Nonlocal and Multipoint Boundary Value Problems for Linear Evolution Equations

@article{Pelloni2015NonlocalAM,
  title={Nonlocal and Multipoint Boundary Value Problems for Linear Evolution Equations},
  author={Beatrice Pelloni and David A. Smith},
  journal={Studies in Applied Mathematics},
  year={2015},
  volume={141}
}
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. The prototypical example of such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third‐order case, which is much less studied and has been shown by the authors to have very… 

Linear evolution equations on the half-line with dynamic boundary conditions

The classical half line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition

The diffusion equation with nonlocal data

The unified transform for evolution equations on the half‐line with time‐periodic boundary conditions *

This paper elaborates on a new approach for solving the generalized Dirichlet‐to‐Neumann map, in the large time limit, for linear evolution PDEs formulated on the half‐line with time‐periodic

Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay

  • G. Hwang
  • Mathematics
    Zeitschrift für Naturforschung A
  • 2020
Abstract Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay (LAD) are studied by utilizing a unified method, known as the Fokas method. The method

Fokas diagonalization

A method for solving linear initial boundary value problems was recently reimplemented as a true spectral transform method. As part of this reformulation, the precise sense in which the spectral

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

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

Kernel density estimation with linked boundary conditions

Kernel density estimation on a finite interval poses an outstanding challenge because of the well‐recognized bias at the boundaries of the interval. Motivated by an application in cancer research, we

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

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

References

SHOWING 1-10 OF 37 REFERENCES

Two-point boundary value problems for linear evolution equations

  • A. FokasB. Pelloni
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2001
We study boundary value problems for a linear evolution equation with spatial derivatives of arbitrary order, on the domain 0 < x < L, 0 < t < T, with L and T positive finite constants. We present a

The Reformulation and Numerical Solution of Certain Nonclassical Initial-Boundary Value Problems

Several physical phenomena are modeled by nonclassical parabolic or hyperbolic initial-boundary value problems in one space variable which involve an integral over the spatial domain of a function of

On the numerical solution of the diffusion equation with a nonlocal boundary condition.

Parabolic partial differential equations with nonlocal boundary specifications feature in the mathematical modeling of many phenomena. In this paper, numerical schemes are developed for obtaining

A hybrid analytical–numerical method for solving evolution partial differential equations. I. The half-line

  • N. FlyerA. S. Fokas
  • Mathematics
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2008
A new method, combining complex analysis with numerics, is introduced for solving a large class of linear partial differential equations (PDEs). This includes any linear constant coefficient PDE, as

The spectral representation of two-point boundary-value problems for third-order linear evolution partial differential equations

  • B. Pelloni
  • Mathematics
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2005
We use a spectral transform method to study general boundary-value problems for third-order, linear, evolution partial differential equations with constant coefficients, posed on a finite space

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

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

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

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.

Evolution PDEs and augmented eigenfunctions. Finite interval

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)

A Unified Approach To Boundary Value Problems

This book presents a new approach to analyzing initial-boundary value problems for integrable partial differential equations (PDEs) in two dimensions, a method that the author first introduced in

Well-posed boundary value problems for linear evolution equations on a finite interval

  • B. Pelloni
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2004
We identify the class of smooth boundary conditions that yield an initial-boundary value problem admitting a unique smooth solution for the case of a dispersive linear evolution PDE of arbitrary