Theory Of Compact Operators Research Articles

On the Behavior Near the Crest of Waves of Extreme Form

The angle $\phi$ which the free boundary of an extreme wave makes with the horizontal is the solution of a singular, nonlinear integral equation that does not fit (as far as we know) into the theory of compact operators on Banach spaces. It has been proved only recently that solutions exist and that (as Stokes suggested in 1880) these solutions represent waves with sharp crests of included angle $2\pi /3$. In this paper we use the integral equation, known properties of solutions and the technique of the Mellin transform to obtain the asymptotic expansion \[ ( * )\qquad \phi (s) = \frac {\pi } {6} + \sum \limits _{n = 1}^k {{a_n}{s^{{\mu _n}}} + o({s^{{\mu _k}}})} \quad {\text {as}} s \downarrow 0\], to arbitrary order; the coordinate $s$ is related to distance from the crest as measured by the velocity potential rather than by length. The first few (and probably all) of the exponents ${\mu _n}$ are transcendental numbers. We are unable to evaluate the coefficients ${a_n}$ explicitly, but define some in terms of global properties of $\phi$, and the others in terms of earlier coefficients. It is proved in [8] that ${a_1} < 0$, and follows here that ${a_2} > 0$. The derivation of (*) includes an assumption about a question in number theory; if that assumption should be false, logarithmic terms would enter the series at very large values of $n$.

On the behavior near the crest of waves of extreme form

The angle ϕ \phi which the free boundary of an extreme wave makes with the horizontal is the solution of a singular, nonlinear integral equation that does not fit (as far as we know) into the theory of compact operators on Banach spaces. It has been proved only recently that solutions exist and that (as Stokes suggested in 1880) these solutions represent waves with sharp crests of included angle 2 π / 3 2\pi /3 . In this paper we use the integral equation, known properties of solutions and the technique of the Mellin transform to obtain the asymptotic expansion \[ ( ∗ ) ϕ ( s ) = π 6 + ∑ n = 1 k a n s μ n + o ( s μ k ) as s ↓ 0 ( * )\qquad \phi (s) = \frac {\pi } {6} + \sum \limits _{n = 1}^k {{a_n}{s^{{\mu _n}}} + o({s^{{\mu _k}}})} \quad {\text {as}}\,s \downarrow 0 \] , to arbitrary order; the coordinate s s is related to distance from the crest as measured by the velocity potential rather than by length. The first few (and probably all) of the exponents μ n {\mu _n} are transcendental numbers. We are unable to evaluate the coefficients a n {a_n} explicitly, but define some in terms of global properties of ϕ \phi , and the others in terms of earlier coefficients. It is proved in [8] that a 1 &gt; 0 {a_1} &gt; 0 , and follows here that a 2 &gt; 0 {a_2} &gt; 0 . The derivation of (*) includes an assumption about a question in number theory; if that assumption should be false, logarithmic terms would enter the series at very large values of n n .

Multiple grid methods for the solution of Fredholm integral equations of the second kind

In this paper multiple grid methods are applied for the fast solution of the large nonsparse systems of equations that arise from the discretization of Fredholm integral equations of the second kind. Various multiple grid schemes, both with Nyström and with direct interpolation, are considered. For these iterative methods, the rates of convergence are derived using the collectively compact operator theory by Anselone and Atkinson. Estimates for the asymptotic computational complexity are given, which show that the multiple grid schemes result in O ( N 2 ) \mathcal {O}({N^2}) arithmetic operations.

Some aspects of the theory of Banach spaces

is defined. A normed space has a natural metric defined on it, viz., d(x, y) = 1) x y 1). If the metric space obtained in this way is complete, i.e., if every Cauchy sequence converges, the normed space is called a Banach space. The basic theory of Banach spaces forms at present an integral part of the graduate (and even advanced undergraduate) program in mathematics. It consists mainly of theorems like the Hahn-Banach theorem, the open mapping theorem, the uniform boundedness principle, and the Riesz theory of compact operators. These important theorems have the characteristics that they are useful for proving many theorems in classical analysis and that they hold for all Banach spaces. In the current investigations in Banach space theory an important role is played by the problem of classifying Banach spaces (i.e., finding invariants under certain equivalence relations). Of course, there is always

The Numerical Solution of Integral Equations on the Half-Line

The Numerical Solution of Integral Equations on the Half-Line