On the eigenvalue problem on a semi infinite interval (original) (raw)
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Numerical methods for approximating eigenvalues of boundary value problems
International Journal of Mathematics and Mathematical Sciences, 1986
This paper describes some new finite difference methods for the approximation of eigenvalues of a two point boundary value problem associated with a fourth order linear differential equation of the type (py') (q y') + (r s)y O. The smallest positive eigenvalue of some typical eigensystems is computed to demonstrate the practical usefulness of the numerical techniques developed. KEY WORDS AND PHRASES. Band matrices, Deflation, Finite-dlfference methods, Generalized matrix eigenvalue problem, Inverse power iteration, The Smallest eigenvalue of a matrix eigenvalue problem, Two-point boundary value problems. 1980 AMS SUBJECT CLASSIFICATION CODE. 65L15.
Boundary-Value Problems for Ordinary Differential Equations on Infinite Intervals
Quarterly Journal of Mathematics, 1977
In a recent paper ([19]), Stuart used the theory of fc-set contractions to establish results for the bounda^-value problem for t e [0, oo), 2/(0) = 0, and y e L 2 [0, oo). Here A is a real parameter and it is assumed that /(0) = 0, /'(0) = 1, and q(t) has a finite limit Q as t -»• oo. In this paper, we generalize Stuart's results. We study in detail the case where 0 ^f{y)/y < 1 for all non-zero y. Stuart's results do not apply to this case. For this case, we also obtain a number of results on the solution structure. These .results show that, in many ways, the above problem behaves like the corresponding boundary-value problem on a compact interval. Finally, we study the solutions near a critical value of A.
On an eigenvalue problem of Ahmad and Lazer for ordinary differential equations
Proceedings of the American Mathematical Society, 1987
Lazer, we show the existence of a class of nonselfadjoint eigenvalue problems related to the equation y(n) + Xp(x)y = 0 for which the general eigenvalues comparison is not true. We use a comparison principle for the zeros of the corresponding Cauchy problem. This paper provides a contribution to the understanding of a problem raised by S. Ahmad and A. C. Lazer [1] in connection with the comparison of the eigenvalues for some multi-point boundary value problems which are not selfadjoint. One is given the equation (1) Lny + Xp(x)y = 0, where p(x) is a continuous function of constant sign on an interval /, A is a parameter, and Lny is a linear differential disconjugate operator of order n, that is, the only solution of Lny = 0 with n zeros on I (counting multiplicity) is y = 0. Let us consider the eigenvalue problem given by equation (1) and the system of boundary conditions ,, Lzy(a)=0, iG{ii,...,ik}, L]V(b)=0, JGiJu.-.Jn-k}, where o, b G I, 1 < k < n-1, Liy, i = 0,..., n-1, are the quasi-derivatives of y(x) (see [7]), and {t'i,..., ¿fc}, {ji, ■ ■ ■ ,jn-k) are two arbitrary sets of indices from the set {0,... ,n-1}. Problems of this type have been studied extensively (cf. [2, 3, 5]). In particular, Elias [5] has shown that if (-l)n_fcp(x) < 0, then the eigenvalues of problems (1) and (2) are real and nonnegative and form a divergence sequence {Am}m£N-Ahmad and Lazer [1] have considered a particular type of boundary condition (2), that is (3) y(a)=y'(a) =-= yik-1\a) = 0, y(b)=y'(b) =-=y(n-k-i\b)=0, and showed that if we set p = Pi, where p¿, i-1,2, are two continuous functions, considering the corresponding sequence of eigenvalues (A¿,m)m6N, i = 1,2, ordered by magnitude, then the condition (4)_ (-l)n-kp2(x) < (-l)"-fepi(x) < 0
FD-method for a nonlinear eigenvalue problem with discontinuous eigenfunctions
Nonlinear Oscillations, 2007
An algorithm for solution of a nonlinear eigenvalue problem with discontinuous eigenfunctions is developed. The numerical technique is based on a perturbation of the coefficients of differential equation combined with the Adomian decomposition method for the nonlinear term of the equation. The proposed approach provides an exponential convergence rate dependent on the index of the trial eigenvalue and on the transmission coefficient. Numerical examples support the theory. Розроблено алгоритм для чисельного розв'язування нелiнiйних задач на власнi значення з розривними власними функцiями. В основi чисельного методу лежить збурення коефiцiєнтiв диференцiального рiвняння в поєднаннi з методом декомпозицiї Адомяна нелiнiйної частини рiвняння. Запропонований пiдхiд забезпечує експоненцiальну швидкiсть збiжностi, яка залежить вiд порядкового номера власного значення та коефiцiєнта трансмiсiї. Наведенi чисельнi розрахунки пiдтверджують теоретичнi висновки.
Boundary-Value Problems for Ordinary Differential Equations on Infinite Intervals II
Quart J Math, 1977
In a recent paper ([19]), Stuart used the theory of fc-set contractions to establish results for the bounda^-value problem for t e [0, oo), 2/(0) = 0, and y e L 2 [0, oo). Here A is a real parameter and it is assumed that /(0) = 0, /'(0) = 1, and q(t) has a finite limit Q as t -»• oo. In this paper, we generalize Stuart's results. We study in detail the case where 0 ^f{y)/y < 1 for all non-zero y. Stuart's results do not apply to this case. For this case, we also obtain a number of results on the solution structure. These .results show that, in many ways, the above problem behaves like the corresponding boundary-value problem on a compact interval. Finally, we study the solutions near a critical value of A.
Some properties of eigenvalues and generalized eigenvectors of one boundary-value problem
2016
We investigate a discontinuous boundary value problem which consists of a Sturm-Liouville equation with piecewise continuous potential together with eigenparameter dependent boundary conditions and supplementary transmission conditions. We establish some spectral properties of the considered problem. In particular, it is shown that the problem under consideration has precisely denumerable many eigenvalues λ 1 , λ 2 , ..., which are real and tends to +∞. Moreover, it is proven that the generalized eigenvectors form a Riesz basis of the adequate Hilbert space.
BIT Numerical Mathematics, 1984
When finite difference and finite element methods are used to approximate continuous (differential) eigenvalue problems, the resulting algebraic eigenvatues only yield accurate estimates for the fundamental and first few harmonics. One way around this difficulty would be to estimate the error between the differential and algebraic eigenvalues by some independent procedure and then use it to correct the algebraic eigenvalues. Such an estimate has been derived by Paine, de Hoog and Anderssen for the Liouville normal form with Dirichlet boundary conditions. In this paper, we extend their result to the Liouville normal form with general boundary conditions.