Non-Self-Adjoint Boundary Eigenvalue ProblemsElsevier, 2003. 6. 26. - 518페이지 This monograph provides a comprehensive treatment of expansion theorems for regular systems of first order differential equations and n-th order ordinary differential equations.In 10 chapters and one appendix, it provides a comprehensive treatment from abstract foundations to applications in physics and engineering. The focus is on non-self-adjoint problems. Bounded operators are associated to these problems, and Chapter 1 provides an in depth investigation of eigenfunctions and associated functions for bounded Fredholm valued operators in Banach spaces. Since every n-th order differential equation is equivalentto a first order system, the main techniques are developed for systems. Asymptotic fundamentalsystems are derived for a large class of systems of differential equations. Together with boundaryconditions, which may depend polynomially on the eigenvalue parameter, this leads to the definition of Birkhoff and Stone regular eigenvalue problems. An effort is made to make the conditions relatively easy verifiable; this is illustrated with several applications in chapter 10.The contour integral method and estimates of the resolvent are used to prove expansion theorems.For Stone regular problems, not all functions are expandable, and again relatively easy verifiableconditions are given, in terms of auxiliary boundary conditions, for functions to be expandable.Chapter 10 deals exclusively with applications; in nine sections, various concrete problems such asthe Orr-Sommerfeld equation, control of multiple beams, and an example from meteorology are investigated.Key features:• Expansion Theorems for Ordinary Differential Equations• Discusses Applications to Problems from Physics and Engineering• Thorough Investigation of Asymptotic Fundamental Matrices and Systems• Provides a Comprehensive Treatment• Uses the Contour Integral Method• Represents the Problems as Bounded Operators• Investigates Canonical Systems of Eigen- and Associated Vectors for Operator Functions |
목차
| 1 | |
Chapter II First order systems of ordinary differential equations | 53 |
Chapter III Boundary eigenvalue problems for first order systems | 101 |
Chapter IV Birkhoff regular and Stone regular boundary eigenvalue problems | 129 |
Chapter V Expansion theorems for regular boundary eigenvalue problems for first order systems | 203 |
Chapter VI nth order differential equations | 249 |
Chapter VII Regular boundary eigenvalue problems for nth order equations | 279 |
Chapter VIII The differential equation KηλHη | 321 |
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A₁ associated functions associated vectors assume assumptions Banach space belongs Birkhoff regular boundary conditions boundary eigenvalue operator boundary eigenvalue problem c₁ C₂ canonical system coefficients consider converges Corollary CSEAV CSRF d₁ defined definition depends holomorphically diagonal differential operators eigenfunction expansions eigenfunctions and associated eigenvalue operator function eigenvalue parameter eigenvalue problem 4.1.1 eigenvectors eigenvectors and associated estimate expansion theorems exponential sum f₁ follows Fredholm operator fulfilled fundamental matrix fundamental system given Hence HÖLDER'S inequality holds infer integral invertible K₁ Lemma linear Lp(a Math multiplicity n-th order differential obtain order systems polynomial problem is Birkhoff Proposition proves r₁ representation root functions s-regular satisfied Sobolev spaces Stone regular subset Suppose system of root tions transl v₁ value problem Y₁ yields zero of order νμ Σ Σ ΣΣ