Orthogonal approximations in Sobolev spaces stability and convergence spectral methods and pseudospectral methods spectral methods for multi-dimensional and high order problems mixed spectral methods combined spectral methods spectral methods on the spherical surface.

Spectral Methods and Their Applications

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

Integral Equations, Origin, and Basic Tools Modeling of Problems as Integral Equations Volterra Integral Equations The Green's Function Fredholm Integral Equations Existence of the Solutions: Basic Fixed Point Theorems Higher Quadrature Rules for the Numerical Solutions Appendices Answers to Exercises References Index.

Introduction to integral equations with applications, by Abdul J. Jerri. Pp 254. $47·50. 1985. ISBN 0-8247-7293-8 (Marcel Dekker)

Legendre multi-Galerkin methods for Fredholm integral equations with weakly singular kernel and the corresponding eigenvalue problem

This paper presents a numerical method for solving a certain class of Fredholm integral equations of the first kind, whose unknown function is singular at the end-points of the integration domain, ...

The economized monic Chebyshev polynomials for solving weakly singular Fredholm integral equations of the first kind

In this paper, we consider the Galerkin method to approximate the solution of Volterra-Urysohn integral equations of second kind with a smooth kernel, using piecewise polynomial bases. We show that the exact solution is approximated with the order of convergence $\mathcal {O}(h^{r})$ for the Galerkin method, whereas the iterated Galerkin solutions converge with the order $\mathcal {O}(h^{2r})$ in uniform norm, where h is the norm of the partition and r is the smoothness of the kernel. For improving the accuracy of the approximate solution of the integral equation, the multi-Galerkin method is also discussed here and we prove that the exact solution is approximated with the order of convergence $\mathcal {O}(h^{3r})$ in uniform norm for iterated multi-Galerkin method. Numerical examples are given to illustrate the theoretical results.

Superconvergence Results for Volterra-Urysohn Integral Equations of Second Kind

In this paper, Galerkin method is applied to approximate the solution of Volterra integral equations of second kind with a smooth kernel, using piecewise polynomial bases. We prove that the approximate solutions of the Galerkin method converge to the exact solution with the order $${\mathcal {O}}(h^{r}),$$
 whereas the iterated Galerkin solutions converge with the order $${\mathcal {O}}(h^{2r})$$
 in infinity norm, where h is the norm of the partition and r is the smoothness of the kernel. We also consider the multi-Galerkin method and its iterated version, and we prove that the iterated multi-Galerkin solution converges with the order $${\mathcal {O}}(h^{3r})$$
 in infinity norm. Numerical examples are given to illustrate the theoretical results.

Superconvergence results for linear second-kind Volterra integral equations

In this paper, we discuss the superconvergence of the Galerkin solutions for second kind nonlinear integral equations of Volterra–Hammerstein type with a smooth kernel. Using Legendre polynomial bases, we obtain order of convergence $${\mathcal{O}}(n^{-r})$$ for the Legendre Galerkin method in both $$L^2$$-norm and infinity norm, where n is the highest degree of the Legendre polynomial employed in the approximation and r is the smoothness of the kernel. The iterated Legendre Galerkin solutions converge with the order $${\mathcal{O}}(n^{-2r}),$$ whose convergence order is the same as that of the multi-Galerkin solutions. We also prove that iterated Legendre multi-Galerkin method has order of convergence $${\mathcal{O}}(n^{-3r})$$ in both $$L^2$$-norm and infinity norm. Numerical examples are given to demonstrate the efficacy of Galerkin and multi-Galerkin methods.

Legendre spectral Galerkin and multi-Galerkin methods for nonlinear Volterra integral equations of Hammerstein type

In this article, a Jacobi spectral Galerkin method is developed for weakly singular Volterra integral equations of the second kind. To obtain the superconvergence results, we transform the domain of integration of Volterra integral equation to the standard interval $$[-1, 1]$$ using variable transformation and function transformation. We obtain the convergence rates both in infinity and weighted $$L^2$$-norm. We prove that the Jacobi spectral iterated Galerkin method shows improvement over the Jacobi spectral Galerkin method. We improve these results further by considering the Jacobi spectral iterated multi-Galerkin method. Theoretical results are justified by the numerical results.

Moumita Mandal

Papers

Superconvergence Results for Volterra-Urysohn Integral Equations of Second Kind

Superconvergence results for linear second-kind Volterra integral equations

Legendre spectral Galerkin and multi-Galerkin methods for nonlinear Volterra integral equations of Hammerstein type

Error Analysis of Jacobi Spectral Galerkin and Multi-Galerkin Methods for Weakly Singular Volterra Integral Equations

Convergence analysis for derivative dependent Fredholm-Hammerstein integral equations with Green’s kernel