## Applied Partial Differential EquationsPartial differential equations are a central concept in mathematics. They arise in mathematical models whose dependent variables vary continuously as functions of several independent variables (usually space and time). Their power lies in their universality: there is a huge and ever- growingrange of real-world problems to which they can be applied, from fluid mechanics and electromagnetism to probability and finance. This is an enthusiastic and clear guide to the theory and applications of PDEs. It deals with questions such as the well-posedness of a PDE problem: when is there aunique solution that changes only slightly when the input data is slightly changed? This is connected to the problem of establishing the accuracy of a numerical solution to a PDE, a problem that become increasingly important as the power of computer software to produce numerical solutionsgrows. |

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### Contents

Introduction | 1 |

Firstorder quasilinear systems | 35 |

Introduction to secondorder scalar equations | 75 |

Copyright | |

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### Common terms and phrases

aerofoil analytic arbitrary behaviour boundary data boundary value problem Cauchy data Cauchy problem Chapter characteristic projections Charpit's coefficients consider constant curve Deduce defined dependent derivatives Dirichlet discontinuity discussion domain du/dt du/dx du/dy dx dy eigenfunctions eigenvalue elliptic equations example Exercise existence first-order flow fluid formula Fourier transform free boundary conditions free boundary problems generalisation geometric gives Green's function Green's theorem heat equation Helmholtz Hence hyperbolic equations independent variables initial integral inversion Laplace's equation linear mathematical maximum principle nonlinear ordinary differential equations parabolic equations partial differential equations plane Poisson's equation prescribed propagation quasilinear Rankine-Hugoniot ray cone region remark result Riemann function Riemann invariants satisfies scalar second-order semilinear shock Show simply singularities smooth solve Stefan problem Suppose surface theorem theory two-dimensional uniqueness vanishes vector velocity wave equation weak solution write zero