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Numerical Computation of Eigenvalues of Differential Operators and the Boundary Value Space Theory for the Self-Adjoint Extensions of Symmetric Operators

Author: ChenJinShe
Tutor: SunJiong
School: Inner Mongolia University
Course: Applied Mathematics
Keywords: Symmetric Operators Differential operator Discontinuous Sturm-Liouville operator Self-conjugate expansion Boundary value space Boundary triple Numerical calculation Eigenvalues Characteristic function Separation characteristic parameters
CLC: O175.3
Type: PhD thesis
Year: 2009
Downloads: 84
Quote: 1
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Abstract

This paper focuses on the characteristics of the (differential) operator field value numerical calculations the symmetric operators since the the conjugate expansion and differential operator self-conjugate domain describe three aspects to carry out research work. Numerical calculation of eigenvalues ??of differential operators, both in theory , and in practice, are of great significance in fact, can be given directly to the analytical solution of the differential operator few categories, but for numerical methods, especially with the emergence of modern high-speed computer and updates, show increasing the power of quantitative solve various technical problems at the same time we should be noted that the numerical results, in turn, can further inspire people to come to a deeper qualitative results of numerical analysis in the theoretical study plays an increasingly more important role in the theory of fluid and magnetic fluid heat conduction or diffusion and boundary conditions in the multilayer dielectric high-end (non-) self-conjugate containing a spectral parameter differential operator spectral theory several important issues. past numerical methods for second-order self-conjugate problem in recent years, Greenberg and Marletta based Atkinson-Pr (u ¨) the fer vibration theory shooting method, but this ways for Differential Equations, using the is piecewise constant factor approximation approach has some shortcomings. solve the above problem, we propose a new approach in dealing with the problem of high vibration coefficient, processing these problems a unified framework by the characteristic equation Ly (x, lambda) = λy (x, lambda) (l said n-order ordinary differential equation, lambda is the characteristic parameter.) general solution of a characteristic parameter power series, we discovered and proved its coefficients satisfy a recurrence relation constituted by the differential equations based on the nature of the Volterra integral operator, we give and prove a solving the coefficient function alpha _{i < / sub> (x), which is constructed for solving the characteristic equation power series solution. Moreover, we prove the stability of this method is numerically and given this power series solution truncation error is estimated by the power series solution, calculate the corresponding characteristic determinant (zero or feature value) and then apply the numerical solution of the root-finding tools can be obtained eigenvalues. further, we give the corresponding feature function calculation method. Finally, through specific examples, verification and analysis of our method of calculating the algorithm is simple, clear thinking, wide applicability, not only can solve the second-order self-conjugate problem, but also for solving the above-mentioned high-end (non-) self-conjugate, discontinuous Sturm-Liouville problem, these new problems the same time, the algorithm also overcome some of the defects of the Greenberg and Marletta method, the symmetric operators in the classical theory of boundary value space, expansion The description is actually based on the theory of linear relationship model, which makes the theory of boundary value space in the application is not very convenient, it is in a position to exercise its power to bring some obstacles to the development of structural boundary value space theory to discuss issues related to described the expansion of self-conjugate. First of all, departure from Neumann formula and the definition of self-conjugate, direct proof boundary mapping and self-conjugate domain contact constructive method proved that all by any self-conjugate domain boundary value space \a structural definition of the boundary value space. further discussed for unitary parameters a Bianalytic mapping exists between self-the unitary transformation expansion conjugate with the Cayley expansion unitary transformation for further study of spectrum The dependence of the boundary conditions played a stepping stone. Finally, we give and prove a general boundary conditions B (x): = the MΓ ₁ x NΓ ₂ x = 0 (M, N is the order of the loss index phalanx) is self-conjugate necessary and sufficient conditions, as well as the corresponding boundary map constructor The result is a closed symmetrical all have an equal loss index (<∞) Operators child are applicable, provides a unified tool for the practical application of various types of Ordinary Differential Operators. article With the development of the general theory of symmetric operators self-conjugate boundary value space expansion, re-given Ordinary Differential Operators self-conjugate Domain analytical description specifically for regular and singular Ordinary Differential Operators and discontinuous Sturm-Liouville operator sub, the use of a simple boundary mapping and the C ^{m (m means that the Deficiency Index of the corresponding operator) unitary transformation on the \background and the main results; second chapter briefly describes several practical background Differential Operators problem; Chapter III presents the separation characteristic parameter method for solving the characteristic equation of the general method of solution containing a parameter, and the corresponding characteristic function method of calculating ; discussed in Chapters IV and V, respectively, from the conjugate and numerical solution of non-self-conjugate eigenvalues ??and the corresponding numerical examples and numerical analysis; Chapter 6 discusses the theory of symmetric operators self-conjugate boundary value space expansion Ordinary Differential Operators; Chapter VII discusses self-conjugate boundary value space domain description; discussed in Chapter 8 discontinuous Sturm-Liouville problem domain description of self-conjugate boundary value space.}}

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Differential operator theory
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