Chapter
2.2 Method of undetermined coefficients
2.4.2 Gauss-Legendre rule
2.4.3 Gauss-Chebyshev open and closed rules
2.4.4 Clenshaw-Curtis quadrature
2.5.1 Mapping the independent variable
2.6 Infinite range integrals
2.6.1 Interval truncation
2.6.2 Infinite range Gauss rules
2.6.3 Mapped finite range rules
2.7 Error estimates for numerical quadrature
2.7.3 Error estimates for repeated (M-panel) rules
2.7.4 Error expansions for the Gauss-Legendre rule
2.7.5 An error expansion for Gauss-Chebyshev quadrature
2.8 High-order vs. low-order rules
2.8.2 Numerical performance
2.9.2 Singular Chebyshev rules
2.9.3 Subtraction of the singularity
2.9.4 Ignoring the singularity
3 Introduction to the theory of linear integral equationsof the second kind
3.1 Regular values and the resolvent equations
3.2 The adjoint kernel and the adjoint equation
3.3 Characteristic values and characteristic functions
3.5 Generalisation of the Neumann series
3.6 The resolvent operator and the resolvent kernel
3.7 Linear Volterra equations of the second kind
4 The Nystrom (quadrature) method for Fredholmequations of the second kind
4.1 Iteration and the Neumann series
4.1.1 A computable error bound
4.1.2 Numerical evaluation of the terms in the Neumann series
4.1.3 Truncation error in the solution of the approximatingequation (6)
4.1.4 Total error in the iterates xn
4.2 The Nystrom or quadrature method
4.2.2 Accuracy of the Nystrom method: qualitative considerations
4.2.3 Convergence and rates of convergence
4.2.5 Sufficient conditions for a smooth solution
4.3 Practical error estimates for the Nystrom method
4.3.1 Brute force error estimates
4.3.2 Comparisons of two rules using the same quadrature points
4.3.3 Direct estimation of the error term
4.3.4 The difference correction
4.3.5 Deferred approach to the limit
4.4 The product Nystrom method
4.5 The method of El-gendi
4.6 Techniques for the solution of the Nystrom equations
4.6.1 Iterative or direct methods?
4.6.2 Alternative iterative schemes
5 Quadrature methods for Volterra equations of the second kind
5.1.1 Quadrature methods for linear equations
5.1.2 Quadrature methods for nonlinear equations
5.1.3 Runge-Kutta methods
5.1.4 Starting procedures
5.1.6 Round-off error and stability
5.3 A Fredholm-type approach for linear Volterra equations
5.4 Variable step methods
5.5 Singular equations: product integration
6 Eigenvalue problems and the Fredholm alternative
6.1 Formal properties of the eigenvalue problem
6.1.3 Non-Hermitian kernels
6.2 Kernels of finite rank (degenerate or separable kernels)
6.2.1 Algebraic solution of finite rank integral equations
6.2.2 Eigenvalues of a kernel of finite rank
6.3 Non-degenerate kernels
7 Expansion methods for Fredholm equations of thesecond kind
7.2 Numerical methods based on approximating the kernel by a separable kernel
7.3 Methods based on an expansion of the solution
7.3.2 Residual minimisation methods
7.3.4 L2 norm: least squares approximation
7.3.5 Method of moments or Ritz-Galerkin
7.3.6 The variational derivation of the Galerkin equations
7.3.7 A relation between the method of moments and the methods of Section 7.2
7.4 Expansion methods for eigenvalue problems
7.4.1 The Ritz-Galerkin method
8 Numerical techniques for expansion methods
8.2 L^ residual minimisation
8.3 Least squares residual minimisation
8.3.1 Setting up the defining equations
8.3.2 Solving the defining equations
8.3.3 Rank deficient systems
8.5 The Fast Galerkin algorithm
8.6 Comparison of methods
9 Analysis of the Galerkin method with orthogonalbasis
9.1 Structure of the equations
9.2 Convergence of Fourier half range cosine expansions
9.3 Expansions in Chebyshev polynomials
9.4 Expansion of functions of two variables
9.5 Estimates for the solution of the Galerkin equations
9.6.1 Approximation and truncation errors
9.7 Quadrature errors for the Fast Galerkin method
9.7.1 Quadrature errors in y
9.7.2 Quadrature errors in B
9.8 Iterative solution of the equations La(N) = y
9.8.1 Iterative scheme for the Fast Galerkin method
10 Numerical performance of algorithms for Fredholm equations of the second kind
10.1 The comparison of algorithms
10.2 Non-automatic routines for second kind Fredholm equations
10.3 Comparing automatic routines
10.4 Numerical comparison of automatic routines
11 Singular integral equations
11.2.1 Direct treatment of the infinite range
11.2.2 Mapping onto a finite interval
11.3 Product integration for singular integrals
11.4 Subtraction of the singularity
11.4.1 Basic subtraction identities
11.4.2 Application to the evaluation of singular integrals
11.4.3 Application to the solution of Fredholm integral equations
11.5 Cauchy integral equations
11.6 The Singular Galerkin method
11.6.2 The basic algorithm
11.6.3 Extensions of the algorithm
12 Integral equations of the first kind
12.2 Eigenfunction expansions
12.2.2 Non-Hermitian kernels
12.3 The method of regularisation
12.4 The Augmented Galerkin method
12.4.2 Stability of the Augmented Galerkin method
12.4.3 Computational details
12.4.4 A numerical criterion for the existence of a solution
12.5 First kind Volterra equations
13 Integro-differential equations
13.2 Quadrature methods for the numerical solution of integro differential equations
13.2.1 Differential equation methods for Volterra-type equations
13.2.2 Integral equation methods for Volterra-type equations
13.2.3 Differential equation methods for Fredholm-type equations
13.3.3 The Fast Galerkin scheme for linear integro-differential equations (Babolian and Delves, 1981)
Appendix: Singular expansions
A.1 Expansions for singular driving terms
A.2 Expansions for singular kernels
Computational Methods for Integral Equations
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