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Numerical Methods for Partial Differential Equations

The subject of partial differential equations holds an exciting and special position in mathematics. Partial differential equations were not consciously created as a subject but emerged in the 18th century as ordinary differential equations failed to describe the physical principles being studied. T...

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Bibliographic Details
Main Authors: Evans, G., Blackledge, J. (Author), Yardley, P. (Author)
Format: eBook
Language:English
Published: London Springer London 2000, 2000
Edition:1st ed. 2000
Series:Springer Undergraduate Mathematics Series
Subjects:
Numerical Analysis
Mathematical Analysis
Analysis
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 1. Background Mathematics
  • 1.1 Introduction
  • 1.2 Vector and Matrix Norms
  • 1.3 Gerschgorin’s Theorems
  • 1.4 Iterative Solution of Linear Algebraic Equations
  • 1.5 Further Results on Eigenvalues and Eigenvectors
  • 1.6 Classification of Second Order Partial Differential Equations
  • 2. Finite Differences and Parabolic Equations
  • 2.1 Finite Difference Approximations to Derivatives
  • 2.2 Parabolic Equations
  • 2.3 Local Truncation Error
  • 2.4 Consistency
  • 2.5 Convergence
  • 2.6 Stability
  • 2.7 The Crank-Nicolson Implicit Method
  • 2.8 Parabolic Equations in Cylindrical and Spherical Polar Coordinates
  • 3. Hyperbolic Equations and Characteristics
  • 3.1 First Order Quasi-linear Equations
  • 3.2 Lax-Wendroff and Wendroff Methods
  • 3.3 Second Order Quasi-linear Hyperbolic Equations
  • 3.4 Reetangular Nets and Finite Difference Methods for Second Order Hyperbolic Equations
  • 4. Elliptic Equations
  • 4.1 Laplace’s Equation
  • 4.2 Curved Boundaries
  • 4.3 Solution of Sparse Systems of Linear Equations
  • 5. Finite Element Method for Ordinary Differential Equations
  • 5.1 Introduction
  • 5.2 The Collocation Method
  • 5.3 The Least Squares Method
  • 5.4 The Galerkin Method
  • 5.5 Symmetrie Variational Forrnulation
  • 5.6 Finite Element Method
  • 5.7 Some Worked Examples
  • 6. Finite Elements for Partial Differential Equations
  • 6.1 Introduction
  • 6.2 Variational Methods
  • 6.3 Some Specific Elements
  • 6.4 Assembly of the Elements
  • 6.5 Worked Example
  • 6.6 A General Variational Principle
  • 6.7 Assembly and Solution
  • 6.8 Solution of the Worked Example
  • 6.9 Further Interpolation Functions
  • 6.10 Quadrature Methods and Storage Considerations
  • 6.11 Boundary Element Method
  • A. Solutions to Exercises
  • References and Further Reading

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