Single Partial Differential Equation Research Articles

Accurate buckling equations for arbitrary and cylindrical elastic shells

Accurate equations for the neutral equilibrium of arbitrary elastic shells are derived. These equations involve no displacement components and thus complement alternate sets of neutral equilibrium equations proposed by Koiter and Budiansky. The equations are specialized to a uniformly loaded circular cylindrical shell and reduced to a single partial differential equation. For a simply supported cylindrical shell, expressions for the buckling loads are obtained which are as simple as those predicted by the Donnell equations, yet as accurate as those predicted by the more elaborate Flügge equations.

The generation of continental shelf waves

Assuming zero divergence, the equations of forced long waves in a uniformly rotating, homogeneous ocean are reduced to a single partial differential equation for the stream function. A shelf of exponential slope between a rigid continent and a sea of uniform depth is taken as a model, and certain other assumptions are made which appear physically reasonable. Calculations made on the basis of this simplified theory are in good qualitative agreement with observations of shelf waves, indicating that these waves are generated by the stress of the longshore component of the geostrophic wind.

On Axially Symmetric Magnetohydrostatic Fields

1. Introductiot.. Solutions of the magnetohydrostatic equations for a medium of large conductivity are of interest in connection with problems in cosmic physics. In this paper a general class of axially symmetric solutions is found to satisfy a single partial differential equation of the second order containing two arbitrary functions. From this equation a family of solutions depending on two parameters a, ,B and belonging to the force-free field class is constructed in which the toroidal field is in general a nonlinear function of the magnetic stream function, but the equation satisfied by the magnetic stream function can be reduced to the solution of a linear partial differential equation. For ,B = 0, the solution reduces to the wellknown force-free field solution suitable for simply connected domains discussed by Chandrasekhar [1], [2] and Woltjer [3]. The equipartition of energy theorem between the poloidal -and toroidal energies is shown to be valid for any simply connected region for which the normal component of the magnetic field vanishes on the boundary. This result was established for the case of the sphere without the assumption of axial symmetry by Chandrasekhar and Kendall [4]. For a = 0, a representation is found suitable for malultiply connected domains in which there is a tripartition of the total energy, the toroidal energy being twice the poloidal. An explicit solution is determined for the case in which the meridian curve is a rectangle, and finally it is shown how the new class of solutions may be characterized by a variational principle.

Propagation of ultrashort optical pulses

Abstract A model for optical pulse propagation in a two-level medium having relaxation times which are long compared to pulse length is summarized. Pulse shapes are described by solutions of a single nonlinear partial differential equation. Particular solutions are obtained by employing a Baecklund transformation.

Null Fields in Einstein-Maxwell Field Theory. II

If the propagation vector of an electromagnetic null field is proportional to a gradient, then the metric tensor can be expressed in terms of only four unknown functions. In the present analysis, it is assumed that the divergence of this vector does not vanish. It is shown that the field equations reduce to a single partial differential equation. Particular solutions which represent electromagnetic and purely gravitational waves are given.

On the Single First-Order Partial Differential Equation with a Small Parameter

Previous article Next article On the Single First-Order Partial Differential Equation with a Small ParameterLarry BobisudLarry Bobisudhttps://doi.org/10.1137/1008103PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Miloš Zlámal, Sur l'équation des télégraphistes avec un petit paramètre, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8), 27 (1959), 324–332 MR0115021 0108.28304 Google Scholar[2] Larry Bobisud, Degeneration of the solutions of certain well-posed systems of partial differential equations depending on a small parameter, J. Math. Anal. Appl., 16 (1966), 419–454 10.1016/0022-247X(66)90155-7 MR0201783 0145.35901 CrossrefISIGoogle Scholar[3] Miloš Zlámal, The mixed problem for hyperbolic equations with a small parameter, Czechoslovak Math. J., 10 (85) (1960), 83–122, (In Russian with English Summary.) MR0147779 0089.07305 Google Scholar[4] M. I. Višik and , L. A. Ljusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Amer. Math. Soc. Transl. (2), 20 (1962), 239–364 MR0136861 0122.32402 CrossrefGoogle Scholar[5] K. O. Friedrichs, Nonlinear hyperbolic differential equations for functions of two independent variables, Amer. J. Math., 70 (1948), 555–589 MR0025659 0039.10601 CrossrefISIGoogle Scholar[6] Earl A. Coddington and , Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955xii+429 MR0069338 0064.33002 Google Scholar[7] I. M. Gel'fand and , G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977]xviii+423 MR0435831 0115.33101 Google Scholar[8] J. J. Levin, First order partial differential equations containing a small parameter, J. Rational Mech. Anal., 4 (1955), 481–501 MR0071615 0067.07006 ISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails The Numerical Solution of Singularly Perturbed Nonlinear Partial Differential Equations in Three Space Variables: The Adaptive Explicit Inverse Preconditioning ApproachModelling and Simulation in Engineering, Vol. 2019 | 2 Jan 2019 Cross Ref Topics in singular perturbationsAdvances in Mathematics, Vol. 2, No. 4 | 1 Oct 1968 Cross Ref Volume 8, Issue 4| 1966SIAM Review427-591 History Submitted:25 April 1966Published online:18 July 2006 InformationCopyright © 1966 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1008103Article page range:pp. 479-493ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

The Hamilton-Jacobi Equations for a Relativistic Charged Particle

In the problem of finding the motion of a classical particle one has the choice of dealing with a system of second order ordinary differential equations (Lagrange's equations) or a single first order partial differential equation (the Hamilton-Jacobi equation, henceforth referred to as the H-J equation). In practice the latter method is less frequently used because of the difficulty in finding complete integrals. When these are obtainable, however, the method leads rapidly to the equations of the trajectories. Furthermore it is of fundamental theoretical importance and it provides a basis for quantum mechanical analogues.

Creation of the Heavy Elements by Neutron Capture on a Fast Time Scale

The mathematical problem concerning the creation of the heavy elements by neutron capture on a fast time scale involves the solution of more than a thousand simultaneous differential equations. This set of equations is solved by replacing them with a mathematical equivalent---a single partial differential equation which truns out to assume the form of the equation of continuity. The stream lines of the analogous hydrodynamical problem are determined, from which the problem of nucleogenesis may be solved once the initial condition is given. By assuming a neutron capture mean lifetime of about 0.1 sec for all nonmagic-number nuclides and by making use of the fact that the neutron magic-number nuclides have smaller neutron-capture cross sections, the position, width, and height of the abundance peaks at Br, Xe, and Pt may be determined; these agree reasonably well with the experimental results.

One-Dimensional, Incompressible, Noncapillary, Two-Phase Fluid Flow in a Porous Medium

Published in Petroleum Transactions, AIME, Volume 216, 1959, pages 290–296. Introduction The fundamental equations that are used to describe two-phase fluid flow in porous media are Darcy's law for each phase and an equation of continuity for each component. The special case of one-dimensional, incompressible, two-phase flow has received much attention in the petroleum engineering literature. The basic paper on the subject is that of Buckley and Leverett. Buckley and Leverett showed that one could eliminate the pressure and obtain a single partial differential equation for the saturation: They solved this equation for the case when gravity and capillary forces are negligible. A frequently encountered property of the solution of the basic partial differential equation is that, as time progresses, the saturation becomes a multiple-valued function of the distance coordinate, x. Buckley and Leverett interpreted the formation of multiple values as an indication that the saturation-distance curve had become discontinuous. They developed a method of determining the position of the discontinuity from a material balance. We solve the Buckley-Leverett partial differential equation using the method of characteristics and the concept of shocks. Our treatment is analogous to treatments used in the theory of supersonic compressible flow. Although the numerical results obtained are the same as those obtained by previous methods, we believe that this approach is more logical and has broader applicability. It can, for example, be generalized to treat the case when the fluids are compressible. It is not clear how one would go about doing this using the method of Buckley and Leverett. To illustrate the power of the method of characteristics with shocks, we solve a problem in gravity segregation which has not been treated previously in the literature. A special case of the general method developed here was used by Welge to solve a particular fluid displacement problem. Fundamental Equations We consider two-phase incompressible flow along the vertical direction x in a porous medium. We assume that there is no flow transverse to x. We measure x positive in the upward direction. The fundamental equations are obtained from Darcy's law and the mass conservation law.

Kinetics and Molecular Weight Distributions for Unsteady-State Polymerizations Involving Termination by Chain Transfer with the Monomer

The formal unsteady-state kinetics of a polymerization reaction with first-order initiation and termination by chain transfer with the monomer are solved, and expressions are obtained for number and weight average degrees of polymerization and polymerization rate as a function of time. L. Gold's results for the case of no termination are included as a special case. The qualitative description of the kinetics by a single partial differential equation is also outlined.

A Refined Theory of Elastic, Orthotropic Plates

Abstract A refined theory of elastic, orthotropic plates is presented. The theory includes the effect of transverse shear deformation and normal stress and may be considered a generalization of the classical theory of von Karman modified by the refinements of the Levy-Reissner-Mindlin theories. A nonlinear system of equations is derived directly from the corresponding equations of the three-dimensional theory of elasticity in which body-force terms have been retained. Next, the system of equations is linearized and reduced to a single sixth-order partial differential equation in a stress function. A Levy-type solution of this equation is discussed.

A Donnell Type Theory for Asymmetrical Bending and Buckling of Thin Conical Shells

Abstract Equations, somewhat more accurate than those recently presented by N. J. Hoff, are derived for bending and buckling of thin circular conical shells under arbitrary loading. These equations reduce to Donnell’s equations for thin cylindrical shells when the cone semivertex angle becomes very small and the minimum radius of curvature of the median surface approaches a constant value. At the other end of the scale the equations reduce to the well-known equations for flat circular plates when the cone semivertex angle approaches a right angle. In addition, for the entire range of cone semivertex angles the equations reduce to the known equations for axisymmetrical bending when variations of the displacements around the circumference vanish. The problem of bending is reduced to the solution of a single fourth-order partial differential equation with variable coefficients.

Rectilinear flow of non-Bingham plastic solids and non-Newtonian viscous liquids. I

In a previous paper on steady rectilinear plastic flow of a Bingham solid (1), the point of view was adopted that the study of all possible velocity distributions was equivalent to a study of the geometry of velocity contours in a section normal to the flow. This led to an analysis of the differential geometry of velocity contours in rectilinear plastic flow under zero pressure gradient, and the problem of finding all possible velocity distributions was eventually reduced to the problem of finding the general integral of a single linear second-order partial differential equation with two independent variables. But the form of this differential equation (equation (25) of (1)) was such that only a very limited number of solutions could be found by standard methods. The particular solutions which were obtained corresponded to those flow patterns in which the velocity contours could be identified with a family of parametric curves in a curvilinear coordinate system obtainable by conformal transformation of a cartesian frame of reference. In effect, this means that the results so far achieved by using the new approach to the problem could have been obtained by more direct methods.

III. On the differential equations of dynamics

It is shown in the general paper that if an integral of any one equation of the peculiar system of (partial differential) equations there discussed be found, then if a certain numerical result of subsequent and always possible operations prove odd , an integral of the entire system can be found by the solution of a single differential equation of the first order. It is shown in the paper now sent that, when the above numerical result is even , we can reduce the original system of partial differential equations into a new system, fewer in number by unity at least, and of the same general character, so as to admit of a repetition of the same procedure. Thus the common integral sought will finally be given either by the solution of a single differential equation of the first order, or by finding one integral of the single partial differential equation, which, in the most unfavourable case conceivable, will remain at last.

XXII. On the differential equations of dynamics. A sequel to a paper on simultaneous differential equations

Jacobi, in a posthumous memoir* which has only this year appeared, has developed two remarkable methods (agreeing in their general character, but differing in details) of solving non-linear partial differential equations of the first order, and has applied them in connexion with that theory of the Differential Equations of Dynamics which was esta­blished by Sir W. B. Hamilton in the Philosophical Transactions for 1834‒35. The knowledge, indeed, that the solution of the equations of a dynamical problem is involved in the discovery of a single central function, defined by a single partial differential equation of the first order, does not appear to have been hitherto (perhaps it will never be) very fruitful in practical results. But in the order of those speculative truths which enable us to perceive unity where it was unperceived before, its place is a high and enduring one. Given a system of dynamical equations, it is possible, as Jacobi had shown, to con­struct a partial differential equation such that from any complete primitive of that equation, i. e . from any solution of it involving a number of constants equal to the number of the independent variables, all the integrals of the dynamical equations can be deduced by processes of differentiation. Hitherto, however, the discovery of the com­plete primitive of a partial differential equation has been supposed to require a previous knowledge of the integrals of a certain auxiliary system of ordinary differential equa­tions; and in the case under consideration that auxiliary system consisted of the dynamical equations themselves. Jacobi’s new methods do not require the preliminary integration of the auxiliary system. They require, instead of this, the solution of certain systems of simultaneous linear partial differential equations. To this object therefore the method developed in my recent paper on Simultaneous Differential Equa­tions might be applied. But the systems of equations in question are of a peculiar form. They admit, in consequence of this, of a peculiar analysis. And Jacobi’s methods of solving them are in fact different from the one given by me, though connected with it by remarkable relations. He does indeed refer to the general problem of the solution of simultaneous partial differential equations, and this in language which does not even suppose the condition of linearity. He says, “Non ego hic immorabor qusestioni generali quando et quomodo duabus compluribusve æquationibus differentialibus partialibus una eademque functione Satisfied possit, sed ad casum propositum investigationem restringam. Quippe quo præclaris uti licet artificiis ad integrationem expediendam commodis. ” But he does not, as far as I have been able to discover, discuss any systems of equations more general than those which arise in the immediate problem before him.

III. On the differential equations of dynamics. A sequel to a paper on simultaneous differential equations

Jacobi in a posthumous memoir, which has only this year appeared, has developed two remarkable methods (agreeing in their general character, but differing in details) of solving non-linear partial differential equations of the first order, and has applied them in connexion with that theory of the differential equations of dynamics which was established by Sir W. R. Hamilton in the 'Philosophical Transactions’ for 1834-35. The knowledge, indeed, that the solution of the equation of a dynamical problem is involved in the discovery of a single central function, defined by a single partial differential equation of the first order, does not appear to have been hitherto (perhaps it will never be) very fruitful in practical results.