Elliptic Partial Differential Equations

The first European woman mathematician to earn a doctoral degree. Worked on partial differential equations, elliptic integrals, mathematical physics, and classical and celestial mechanics. 

Regarded as an equation for e, this is a member of the class of elliptic partial differential equations for which a maximum principle is satisfied [76], SO e is required to take its greatest and least values on the... 

Differential methods - in these techniques the internal grid coordinates are found via the solution of appropriate elliptic, parabolic or hyperbolic partial differential equations. 

Given a partial differential equation of the elliptic form... 

A generalized partial differential equation solver which handles simultaneous parabolic, one dimensional elliptic, ordinary and integral equations and uses B-splines with an adaptive grid was written to solve the model. Further details on the model and solution method can be found in Reference 14. 

The description of phenomena in a continuous medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of wave propagation are described by a class of partial differential equations called hyperbolic, and these are essentially different in their properties from other classes such as those that describe equilibrium ( elliptic ) or diffusion and heat transfer ( parabolic ). Prototypes are ... 

The numerical methods for partial differential equations can be classified according to the type of equation (see Partial Differential Equations ) parabolic, elliptic, and hyperbolic. This section uses the finite difference method to illustrate the ideas, and these results can be programmed for simple problems. For more complicated problems, though, it is common to rely on computer packages. Thus, some discussion is given to the issues that arise when using computer packages. 

S. V. Parter, Mildly nonlinear elliptic partial differential equations and their numerical solutions, I, Numero. Math., 7 (1965), pp. 113-128. 

One important use of the stream function is for the visualization of flow fields that have been determined from the solution of Navier-Stokes equations, usually by numerical methods. Plotting stream function contours (i.e., streamlines) provides an easily interpreted visual picture of the flow field. Once the velocity and density fields are known, the stream function field can be determined by solving a stream-function-vorticity equation, which is an elliptic partial differential equation. The formulation of this equation is discussed subsequently in Section 3.13.1. Solution of this equation requires boundary values for l around the entire domain. These can be evaluated by integration of the stream-function definitions, Eqs. 3.14, around the boundaries using known velocities on the boundaries. For example, for a boundary of constant z with a specified inlet velocity u(r),... 

The theory for classifying linear, second-order, partial-differential equations is well established. Understanding the classification is quite important to understanding solution algorithms and where boundary conditions must be applied. Partial differential equations are generally classified as one of three forms elliptic, parabolic, or hyperbolic. Model equations for each type are usually stated as... 

The steady-state heat equation (Eq. 3.284) is often used as the model equation for an elliptic partial-differential equation. An important property of elliptic equations is that the solution at any point within the domain is influenced by every point on the boundary. Thus boundary conditions must be supplied everywhere on the boundaries of the solution domain. The viscous terms in the Navier-Stokes equations clearly have elliptic characteristics. 

Steady parallel flow can be realized in ducts of essentially arbitrary cross section. A linear elliptic partial differential equation must be solved to determine the velocity field and the shear stresses on the walls. For an incompressible, constant-viscosity fluid, the axial momentum equation states that... 

The circumferential ((j>) momentum equation is a partial differential equation. Identify some of its basic properties. Is it elliptic, parabolic, or hyperbolic Is it linear of nonlinear What is its order ... 

This appendix provides a detailed description of how to build Excel spreadsheet solutions for several of the problems that were presented and solved in Chapter 4. Generically, these include an ordinary-differential-equation boundary-value problem, a one-dimensional parabolic partial differential equation, and a two-dimensional elliptic partial differential equation. 

Fig. D.6 Spreadsheet using an iterative method to solve the two-dimensional elliptic partial differential equations describing the axial velocity in a rectangular channel. The problem is described and discussed in Section 4.4.

Allen and Severn (A3, A4) demonstrate how relaxation methods, originally developed for elliptic partial differential equations, can be extended to the heat conduction equation. With elliptic equations, the value of the dependent variable at any mesh point is determined by all... 

E.J. Kansa. Multiquadratics- a scattered data approximation scheme with applications to computational fluid mechanics, ii-solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers Math. Applic., 19 147, 1990. 

Poisson-Boltzmann equation — The Poisson-Boltz-mann equation is a nonlinear, elliptic, second-order, partial differential equation which plays a central role, e.g., in the Gouy-Chapman (- Gouy, - Chapman) electrical -> double layer model and in the - Debye-Huckel theory of electrolyte solutions. It is derived from the classical -> Poisson equation for the electrostatic potential... 

P. Concus, G. H. Golub, and D. P. O Leary, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, Eds., Academic Press, New York, 1976, pp. 309—332. A Generalized Conjugate Gradient Method for the Numerical Solution of Elliptic Partial Differential Equations. 

An orthogonal collocation method for elliptic partial differential equations is presented and used to solve the equations resulting from a two-phase two-dimensional description of a packed bed. Comparisons are made between the computational results and experimental results obtained from earlier work. Some qualitative discrimination between rival correlations for the two-phase model parameters is possible on the basis of these comparisons. The validity of the numerical method is shown by applying it to a one-phase packed-bed model for which an analytical solution is available problems arising from a discontinuity in the wall boundary condition and from the semi-infinite domain of the differential operator are discussed. 

In this paper the coupled elliptic partial differential equations arising from a two-phase homogeneous continuum model of heat transfer in a packed bed are solved, and some attempt is made to discriminate between rival correlations for those parameters not yet well-established, by means of a comparison with experimental results from a previous study (, 4). 

The use of piecewise bicubic Hermite functions in collocation schemes for the solution of elliptic partial differential equations has been described by Prenter (13,17) a short outline is presented here. 

The orthogonal collocation method using piecewise cubic Her-mite polynomials has been shown to give reasonably accurate solutions at low computing cost to the elliptic partial differential equations resulting from the inclusion of axial conduction in models of heat transfer in packed beds. The method promises to be effective in solving the nonlinear equations arising when chemical reactions are considered, because it allows collocation points to be concentrated where they are most effective. 

Systems of (elliptic) partial differential equations (Thompson et al, 1982, 1985 Thompson, 1982)... 

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