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基本説明
Provides a clear, comprehensive guide to first- and second-order ordinary and partial differential equations, whilst introducing important and useful basic material on integral equations.
Full Description
Differential and integral equations involve important mathematical techniques, and as such will be encountered by mathematicians, and physical and social scientists, in their undergraduate courses. This text provides a clear, comprehensive guide to first- and second-order ordinary and partial differential equations, whilst introducing important and useful basic material on integral equations. Readers will encounter detailed discussion of the wave, heat and Laplace equations, of Green's functions and their application to the Sturm-Liouville equation, and how to use series solutions, transform methods and phase-plane analysis. The calculus of variations will take them further into the world of applied analysis.
Providing a wealth of techniques, but yet satisfying the needs of the pure mathematician, and with numerous carefully worked examples and exercises, the text is ideal for any undergraduate with basic calculus to gain a thorough grounding in 'analysis for applications'.
Contents
Preface ; How to use this book ; Prerequisites ; 1. Integral equations and Picard's method ; 2. Existence and uniqueness ; 3. The homogeneous linear equation and Wronskians ; 4. The non-homogeneous linear equation: Variations of parameters and Green's functions ; 5. First-order partial differential equations ; 6. Second-order partial differential equations ; 7. The diffusion and wave equations and the equation of Laplace ; 8. The Fredholm alternative ; 9. Hilbert-Schmidt theory ; 10. Iterative methods and Neumann series ; 11. The calculus of variations ; 12. The Sturm-Liouville equation ; 13. Series solutions ; 14. Transform methods ; 15. Phase-plane analysis ; Appendix: The solution of some elementary ordinary differential equations ; Bibliography ; Index
