# Spectral theory of ordinary and partial linear dierential operators on nite intervals

@inproceedings{Smith2011SpectralTO, title={Spectral theory of ordinary and partial linear dierential operators on nite intervals}, author={D. A. Smith}, year={2011} }

ii Abstract A new, unified transform method for boundary value problems on linear and integrable nonlinear partial differential equations was recently introduced by Fokas. We consider initialboundary value problems for linear, constant-coefficient evolution equations of arbitrary order on a finite domain. We use Fokas’ method to fully characterise well-posed problems. For odd order problems with non-Robin boundary conditions we identify sufficient conditions that may be checked using a simple…

## 15 Citations

### The unified transform method for linear initial-boundary value problems: a spectral interpretation

- Mathematics
- 2014

AbstractIt is known that the uni ed transform method may be used to solve any well-posed initial-boundary value problem for a linear constant-coecient evolution equation on the nite in-terval or the…

### Spectral theory of some non-selfadjoint linear differential operators

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2013

We give a characterization of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear…

### The unified method: I. Nonlinearizable problems on the half-line

- Mathematics
- 2012

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the…

### Nonlocal and Multipoint Boundary Value Problems for Linear Evolution Equations

- Mathematics
- 2015

We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution partial differential equations (PDE) with constant coefficients in one space variable.…

### Fokas Diagonalization of Piecewise Constant Coefficient Linear Differential Operators on Finite Intervals and Networks

- MathematicsActa Applicandae Mathematicae
- 2022

We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense…

### Fokas Diagonalization of Piecewise Constant Coefficient Linear Differential Operators on Finite Intervals and Networks

- MathematicsActa Applicandae Mathematicae
- 2022

We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense…

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

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2011

Abstract 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…

### A numerical implementation of the unified Fokas transform for evolution problems on a finite interval

- Mathematics, Computer ScienceEuropean Journal of Applied Mathematics
- 2017

This work evaluates the novel solution representation formula obtained by the unified transform, also known as Fokas transform, representing a linear evolution in one space dimension, and formulate a strategy to implement effectively this deformation, which allows for accurate numerical results.

### The Dirichlet-to-Neumann map for the elliptic sine-Gordon equation

- Mathematics
- 2012

We analyse the Dirichlet problem for the elliptic sine-Gordon equation in the upper half plane. We express the solution q(x, y) in terms of a Riemann–Hilbert problem whose jump matrix is uniquely…

### Synthesis, as Opposed to Separation, of Variables

- MathematicsSIAM Rev.
- 2012

It is shown here that Lax pairs provide the generalization of the divergence formulation from a separable linear to an integrable nonlinear PDE, whose crucial feature is the existence of a Lax pair formulation.

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