Central Library, Indian Institute of Technology Delhi
केंद्रीय पुस्तकालय, भारतीय प्रौद्योगिकी संस्थान दिल्ली

Select ideas in partial differential equations / Peter J. Costa.

By: Costa, Peter J [author.]Material type: TextTextSeries: Synthesis lectures on mathematics and statistics ; #40. | Synthesis digital library of engineering and computer sciencePublisher: San Rafael, California (1537 Fourth Street, 1537 Fourth Street, San Rafael, CA 94901 USA) : Morgan & Claypool Publishers, [2021]Description: 1 PDF (xx, 214 pages) : illustrations (some color)Content type: text Media type: electronic Carrier type: online resourceISBN: 9781636390963Subject(s): Differential equations, Partial | Mathematical physics | Partial differential equations | MATLAB | Linear partial differential equations | Maxwell's equations | Nonlinear partial differential equations | physics | engineeringGenre/Form: Electronic books.Additional physical formats: Print version:: No titleDDC classification: 515/.353 LOC classification: QA377 | .C677 2021ebOnline resources: Abstract with links to resource | Abstract with links to full text Also available in print.
Contents:
1. The equations of Maxwell -- 1.1. grad, div, and curl -- 1.2. Faraday's law -- 1.3. Coulomb's law -- 1.4. Ampere's law
2. Laplace's equation -- 2.1. Laplace's equation and the classical approach (part 1) -- 2.2. Green's theorem and the modern approach -- 2.3. The classical approach (part 2) -- 2.4. Some examples
3. Fourier series, Bessel functions, and mathematical physics -- 3.1. Fourier series -- 3.2. Fourier series in two dimensions -- 3.3. A return to Laplace's equation (Part 2) -- 3.4. Poisson's equation (modern methods revisited) -- 3.5. The heat equation (a blend of modern and classical)
4. Fourier transform, heat conduction, and the wave equation -- 4.1. Fourier transform -- 4.2. Examples of the Fourier transform -- 4.3. The Fourier transform and the heat equation -- 4.4. The Fourier transform and the wave equation -- 4.5. D'Alembert's solution of the wave equation -- 4.6. Examples of the wave equation : finite domains
5. The three-dimensional wave equation -- 5.1. Fourier transform solution -- 5.2. The three-dimensional wave kernels -- 5.3. Huygen's principle and Duhamel's principle -- 5.4. The method of descent
6 an introduction to nonlinear partial differential equations -- 6.1. Nonlinear Klein-Gordon equation -- 6.2. Change of variables -- 6.3. Separation of variables -- 6.4. Burgers' equation and a nonlinear transformation -- 6.5. Hopf-Cole transformation -- 6.6. The Korteweg de Vries equation and inverse scattering
7. Raman scattering and numerical methods -- 7.1. The stimulated Raman scattering laser model -- 7.2. A quasi-implicit finite difference scheme -- 7.3. Stability, consistency, and convergence -- 7.4. Some results -- 7.5. Alternating direction implicit methods
8. The Hartman-Grobman theorem -- 8.1. Introduction -- 8.2. Definitions, notation, and key conditions -- 8.3. The mapping theorem -- 8.4. The flow theorem -- 8.5. Applications.
Summary: This text provides an introduction to the applications and implementations of partial differential equations. The content is structured in three progressive levels which are suited for upper-level undergraduates with background in multivariable calculus and elementary linear algebra (chapters 1-5), first- and second-year graduate students who have taken advanced calculus and real analysis (chapters 6-7), as well as doctoral-level students with an understanding of linear and nonlinear functional analysis (chapters 7-8) respectively. Level one gives readers a full exposure to the fundamental linear partial differential equations of physics. It details methods to understand and solve these equations leading ultimately to solutions of Maxwell's equations. Level two addresses nonlinearity and provides examples of separation of variables, linearizing change of variables, and the inverse scattering transform for select nonlinear partial differential equations. Level three presents rich sources of advanced techniques and strategies for the study of nonlinear partial differential equations, including unique and previously unpublished results. Ultimately the text aims to familiarize readers in applied mathematics, physics, and engineering with some of the myriad techniques that have been developed to model and solve linear and nonlinear partial differential equations.
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Mode of access: World Wide Web.

System requirements: Adobe Acrobat Reader.

Part of: Synthesis digital library of engineering and computer science.

Includes bibliographical references (pages 205-212) and index.

1. The equations of Maxwell -- 1.1. grad, div, and curl -- 1.2. Faraday's law -- 1.3. Coulomb's law -- 1.4. Ampere's law

2. Laplace's equation -- 2.1. Laplace's equation and the classical approach (part 1) -- 2.2. Green's theorem and the modern approach -- 2.3. The classical approach (part 2) -- 2.4. Some examples

3. Fourier series, Bessel functions, and mathematical physics -- 3.1. Fourier series -- 3.2. Fourier series in two dimensions -- 3.3. A return to Laplace's equation (Part 2) -- 3.4. Poisson's equation (modern methods revisited) -- 3.5. The heat equation (a blend of modern and classical)

4. Fourier transform, heat conduction, and the wave equation -- 4.1. Fourier transform -- 4.2. Examples of the Fourier transform -- 4.3. The Fourier transform and the heat equation -- 4.4. The Fourier transform and the wave equation -- 4.5. D'Alembert's solution of the wave equation -- 4.6. Examples of the wave equation : finite domains

5. The three-dimensional wave equation -- 5.1. Fourier transform solution -- 5.2. The three-dimensional wave kernels -- 5.3. Huygen's principle and Duhamel's principle -- 5.4. The method of descent

6 an introduction to nonlinear partial differential equations -- 6.1. Nonlinear Klein-Gordon equation -- 6.2. Change of variables -- 6.3. Separation of variables -- 6.4. Burgers' equation and a nonlinear transformation -- 6.5. Hopf-Cole transformation -- 6.6. The Korteweg de Vries equation and inverse scattering

7. Raman scattering and numerical methods -- 7.1. The stimulated Raman scattering laser model -- 7.2. A quasi-implicit finite difference scheme -- 7.3. Stability, consistency, and convergence -- 7.4. Some results -- 7.5. Alternating direction implicit methods

8. The Hartman-Grobman theorem -- 8.1. Introduction -- 8.2. Definitions, notation, and key conditions -- 8.3. The mapping theorem -- 8.4. The flow theorem -- 8.5. Applications.

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This text provides an introduction to the applications and implementations of partial differential equations. The content is structured in three progressive levels which are suited for upper-level undergraduates with background in multivariable calculus and elementary linear algebra (chapters 1-5), first- and second-year graduate students who have taken advanced calculus and real analysis (chapters 6-7), as well as doctoral-level students with an understanding of linear and nonlinear functional analysis (chapters 7-8) respectively. Level one gives readers a full exposure to the fundamental linear partial differential equations of physics. It details methods to understand and solve these equations leading ultimately to solutions of Maxwell's equations. Level two addresses nonlinearity and provides examples of separation of variables, linearizing change of variables, and the inverse scattering transform for select nonlinear partial differential equations. Level three presents rich sources of advanced techniques and strategies for the study of nonlinear partial differential equations, including unique and previously unpublished results. Ultimately the text aims to familiarize readers in applied mathematics, physics, and engineering with some of the myriad techniques that have been developed to model and solve linear and nonlinear partial differential equations.

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