## Fundamental Solutions for Differential Operators and ApplicationsOverview Many problems in mathematical physics and applied mathematics can be reduced to boundary value problems for differential, and in some cases, inte grodifferential equations. These equations are solved by using methods from the theory of ordinary and partial differential equations, variational calculus, operational calculus, function theory, functional analysis, probability theory, numerical analysis and computational techniques. Mathematical models of quantum physics require new areas such as generalized functions, theory of distributions, functions of several complex variables, and topological and al gebraic methods. The main purpose of this book is to provide a self contained and system atic introduction to just one aspect of analysis which deals with the theory of fundamental solutions for differential operators and their applications to boundary value problems of mathematical physics, applied mathematics, and engineering, with the related applicable and computational features. The sub ject matter of this book has its own deep rooted theoretical importance since it is related to Green's functions which are associated with most boundary value problems. The application of fundamental solutions to a recently devel oped area of boundary element methods has provided a distinct advantage in that an integral equation representation of a boundary value problem is often x PREFACE more easily solved by numerical methods than a differential equation with specified boundary and initial conditions. This situation makes the subject more attractive to those whose interest is primarily in numerical methods. |

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### Contents

1 | |

Some Basic Concepts | 11 |

Linear Elliptic Operators | 37 |

Linear Parabolic Operators | 60 |

Linear Hyperbolic Operators | 85 |

Nonlinear Operators | 117 |

Elastostatics | 138 |

Elastodynamics | 162 |

Boundary Element Methods | 231 |

Domain Integrals | 266 |

Finite Deflection of Plates | 292 |

Miscellaneous Topics | 307 |

Quasilinear Elliptic Operators | 338 |

Transforms of Distributions | 364 |

Computational Aspects | 376 |

List of Differential Operators | 387 |

### Other editions - View all

Fundamental Solutions for Differential Operators and Applications Prem Kythe No preview available - 2012 |

### Common terms and phrases

applied arbitrary biharmonic black hole body force boundary conditions boundary element method boundary integral equation boundary value problem Brebbia Cauchy problem coefficients components computed constant convolution coordinate defined denotes density derivatives differential equations differential operator Dirichlet displacement distribution divergence theorem domain integral domain Q dS(x elastic elastostatic electric elliptic Example field finite flow fluid formula Fourier transform fundamental solution given Green's function Green’s half-space harmonic heat conduction Hence initial conditions integral equation interior points isotropic known Kythe Laplace transform Laplacian linear locally integrable matrix maximum principle medium neutron node nonlinear Note obtain particles piezoelectric Poisson reduces region Q respectively satisfies the equation singular solution of Eq solve source point spherical stress surface temperature tensor test functions Theorem theory tion tractions vanishes variable vector velocity wave equation wave propagation zero