Integral Equations, Boundary Value Problems And Related Problems

Xing Li

• Publisher: WSPC
• Year: 2013
• ISBN:
  9789814452892
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

In this volume, we report new results about various theories and methods of integral equation, boundary value problems for partial differential equations and functional equations, and integral operators including singular integral equations, applications of boundary value problems and integral equations to mechanics and physics, numerical methods of integral equations and boundary value problems, theories and methods for inverse problems of mathematical physics, Clifford analysis and related problems.

Contents:

• Some Properties of a Kind of Singular Integral Operator for K-Monogenic Function in Clifford Analysis (L P Wang, Z L Xu and Y Y Qiao)
• Some Results Related with Möbius Transformation in Clifford Analysis (Z X Zhang)
• The Scattering of SH Wave on the Array of Periodic Cracks in a Piezoelectric Substrate Bonded a Half-Plane of Functionally Graded Materials (J Q Liu, X Li, S Z Dong, X Y Yao and C F Wang)
• Anti-Plane Problem of Two Collinear Cracks in a Functionally Graded Coating–Substrate Structure (S H Ding and X Li)
• A Kind of Riemann Boundary Value Problem for Triharmonic Functions in Clifford Analysis (L F Gu)
• A New Dynamical Systems Method for Nonlinear Operator Equations (X J Luo, F C Li and S H Yang)
• A Class of Integral Inequality and Application (W S Wang)
• An Efficient Spectral Boundary Integral Equation Method for the Simulation of Earthquake Rupture Problems (W S Wang and B W Zhang)
• High-Frequency Asymptotics for the Modified Helmholtz Equation in a Half-Plane (H M Huang)
• An Inverse Boundary Value Problem Involving Filtration for Elliptic Systems of Equations (Z L Xu and L Yan)
• Fixed Point Theorems of Contractive Mappings in Extended Cone Metric Spaces (H P Huang and X Li)
• Positive Solutions of Singular Third-Order Three-Point Boundary Value Problems (B Q Yan and X Liu)
• Modified Neumann Integral and Asymptotic Behavior in the Half-Space (Y H Zhang, G T Deng and Z Z Wei)
• Piecewise Tikhonov Regularization Scheme to Reconstruct Discontinuous Density in Computerized Tomography (J Cheng, Y Jiang, K Lin and J W Yan)
• About the Quaternionic Jacobian Conjecture (H Liu)
• Interaction Between Antiplane Circular Inclusion and Circular Hole of Piezoelectric Materials (L H Chang and X Li)
• Convergence of Numerical Algorithm for Coupled Heat and Mass Transfer in Textile Materials (M B Ge, J X Cheng and D H Xu)
• Haversian Cortical Bone with a Radial Microcrack (X Wang)
• Spectra of Unitary Integral Operators on L 2 (ℝ) with Kernels k ( xy ) (D W Ma and G Chen)
• The Numerical Simulation of Long-Period Ground Motion on Basin Effects (Y Q Li and X Li)
• Complete Plane Strain Problem of a One-Dimensional Hexagonal Quasicrystals with a Doubly-Periodic Set of Cracks (X Li and P P Shi)
• The Problem About an Elliptic Hole with III Asymmetry Cracks in One-Dimensional Hexagonal Piezoelectric Quasicrystals (H S Huo and X Li)
• The Second Fundamental Problem of Periodic Plane Elasticity of a One-Dimensional Hexagonal Quasicrystals (J Y Cui, P P Shi and X Li)
• The Optimal Convex Combination Bounds for the Centroidal Mean (H Liu and X J Meng)
• The Method of Fundamental Solution for a Class of Elliptical Partial Differential Equations with Coordinate Transformation and Image Technique (L N Wu and Q Jiang)
• Various Wavelet Methods for Solving Fractional Fredholm–Volterra Integral Equations (P P Shi, X Li and X Li)

Readership: Researchers in analysis and differential equations.
Key Features:

• This book assumes minimal background in mathematical logic and takes the reader all the way to current research in a highly active area
• It is the first detailed introduction to this particular approach to this area of research
• The combination of fully worked out arguments and exercises make this book well suited to self-study by graduate students and other researchers unfamiliar with the area

SPECTRA OF UNITARY INTEGRAL OPERATORS ON

WITH KERNELS k(xy)

DAOWEI MA

Department of Mathematics, Wichita State University,

Wichita, KS 67260-0033, USA

[email protected]

GOONG CHEN

Department of Mathematics, Texas A&M University, College Station, TX 77843,

USA, and Science Program, Texas A&M University at Qatar, Education City,

Doha, Qatar

[email protected]

Unitary integral transforms play an important role in mathematical physics. A primary example is the Fourier transform whose kernel is of the form k(x, y) = k(xy), i.e., of the product type. Here we consider the determination of spectrum for such unitary operators as the issue is important in the solvability of the corresponding inhomogeneous Fredholm integral equation of the second kind. A Main Theorem is proven that characterizes the spectral set. Properties of eigenfunctions and eigenspace dimensions are further derived as consequences of the Main Theorem. Concrete examples are also offered as applications.

2010 Mathematics Subject Classification. Primary: 42A38

Keywords and phrases: Unitary transformation, convolution operator, Spectrum.

1. Introduction

The authors are honored to dedicate this article to their teacher and friend, Professor Lu Jianke. The first named author is Professor Lu’s former student, and both authors have been Professor Lu’s friends for decades. Professor Lu has devoted more than sixty years to the research and teaching of mathematics in China. The authors have immense respect for Professor Lu, and great admiration for his work and his enthusiasm and finesse in the study of linear elas...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• Preface
• Organizing Committees
• Some Properties of a Kind of Singular Integral Operator for K-Monogenic Function in Clifford Analysis
• Some Results Related with Mobius Transformation in Clifford Analysis
• The Scattering of SH Wave on the Array of Periodic Cracks in a Piezoelectric Substrate Bonded a Half-Plane of Functionally Graded Materials
• Anti-Plane Problem of Two Collinear Cracks in a Functionally Graded Coating-Substrate Structure
• A Kind of Riemann Boundary Value Problem for Triharmonic Functions in Clifford Analysis
• A New Dynamical Systems Method for Nonlinear Operator Equations
• A Class of Integral Inequality and Application
• An Efficient Spectral Boundary Integral Equation Method for the Simulation of Earthquake Rupture Problems
• High-Frequency Asymptotics for the Modified Helmholtz Equation in a Half-Plane
• An Inverse Boundary Value Problem Involving Filtration for Elliptic Systems of Equations
• Fixed Point Theorems of Contractive Mapping in Extended Cone Metric Spaces
• Positive Solutions of Singular Third-Order Three-Point Boundary Value Problems
• Modified Neumann Integral and Asymptotic Behavior in the Half Space
• Piecewise Tikhonov Regularization Scheme to Reconstruct Discontinuous Density in Computerized Tomography
• About the Quaternionic Jacobian Conjecture
• Interaction between Antiplane Circular Inclusion and Circular Hole of Piezoelectric Materials
• Convergence of Numerical Algorithm for Coupled Heat and Mass Transfer in Textile Materials
• Haversian Cortical Bone with a Radial Microcrack
• Spectra of Unitary Integral Operators on L2(R) with Kernels k(xy)
• The Numerical Simulation of Long-Period Ground Motion on Basin Effects
• Complete Plane Strain Problem of a One-Dimensional Hexagonal Quasicrystals with a Doubly-Periodic Set of Cracks
• The Problem about an Elliptic Hole with III Asymmetry Cracks in One-Dimensional Hexagonal Piezoelectric Quasicrystals
• The Second Fundamental Problem of Periodic Plane Elasticity of a One-Dimensional Hexagonal Quasicrystals
• The Optimal Convex Combination Bounds for the Centroidal Mean
• The Method of Fundamental Solution for a Class of Elliptical Partial Differential Equations with Coordinate Transformation and Image Technique
• Various Wavelet Methodsolving Fractional Fredholm-Volterra Integral Equations