## Partial Differential EquationsThis book provides a basic introductory course in partial differential equations, in which theory and applications are interrelated and developed side by side. Emphasis is on proofs, which are not only mathematically rigorous, but also constructive, where the structure and properties of the solution are investigated in detail. The authors feel that it is no longer necessary to follow the tradition of introducing the subject by deriving various partial differential equations of continuum mechanics and theoretical physics. Therefore, the subject has been introduced by mathematical analysis of the simplest, yet one of the most useful (from the point of view of applications), class of partial differential equations, namely the equations of first order, for which existence, uniqueness and stability of the solution of the relevant problem (Cauchy problem) is easy to discuss. Throughout the book, attempt has been made to introduce the important ideas from relatively simple cases, some times by referring to physical processes, and then extending them to more general systems. |

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### Contents

Single First Order Partial Differential Equations 152 | 1 |

First Order Nonlinear Equations in Two Independent | 17 |

Complete Integral | 30 |

5 First Order Equations in More Than Two Independent | 36 |

6 Applications of the Theory of a Single First Order Equation | 46 |

Linear Second Order Partial Differential Equations 53149 | 53 |

Potential Theory and Elliptic Differential Equations | 67 |

The Diffusion Equation and Parabolic Differential | 91 |

Hyperbolic Partial Differential Equations 150248 | 150 |

Hyperbolic System of Two First Order Quasilinear | 176 |

4 General Theory of a Simple Wave | 191 |

EQUATIONS IN MORE THAN Two INDEPENDENT VARIABLES | 216 |

7 The Wave Equation | 222 |

on a Manifold Which is not Spacelike | 229 |

8 Hyperbolic System of First Order Equations | 235 |

249 | |

### Common terms and phrases

assume boundary value problem Cauchy data Cauchy problem characteristic conoid characteristic curves characteristic equation characteristic manifold characteristic surface characteristic velocity Charpit's equations coefficients compatibility condition conservation law Consider constant continuous datum curve defined denote determined dimensional Dirichlet problem discontinuity equa exists family of characteristic finite genuine solution given Green's function hence hyperbolic equations hyperbolic system independent variables initial conditions initial value problem inner derivative integral surface intersection Laplace's equation linear matrix mean value property Monge cone Monge curves Monge strips nonlinear normal conoid normal form obtained order equation order partial differential ordinary differential equations parameter family partial derivatives partial differential equation plane prescribed Proof quasilinear equation region represents Riemann invariants satisfies the equation second order semilinear shock simple wave solve space space-like sphere straight line tangent theorem tion transformation unique vector wave equation weak solution zero