Mathematical Foundation of the Boundary Integro-Differential Equation Method

The book focuses on  the mathematical foundations of boundary integro-differential equation method, with a primary focus on reducing the hypersingular integrals in traditional boundary integral equations into boundary integro-differential equations with weak singularities. It briefly introduces the theory of distributions, while the boundary integral equations method is grounded in the fundamental solutions of linear partial differential equations, hence a relatively detailed exposition of the fundamental solutions of differential equations is also provided. In the subsequent chapters, the authors sequentially discuss the boundary integro-differential equation methods and theories for Laplace equation, Helmholtz equation, Navier equations, Stokes equations, among others. Furthermore, the book addresses the boundary integro-differential equation method for certain nonlinear problems, such as thermal radiation, variational inequalities, and Steklov eigenvalue problems. Lastly, it explores the symmetric coupling issues between finite element and boundary element methods.

Chapter 1  Distributions.- Chapter 2  Fundamental Solutions of Linear Differential Operators.- Chapter 3  Boundary Value Problems of the Laplace Equations.- Chapter 4  Boundary Value Problem of Modified Helmholtz Equation.- Chapter 5  Boundary Value Problems of Helmholtz Equation.- Chapter 6  Boundary Value Problems of the Navier Equations.- Chapter 7  Boundary Value Problems of the Stokes Equations.- Chapter 8  Some Nonlinear Problems.- Chapter 9  Coercive and Symmetrical Coupling Methods of Finite Element Method and Boundary Element Method.

Han Houde is a professor at Tsinghua University who has devoted himself to the teaching and scientific research of computational mathematics. He has achieved significant breakthroughs in the study of numerical solutions of partial differential equations, demonstrating a series of creative research results. In particular, he has made important contributions to the numerical solutions of partial differential equations on unbounded domains and the coupling method of the finite element method and boundary element method. Additionally, he has contributed to the numerical solutions of boundary integral-differential equations and variational inequalities problems, as well as the numerical solutions of singular perturbation problems, ill-posed problems, and infinite element methods.

Yin Dongsheng is an associate professor at Tsinghua University whose research interest is mainly focused on high frequency waves, partial differential equations on unbounded domains, and fractional differential equations.