE with a space-dependent coefficient. The analysis is presented in such a way that it can be applied to PDEs with higher derivatives, and thus provides a method for solving initial-boundary value problems for a certain class of linear evolution equations with variable coefficients of arbitrary order.

On nonclassical problems for first-order evolution equations

2011 ◽  
Vol 18 (3) ◽  
pp. 441-463
Author(s):  
Gia Avalishvili ◽  
Mariam Avalishvili
Keyword(s):  
Boundary Value Problems ◽  
Parabolic Equations ◽  
Evolution Equations ◽  
Initial Conditions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
First Order ◽  
Uniqueness Results ◽  
Initial Boundary Value

Abstract The present paper deals with nonclassical initial-boundary value problems for parabolic equations and systems and their generalizations in abstract spaces. Nonclassical problems with nonlocal initial conditions for an abstract first-order evolution equation with time-dependent operator are considered, the existence and uniqueness results are proved and the algorithm of approximation of nonlocal problems by a sequence of classical problems is constructed. Applications of the obtained general results to initial-boundary value problems for parabolic equations and systems are considered.

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