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Computing Eigenvalues of Singular SturmLiouville Problems
, 1999
"... We describe a new algorithm to compute the eigenvalues of singular SturmLiouville problems with separated selfadjoint boundary conditions for both the limitcircle nonoscillatory and oscillatory cases. Also described is a numerical code implementing this algorithm and how it compares with SLEIGN. ..."
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Cited by 18 (9 self)
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We describe a new algorithm to compute the eigenvalues of singular SturmLiouville problems with separated selfadjoint boundary conditions for both the limitcircle nonoscillatory and oscillatory cases. Also described is a numerical code implementing this algorithm and how it compares with SLEIGN. The latter is the only effective general purpose software available for the computation of the eigenvalues of singular SturmLiouville problems.
Asymptotic first eigenvalue estimates for the biharmonic operator on a rectangle, preprint
, 1996
"... We find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle. This is done by finding lower and upper bounds which become increasingly accurate with increasing length. The lower bound is found by algebraic manipulation of the operator, and the upper b ..."
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Cited by 7 (0 self)
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We find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle. This is done by finding lower and upper bounds which become increasingly accurate with increasing length. The lower bound is found by algebraic manipulation of the operator, and the upper bound is found by minimising the quadratic form for the operator over a test space consisting of separable functions. These bounds can be used to show that the negative part of the groundstate is small. 1.
Bounds for the N lowest eigenvalues of fourthorder boundary value problems
"... . We describe a method for the calculation of the N lowest eigenvalues of fourthorder problems in H 2 0 (\Omega ). In order to obtain small error bounds, we compute the defects in H \Gamma2 (\Omega ) and, to obtain a bound for the rest of the spectrum, we use a boundary homotopy method. As an e ..."
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Cited by 3 (0 self)
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. We describe a method for the calculation of the N lowest eigenvalues of fourthorder problems in H 2 0 (\Omega ). In order to obtain small error bounds, we compute the defects in H \Gamma2 (\Omega ) and, to obtain a bound for the rest of the spectrum, we use a boundary homotopy method. As an example, we compute strict error bounds (using interval arithmetic to control rounding errors) for the 100 lowest eigenvalues of the clamped plate problem in the unit square. Applying symmetry properties, we prove the existence of double eigenvalues. AMS Symbol classification: 65N25 Key words: biharmonic operator, eigenvalue enclosures, plate equation, boundary homotopy, double eigenvalues There are various methods for the inclusion of eigenvalues for the equations Tu = u resp. T Tu = u, where T is a secondorder elliptic operator (e. g. see BehnkeGoerisch [1] for a review on this topic). To obtain strict bounds for the N lowest eigenvalues, additional information on the rest of the ...
MAURO PICONE, SANDRO FAEDO, AND THE NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS IN ITALY (1928–1953)
"... Abstract. In this paper we revisit the pioneering work on the numerical analysis of partial differential equations (PDEs) by two Italian mathematicians, Mauro Picone (1885–1977) and Sandro Faedo (1913–2001). We argue that while the development of constructive methods for the solution of PDEs was cen ..."
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Abstract. In this paper we revisit the pioneering work on the numerical analysis of partial differential equations (PDEs) by two Italian mathematicians, Mauro Picone (1885–1977) and Sandro Faedo (1913–2001). We argue that while the development of constructive methods for the solution of PDEs was central to Picone’s vision of applied mathematics, his own work in this area had relatively little direct influence on the emerging field of modern numerical analysis. We contrast this with Picone’s influence through his students and collaborators, in particular on the work of Faedo which, while not the result of immediate applied concerns, turned out to be of lasting importance for the numerical analysis of timedependent PDEs.
ON THE DIFFERENTIAL EQUATION UXXXX + UY Y Y Y = F FOR AN ANISOTROPIC STIFF MATERIAL
"... Abstract. We study the differential operator L = ∂4 ∂x4 + ∂4 ∂y4 and investigate positivity preserving properties in the sense that f ≥ 0 implies that solutions u of Lu − λu = f are nonnegative. Since the operator is of fourth order we have no maximum principle at our disposal. The operator models t ..."
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Abstract. We study the differential operator L = ∂4 ∂x4 + ∂4 ∂y4 and investigate positivity preserving properties in the sense that f ≥ 0 implies that solutions u of Lu − λu = f are nonnegative. Since the operator is of fourth order we have no maximum principle at our disposal. The operator models the deformation of an anisotropic stiff material like a wire fabric, and it has to be complemented by appropriate boundary conditions. Our operator was introduced by Jacob II Bernoulli as the operator that supposedly models the vibrations of an elastic plate. This model was later revised by Kirchhoff, because the operator and its solutions were anisotropic. Modern materials, however, are often anisotropic and therefore the old model of Bernoulli deserves an updated investigation. It turns out that even our simple looking model problem contains some hard analytical challenges.
unknown title
, 1999
"... A numerical investigation of some problems associated with the biharmonic operator ..."
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A numerical investigation of some problems associated with the biharmonic operator
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"... Dedicated to the memory of John Barrett. ABSTRACT. Regular and singular SturmLiouville problems (SLP) are studied including the continuous and differentiable dependence of eigenvalues on the problem. Also initial value problems (IVP) are considered for the SL equation and for general first order sy ..."
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Dedicated to the memory of John Barrett. ABSTRACT. Regular and singular SturmLiouville problems (SLP) are studied including the continuous and differentiable dependence of eigenvalues on the problem. Also initial value problems (IVP) are considered for the SL equation and for general first order systems. 1