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Method for solving moving boundary value problems for linear evolution equations

Fokas1, Pelloni

  • 1Department of Mathematics, Imperial College, London SW7 2BZ, United Kingdom.

Physical Review Letters
|September 16, 2000
PubMed
Summary

We present a novel method for solving linear evolution equations in time-varying domains. This approach yields integral representations for solutions, applicable to complex mathematical problems.

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Area of Science:

  • Mathematical Physics
  • Partial Differential Equations
  • Complex Analysis

Background:

  • Initial boundary value problems (IBVPs) are fundamental in modeling physical phenomena.
  • Solving IBVPs in time-dependent domains presents significant analytical challenges.
  • Linear evolution equations with time-varying domains require specialized solution techniques.

Purpose of the Study:

  • To develop a new method for solving linear evolution equations in time-dependent domains.
  • To provide an integral representation for the solutions of such problems.
  • To demonstrate the applicability of the method to specific equations and explore its generalization.

Main Methods:

  • The study introduces a novel analytical method for IBVPs.
  • It applies the method to a linear evolution equation with a specific dispersion relation in a semi-infinite time-dependent domain.

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  • The core technique involves deriving integral representations in the complex k-plane.

    • Main Results:

    • The solution is shown to admit an integral representation in the complex k-plane.
    • This representation involves integrals of specific functions (rho(k) or rho(k, kmacr;)) along time-dependent contours or over fixed 2D domains.
    • The functions rho(k) and rho(k, kmacr;) are computable via Volterra linear integral equations.

    Conclusions:

    • The developed method provides a powerful tool for analyzing linear evolution equations in complex domains.
    • The integral representation offers new insights into the structure and behavior of solutions.
    • The technique is generalizable to nonlinear integrable partial differential equations, broadening its impact.