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In this short communication, we announce an algorithmic procedure for constructing non-uniqueness counter-examples of classical solutions to initial-boundary-value problems for a wide class of linear evolution partial differential…
The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the…
In this note, we announce a systematic analysis of continuous dependence on the data in classical spaces for the initial-boundary-value problem of the diffusion equation on the half-line, with data that are not necessarily compatible at the…
We present the numerical solution of two-point boundary value problems for a third order linear PDE, representing a linear evolution in one space dimension. The difficulty of this problem is in the numerical imposition of the boundary…
We 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…
We examine the analytic extension of solutions of linear, constant-coefficient initial-boundary value problems outside their spatial domain of definition. We use the Unified Transform Method or Method of Fokas, which gives a representation…
We present a method to solve initial-boundary value problems for linear and integrable nonlinear differential-difference evolution equations. The method is the discrete version of the one developed by A. S. Fokas to solve initial-boundary…
We study initial boundary value problems for linear scalar partial differential equations with constant coefficients, with spatial derivatives of {\em arbitrary order}, posed on the domain $\{t>0, 0<x<L\}$. We first show that by analysing…
A new method for the solution of initial-boundary value problems for \textit{linear} and \textit{integrable nonlinear} evolution PDEs in one spatial dimension was introduced by one of the authors in 1997 \cite{F1997}. This approach was…
We study a semidiscrete analogue of the Unified Transform Method introduced by A. S. Fokas, to solve initial-boundary-value problems for linear evolution partial differential equations with constant coefficients on the finite interval $x…
We represent an integration algorithm combining the characteristics method and Hopf-Cole transformation. This algorithm allows one to partially integrate a large class of multidimensional systems of nonlinear Partial Differential Equations…
We implement the Unified Transform Method of Fokas as a numerical method to solve linear partial differential equations on the half-line. The method computes the solution at any x and t without spatial discretization or time stepping. With…
We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schr\"odinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for…
We analytically derive novel explicit integral representations for the solution of nonhomogeneous initial-boundary-value problems for a large category of evolution partial differential equations of Sobolev-Galpern type with generic…
We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution PDEs with constant coefficients in one space variable. The prototypical such PDE is the heat equation, for which problems of…
The initial-boundary value problem (ibvp) for the $m$-th order dispersion Korteweg-de Vries (KdV) equation on the half-line with rough data and solution in restricted Bourgain spaces is studied using the Fokas Unified Transform Method…
The so-called unified method expresses the solution of an initial-boundary value problem (IBVP) for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple IBVP, which will be referred…
We present an approach for analyzing initial-boundary value problems for integrable equations whose Lax pairs involve $3 \times 3$ matrices. Whereas initial value problems for integrable equations can be analyzed by means of the classical…
We discuss a semi-discrete analogue of the Unified Transform Method, introduced by A. S. Fokas, to solve initial-boundary-value problems for linear evolution partial differential equations of constant coefficients. The semi-discrete method…
In this paper, we solve explicitly and analyze rigorously inhomogeneous initial-boundary-value problems (IBVP) for several fourth-order variations of the traditional diffusion equation and the associated linearized Cahn-Hilliard (C-H) model…