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Essential Partial Differential Equations 1st Edition by David F Griffiths, David J Silvester, John W Dold ISBN B015SC0IX8 9783319225692

Name: Essential Partial Differential Equations 1st Edition by David F Griffiths, David J Silvester, John W Dold ISBN B015SC0IX8 9783319225692
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Essential Partial Differential Equations Analytical and Computational Aspects Instructor Solution Manual Solutions 1st Edition David F F Griffiths David J Silvester John W Dold Digital Instant Download

Author(s): David F F Griffiths David J Silvester John W Dold

Edition: 1

File Details: PDF, 1.32 MB

Year: 2015

Language: english

SKU: EB-49479232 Category: Uncategorized Tag: David F F Griffiths David J Silvester John W Dold

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Essential Partial Differential Equations 1st Edition by David F Griffiths, David J Silvester, John W Dold – Ebook PDF Instant Download/Delivery: B015SC0IX8 ,9783319225692
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ISBN 10: B015SC0IX8
ISBN 13: 9783319225692
Author: David F Griffiths, David J Silvester, John W Dold

This volume provides an introduction to the analytical and numerical aspects of partial differential equations (PDEs). It unifies an analytical and computational approach for these; the qualitative behaviour of solutions being established using classical concepts: maximum principles and energy methods. Notable inclusions are the treatment of irregularly shaped boundaries, polar coordinates and the use of flux-limiters when approximating hyperbolic conservation laws. The numerical analysis of difference schemes is rigorously developed using discrete maximum principles and discrete Fourier analysis. A novel feature is the inclusion of a chapter containing projects, intended for either individual or group study, that cover a range of topics such as parabolic smoothing, travelling waves, isospectral matrices, and the approximation of multidimensional advection–diffusion problems. The underlying theory is illustrated by numerous examples and there are around 300 exercises, designed to promote and test understanding. They are starred according to level of difficulty. Solutions to odd-numbered exercises are available to all readers while even-numbered solutions are available to authorised instructors. Written in an informal yet rigorous style, Essential Partial Differential Equations is designed for mathematics undergraduates in their final or penultimate year of university study, but will be equally useful for students following other scientific and engineering disciplines in which PDEs are of practical importance. The only prerequisite is a familiarity with the basic concepts of calculus and linear algebra.

Essential Partial Differential Equations 1st Edition Table of contents:

1 Setting the Scene

1.1 Some Classical PDEs

1.2 and Some Classical Solutions

2 Boundary and Initial Data

2.1 Operator Notation

2.2 Classification of Boundary Value Problems

2.2.1 Linear Problems

2.2.2 Nonlinear Problems

2.2.3 Well-Posed Problems

3 The Origin of PDEs

3.1 Newton’s Laws

3.1.1 The Wave Equation for a String

3.2 Conservation Laws

3.2.1 The Heat Equation

3.2.2 Laplace’s Equation and the Poisson Equation

3.2.3 The Wave Equation in Water

3.2.4 Burgers’ Equation

4 Classification of PDEs

4.1 Characteristics of First-Order PDEs

4.2 Characteristics of Second-Order PDES

4.2.1 Hyperbolic Equations

4.2.2 Parabolic Equations

4.2.3 Elliptic Equations

4.3 Characteristics of Higher-Order PDEs

4.4 Postscript

5 Boundary Value Problems in mathbbR1

5.1 Qualitative Behaviour of Solutions

5.2 Comparison Principles and Well-Posedness

5.3 Inner Products and Orthogonality

5.3.1 Self-adjoint Operators

5.3.2 A Clever Change of Variables

5.4 Some Important Eigenvalue Problems

6 Finite Difference Methods in mathbbR1

6.1 The Approximation of Derivatives

6.2 Approximation of Boundary Value Problems

6.3 Convergence Theory

6.4 Advanced Topics and Extensions

6.4.1 Boundary Conditions with Derivatives

6.4.2 A Fourth-Order Finite Difference Method

7 Maximum Principles and Energy Methods

7.1 Maximum Principles

7.2 Energy Methods

8 Separation of Variables

8.1 The Heat Equation Revisited

8.1.1 Extension to Other Parabolic Problems …

8.1.2 … and to Inhomogeneous Data

8.2 The Wave Equation Revisited

8.3 Laplace’s Equation

9 The Method of Characteristics

9.1 First-Order Systems of PDEs

9.2 Second-Order Hyperbolic PDEs

9.3 First-Order Nonlinear PDES

9.3.1 Characteristics of Burgers’ Equation

9.3.2 Shock Waves

9.3.3 Riemann Problems and Expansion Fans

10 Finite Difference Methods for Elliptic PDEs

10.1 A Dirichlet Problem in a Square Domain

10.1.1 Linear Algebra Aspects

10.1.2 Convergence Theory

10.1.3 Improving the Solution Accuracy

10.2 Advanced Topics and Extensions

10.2.1 Neumann and Robin Boundary Conditions

10.2.2 Non-Rectangular Domains

10.2.3 Polar Coordinates

10.2.4 Regularity of Solutions

10.2.5 Anisotropic Diffusion

10.2.6 Advection–Diffusion

11 Finite Difference Methods for Parabolic PDEs

11.1 Time Stepping Algorithms

11.1.1 An Explicit Method (FTCS)

11.1.2 An Implicit Method (BTCS)

11.1.3 The θ-Method

11.2 Von Neumann Stability

11.3 Advanced Topics and Extensions

11.3.1 Neumann and Robin Boundary Conditions

11.3.2 Multiple Space Dimensions

11.3.3 The Method of Lines

12 Finite Difference Methods for Hyperbolic PDEs

12.1 Explicit Methods

12.1.1 Order Conditions

12.1.2 Stability Conditions

12.1.3 First-Order Schemes

12.1.4 Second-Order Schemes

12.1.5 A Third-Order Scheme

12.1.6 Quasi-implicit Schemes

12.2 The Two-Way Wave Equation

12.3 Convergence Theory

12.4 Advanced Topics and Extensions

12.4.1 Dissipation and Dispersion

12.4.2 Nonperiodic Boundary Conditions

12.4.3 Nonlinear Approximation Schemes

13 Projects

Appendix AGlossary and Notation

Appendix BSome Linear Algebra

Appendix CIntegral Theorems

Appendix DBessel Functions

Appendix EFourier Series

References

Index

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