## The One-Dimensional Heat EquationThis is a version of Gevrey's classical treatise on the heat equations. Included in this volume are discussions of initial and/or boundary value problems, numerical methods, free boundary problems and parameter determination problems. The material is presented as a monograph and/or information source book. After the first six chapters of standard classical material, each chapter is written as a self-contained unit except for an occasional reference to elementary definitions, theorems and lemmas in previous chapters. |

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### Contents

Preliminaries | 1 |

Analy ticiry of Solutions | 10 |

Introduction | 13 |

The Cauchy Problem | 25 |

The InitialValue Problem | 32 |

The InitialBoundaryValue Problem for the Quarter | 45 |

The InitialBoundaryValue Problem for the Quarter | 54 |

The Reduction of Some InitialBoundaryValue | 67 |

Determination of an Unknown TimeDependent | 187 |

Initial andor BoundaryValue Problems | 213 |

Some Properties of Solutions in General Domains | 252 |

The Solution in a General Region with | 265 |

The OnePhase Stefan Problem with | 281 |

The OnePhase Stefan Problem with | 313 |

The Inhomogeneous Heat Equation u wtr + | 327 |

Some References to the Literature on JFu uII u | 357 |

Integral Equations | 82 |

Solutions of BoundaryValue Problems | 101 |

Continuous Dependence upon the Data | 128 |

Some Numerical Methods for Some StateEstimation | 152 |

19501973 | 368 |

19731981 | 408 |

Symbol Index | 478 |

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### Common terms and phrases

Acad Akad Anal analysis Appl application approximation argument assume Assumption behavior boundary boundary conditions boundary-value problem bounded Cannon Cauchy Chapter compact condition Consequently consider convergence Corollary defined denote dependence derivatives determining diffusion Dokl domain English summary estimate example Exercise exists extended following result follows free boundary function given heat conduction heat equation Hence implies initial integral equation Lemma limit linear Lipschitz-continuous Math Mathematical Maximum Principle Mech method Moreover Nauk nonlinear Note numerical obtain parabolic equations Partial Differential Equations periodic piecewise-continuous positive constant possesses preceding Proc Proof Property reader Remark representation respect result Russian satisfies sequence solution SSSR statement Stefan problem sufficiently temperature term Theorem theory tion transformation uniform uniformly unique unique solution Univ variable

### Popular passages

Page 451 - Target patterns and horseshoes from a perturbed central-force problem: some temporally periodic solutions to reaction-diffusion equations, Stud. Appl. Math. 64 (1981), No.

Page 395 - Free boundary problems for the heat equation in which the moving interface coincides initially with the fixed face.