Related Partial Differential Equations Research Articles

From individuals to epidemics.

Heterogeneous mixing fundamentally changes the dynamics of infectious diseases; finding ways to incorporate it into models represents a critical challenge. Phenomenological approaches are deficient in their lack of attention to underlying process; individual-based models, on the other hand, may obscure the essential interactions in a sea of detail. The challenge then is to find ways to bridge these levels of description, starting from individual-based models and deriving macroscopic descriptions from them that retain essential detail, and filter out the rest. In this paper, attempts to achieve this transformation are described for a class of models where nonrandom mixing arises from the spatial localization of interactions. In general, the epidemic threshold is found to be larger owing to spatial localization than for a homogeneously mixing population. An improved estimate of the dynamics is developed by the use of moment equations, and a simple estimate of the threshold in terms of a 'dyad heuristic'. For more general models in which local infection is not described by mass action, the connection with related partial differential equations is investigated.

On Hele–Shaw flow of power-law fluids

This paper reviews the governing equations for a plane Hele–Shaw flow of a power-law fluid. We find two closely related partial differential equations, one for the pressure and one for the stream function. Some mathematical results for these equations are presented, in particular some exact solutions and a representation theorem. The results are applied to Hele–Shaw flow. It is then possible to determine the flow near an arbitrary corner for any power-law fluid. Other examples are also given.

Convergence to equilibrium of critical branching particle systems and superprocesses, and related nonlinear partial differential equations

We consider a class of multitype particle systems in ℝ d undergoing spatial diffusion and critical stable multitype branching, and their limits known as critical stable multitype Dawson-Watanabe processes, or superprocesses. We show that for large classes of initial states, the particle process and the superprocess converge in distribution towards known equilibrium states as time tends to infinity. As an application we obtain the asymptotic behavior of a system of nonlinear partial differential equations whose solution is related to the distribution of both the particle process and the superprocess.

On manpower planning in the presence of learning

Manpower planning models rarely incorporate effects of learning on production capacity. Here those effects are explored through some models. First, we discuss a discrete-time model in which capacity is maintained at a desired level through appropriate recruitment and layoffs. Then we propose a related continuous-time partial differential equation model, and derive the recruitment rate level which will maintain capacity. Some examples are provided.

Parabolic decomposition of the separable Helmholtz equation

The solution of the separable Helmholtz equation is represented as the convolution of solutions of two related (formally) parabolic partial differential equations. Separable means that the square of the index of refraction is a function of depth plus a function of range and that the depth of the ocean is assumed to be constant. This representation decomposes the separable Helmholtz equation into two parabolic partial differential equations. One equation is the usual Fock-Tappert parabolic equation for sound propagation. The solutions of the second parabolic equation may be regarded as candidate kernels of special integral transformations, or transmutations. All of these equations have many solutions, of course, so initial and/or boundary conditions are presented that will determine solutions of the parabolic equations so that their convolution will give the Green's function for the Helmholz equation. Examples of such transmutation representations will be presented.

Parabolic decomposition of the separable Helmholtz equation

The solution of the separable Helmholtz equation is represented as the convolution of solutions of two related (formally) parabolic partial differential equations. Separable means that the square of the index of refraction is a function of depth plus a function of range and that the depth of the ocean is assumed to be constant. This representation decomposes the separable Helmholtz equation into two parabolic partial differential equations. One equation is the usual Fock‐Tappert parabolic equation for sound propagation. The solutions of the second parabolic equation may be regarded as candidate kernels of special integral transformations, or transmutations. All of these equations have many solutions, of course, so we present initial and/or boundary conditions that will determine solutions of the parabolic equations so that their convolution will give the Green's function for the Helmholtz equation. Examples of such transmutation representations will be presented.

Stable equilibrium configurations of elastic crystals

Nonlinear thermoelasticity theory for crystals is generating some knotty questions in the calculus of variations and in the theory of related partial differential equations. This stems from invariance assumptions which are suggested by molecular theory, and seem to be needed to analyze some commonly observed phenomena, such as twinning. What might be regarded as the simplest problem is to characterize the most stable configurations of unloaded crystals. Even this is difficult, because these are not all unique, but form infinite sets. A number of these do seem to match configurations observed in real crystals. After elaborating this a bit, I will present methods for constructing solutions which are special, but resemble configurations which are observed.

Nonlinear lattice and soliton theory

Since the notion of stable pulses, known as solitons, plays a central role in the phenomena of wave propagation in nonlinear systems, an exposition of this topic is developed in some detail. It is known that the equations of motion of the one-dimensional lattice of particles with exponential interaction are integrable, namely, they admit exact solutions, and this system is equivalent to an LC circuit with certain nonlinear capacitance. In addition, a closely related partial differential equation called the Korteweg-de Vries (KdV) equation is also integrable. Special emphasis is placed on these integrable systems.

A systematic method for the solution of some nonlinear evolution equations. I. The Burgers equations

The eigenfunction method introduced by Van Kampen (1977) to solve a Fokker-Planck equation is extended and applied to a related partial differential equation. From this analysis we are able to obtain the solution of the Burgers and the damped Burgers equation in a systematic way.

Operator waves in Hilbert space and related partial differential equations

The success of the theory of characteristic functions of nonselfadjoint operators and its applications to the System Theory [1–17] is the inspiration for attempts towards creating a general theory in the much more complicated case of several commuting nonselfadjoint operators. In this paper we study the close relations between sets of commuting operators in Hilbert space and related systems of partial differential equations. At the same time a generalization of the classical Cayley Hamilton Theorem, in the case of two commuting operators, is obtained which leads to unexpected connections with the theory of algebraic curves.

Prolongation structures of nonlinear evolution equations

A technique is developed for systematically deriving a ’’prolongation structure’’—a set of interrelated potentials and pseudopotentials—for nonlinear partial differential equations in two independent variables. When this is applied to the Korteweg−de Vries equation, a new infinite set of conserved quantities is obtained. Known solution techniques are shown to result from the discovery of such a structure: related partial differential equations for the potential functions, linear ’’inverse scattering’’ equations for auxiliary functions, Bäcklund transformations. Generalizations of these techniques will result from the use of irreducible matrix representations of the prolongation structure.

Solution of a nonlinear partial differential equation with initial conditions

The exact solution ϕ \phi of a particular nonlinear partial differential equation is obtained in terms of solution u of a related linear partial differential equation. It is noted that solution ϕ \phi may be found subject to initial conditions if certain initial conditions can be determined for solution u. Two examples are solved explicitly.

Errata to “Hypergeometric operator series and related partial differential equations”

Errata to “Hypergeometric operator series and related partial differential equations”

Two- and three-dimensional potential flow analysis by the method of singularities

The principles of a finite element technique for the computation of two- and three-dimensional potential flows, based on the method of singularities are outlined. The geometry of the surface is approximated by curved elements and the source density distribution is represented by Lagrangian interpolation polynomials. The methods described are applicable to any field problems defined by Poisson's, Helmholtz's or related elliptic partial differential equations.

High-order polynomial triangular finite elements for potential problems

An analytic derivation is given for high-accuracy triangular finite elements useful for numerical solution of field problems involving Laplace's, Poisson's, Helmholtz's, or related elliptic partial differential equations in two dimensions. General expressions are developed for complete polynomial fields of arbitrarily high order, and the method for obtaining element describing matrices is shown. These matrices can always be written in terms of trigonometric functions of the vertex angles, and the triangle area, multiplied by certain numerical coefficient matrices which are the same for any triangle. For polynomial fields up to fourth order, the numerical coefficient matrices are given, so that the element matrices for any triangle can be found easily. Use of these new elements is illustrated by a simple vibration problem.

Hypergeometric operator series and related partial differential equations

Hypergeometric operator series and related partial differential equations

An operator calculus for related partial differential equations

An operator calculus for related partial differential equations