## Partial Differential EquationsIn this book, Professor Copson gives a rigorous account of the theory of partial differential equations of the first order and of linear partial differential equations of the second order, using the methods of classical analysis. In spite of the advent of computers and the applications of the methods of functional analysis to the theory of partial differential equations, the classical theory retains its relevance in several important respects. Many branches of classical analysing have their origins in the rigourous discussion of problems in applies mathematics and theoretical physics, and the classical treatment of the theory of partial differential equations still provides the best method of treating many physical problems. A knowledge of the classical theory is essential for pure mathematics who intend to undertake research in this field, whatever approach they ultimately adopt. The numerical analyst needs a knowledge of classical theory in order to decide whether a problem has a unique solution or not. |

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### Contents

Partial differential equations of the first order | 1 |

Characteristics of equations of the second order | 24 |

Boundary value and initial value problems | 44 |

Equations of hyperbolic type | 54 |

Riemanns method | 77 |

The equation of wave motions | 90 |

Marcel Rieszs method | 107 |

Potential theory in the plane | 131 |

Subharmonic functions and the problem of Dirichlet | 175 |

Equations of elliptic type in the plane | 186 |

Equations of elliptic type in space | 207 |

The equation of heat | 238 |

Appendix | 271 |

Books for further reading | 277 |

279 | |

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### Common terms and phrases

analytic function assume belongs boundary values bounded bounded domain called Cauchy characteristic choose coefficients consider constant contained continuous second continuously differentiable convergent defined denote depend determined differentiable differential equation direction disc equal equation everywhere example exists expression finite follows formula function given gives Green's function harmonic Hence identically zero independent integral Laplace's equation lies linear mean value theorem method normal obtain origin partial particular plane polar coordinates positive potential problem of Dirichlet prove reduced regular regular closed curve respect result satisfies second derivatives sequence Similarly singularity solution solve strip Suppose tends to zero term theorem theory transformation u(xo uniformly unique solution value problem vanishes variables zero ди ду дх