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The Initial-Boundary Value Problem

  • J. A. Leach
  • D. J. Needham
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter, we extend the analysis of Chapter 7 and consider the evolution of the scalar initial-boundary value problem (6.16)–(6.20), namely,
$$ u_t = u_{xx} + f(u), x,t > 0, $$
(1)
$$ f(u) = \left\{ {\begin{array}{*{20}c} {(1 - u)u^m - ku^n ,u > 0,} \\ {0, u \leqslant 0,} \\ \end{array} } \right. $$
(2)
$$ u(x,0) = \left\{ {\begin{array}{*{20}c} {u_0 g(x), 0 < x \leqslant \sigma ,} \\ {0, x > \sigma , } \\ \end{array} } \right. $$
(3)
$$ u(x,t) \to 0 as x \to \infty , t \geqslant 0, $$
(1)
$$ u_x (0,t) = 0, t > 0. $$
(5)
where g(x) is a prescribed, positive function for x < a with a maximum value of unity, and which is analytic on 0 ≤ x ≤ σ.

Keywords

Asymptotic Expansion Asymptotic Solution Steady State Solution Stable Manifold Comparison Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London 2004

Authors and Affiliations

  • J. A. Leach
    • 1
  • D. J. Needham
    • 1
  1. 1.Department of MathematicsThe University of ReadingReadingUK

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