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The heating (or cooling) of the rock adjacent to water flowing through a crevice is of interest in certain geothermal studies. For many purposes, only the rock temperature at the rock-water interface is needed. The fact that the convective heat transfer coefficient is a function of time in the case of unsteady flow takes this problem out of the strictly classical domain. Several methods of solution are described. An approach in which the heat equation is solved, under appropriate side conditions, by standard finite difference techniques is shown to be satisfactory but rather wasteful because much unneeded information concerning internal rock temperatures must be obtained. Several integral equations of Volterra type are derived that provide the temperature only at the interface. Two numerical approaches to their solution are described: one quite classical and only partially effective; the other an apparently new algorithm that is fast and efficient. Analytical upper and lower bounds on the temperature are obtained and serve to demonstrate that this numerical device is also very accurate. The algorithm is applicable to a wide class of Volterra integral equations of the second kind. A brief discussion of the possible future uses of this scheme and the need for additional research is given.

In the critical state theory of superconductors the magnetic field can be given as the solution of a nonlinear integral equation of Uryson type. It is shown that for sufficiently small values of physical and geometric parameters the equation always has a unique solution. A sufficient condition for the nonvanishing of the B‐field is derived and application is made to several models, two of which result from experiments with NbTi and Nb3Sn. The investigation seems to cast doubt on the validity of some aspects of the critical state theory.

The present paper extends the synthetic method of transport theory to a large class of integral equations. Convergence and divergence properties of the algorithm are studied analytically, and numerical examples are presented which demonstrate the expected theoretical behavior. It is shown that, in some instances, the computational advantage over the familiar Neumann approach is substantial.

The Taylor theory of toroidal plasma relaxation is considered as a nonlinear eigenvalue problem. The analysis is rigorous and applies to quite general toroidal cross sections. Emphasis is placed on the symmetric state case, where the existence of field reversal and flux free states is demonstrated. Certain anomalies are revealed by the mathematical treatment and their significance is studied. Existence of a solution to the Taylor problem in the symmetric state is proved and the location of the eigenvalue of this solution state relative to other states is examined. The question of the existence of helical states is not resolved, but it is shown that in many respects any helical states behave like the anomalous cases in the symmetric problem and hence do not significantly affect the theory.

A new technique based upon an invariant imbedding-Ricatti transformation approach is presented for the calculation of eigenvalues ofSturm-Liouville type systems of differential equations. A very simple numerical procedure is developed which is easily programmed and which uses reliable subroutines. The method is capable of handling a large class of problems. Included among these are problems in which the eigenvalues appears in a non-linear fashion, cases in which the eigenvalue occurs in the boundary condition, and equations which have singularities. The numerical computations are generally well-conditioned and very accurate results were obtained. Es wird eine neue Technik, basierend auf einer Methode, welche invariante Einbettung (Invariant Imbedding) mit derRicatti-Transformation kombiniert, dargestellt zum Zwecke der Berechnung der Eigenwerte von Differentialgleichungssystemen, welche vomSturm-Liouville-Typ sind. Eine sehr einfache numerische Prozedur ist entwickelt worden, die leicht zu programmieren ist und verlaβliche Unterprogramme benutzt. Dieses Verfahren vermag eine umfangreiche Klasse von Problemen zu handhaben. Unter ihnen sind solche inbegriffen, bei denen der Eigenwert in einer nichtlinearen Form auftritt; auch Falle, wo der Eigenwert in der Randbedingung vorkommt; und schlieβlich auch Gleichungen, die Singularitaten aufweisen. Es sei betont, daβ die hier von uns behandelten numerischen Berechnungen im allgemeinen stabil sind und sehr genaue Ergebnisse hervorzubringen vermogen.

No abstract. 13:000396A procedure is developed for obtaining, in nuclear shell theory, the matrix elements associated with short range interactions between nucleons. It is shown that potentials of arbitrary range, both central and spin dependent, can be developed in terms of a series of velocity dependent potentials in increasing powers of the momentum, but all of zero range. The coefficients of the terms in this series are the successive moments of the potential; thus. for short range forces, only the first few terms of the series will be relevant. All the matrix elements for two nucleons in the same shell and in LS coupling, up to terms containing the first moment of the potentgnal, are obtained. The matrix elements for central spin-orbit coupling and tcnsor forces, are compared, taking for the interaction potential between nucleons, a value recently proposed. (auth)

in the presence of appropriate auxiliary hypotheses. We show that the result is not valid without some additional hypothesis; and that either measurability of f or boundedness of f on an appropriate set of intervals is adequate. In the presence of auxiliary hypotheses, the requirement that (1) hold for all s is apt to be unnecessarily strong. Theorem 1 covers a situation in which (1) is assumed for some nonzero s; while Theorems 2 and 3 cover situations, by reduction to Theorem 1, in which (1) is assumed for a more plentiful set but in which weaker side conditions are assumed. Functions satisfying (1) arise in renewal theory [1 ]. In such cases, the functions usually satisfy stronger additional conditions than any we consider here, and (2) may often be obtained from such conditions instead of (1). In what follows we assume that f is a real-valued function defined on a set containing a right ray and X is a real number. It is clear that if (1) holds for s in some set S, then (1) holds for s in the additive subgroup of the real numbers generated by S; and we shall only consider such subgroups in our hypotheses.

With the use of invariance principles in a systematic fashion, we shall derive not only new analytic formulations of the classical particle processes, those of transport theory, radiative transfer, random walk, multiple scattering, and diffusion theory, but, in addition, new computational algorithms which seem well fitted to the capabilities of digital computers. Whereas the usual methods reduce problems to the solution of systems of linear equations, we shall try to reduce problems to the iteration of nonlinear transformations.Although we have analogous formulations of wave processes, we shall reserve for a second paper in this series a detailed and extensive treatment of this part of mathematical physics.

Diffusion theory classically has been regarded as an approximation to the more rigorous (but, of course, not completely rigorous) transport theory under the assumption of high velocity and small mean free path. Furthermore, passage to the limit in the 'telegrapher's equation,' a linear partial differential equation of hyperbolic type, has been carried out. The limits are studied of the non-linear functional equations obtained from the transport processes with finite velocity as the velocity increases without bound. Corresponding results are obtained for heat or diffusion processes, where the physical picture is not as clear. The equations can be interpreted in such a way as to be derived by invariant imbedding techniques. In all cases, the equations are of the generalized Riccati type which are characteristic of these processes of mathematical physics.

Nonlinear initial value problems (of a rather unusual nature) will be obtained which have as their solutions so(x) and so(y). From these values Sp(z) may be found by solving another related initial value problem. The equations derived are amenable to numerical treatment and may very well provide advantages over the more customary direct methods of computing $o(z) from (1.1). The representation (1.2) actually encompasses a very large class of kernels. If, for example, a(s) = s, we have the usual Laplace transform. Kernels expressed in terms of such Laplace integrals arise often in transport theory. A problem in gas dynamics analyzed in [10] involves a kernel represenited in this way. Again, if a(s) = -log s, then K is a Mellin transform. We shall also find that a(s) = i log s is admissible in our theory, and this leads to a Fourier integral

The method of invariant imbedding was used to investigate neutron behavior in fixed media. Equations are formulated for the case in which the region changes in length as the neutron process continues.

In a series of papers dating from 1956, the authors have used the theory of invariant imbedding to derive a variety of nonlinear functional equations for the description of radiative transfer, neutron transport, diffusion and heat conduction, scattering and random walk, and wave propagation. In this paper a new method is presented for establishing the existence of solutions of these equations in the cases where no creation of matter is involved. This method is based upon the introduction of a new class of physically important functions, the dissipation functions. Combining these new functions with the functions utilized in the past, the reflection and transmission functions, we easily obtain a basic conservation relation upon which all else hinges. The uniqueness proofs follow conventional lines.