Equation elliptic differential

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. 

Chapters 2-5 are concerned with concrete difference schemes for equations of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on homogeneous difference schemes for ordinary differential equations, by means of which we try to solve the canonical problem of the theory of difference schemes in which a primary family of difference schemes is specified (in such a case the availability of the family is provided by pattern functionals) and schemes of a desired quality should be selected within the primary family. This problem is solved in Chapter 3 using a particular form of the scheme and its solution leads us to conservative homogeneous schemes. 

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

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. 

The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

In these formulas the symbol Za(co) stands for the Bessel function, sn (to), dn ( ), cn ( ) are the Jacobi elliptic functions having the module /(xvxv) is the general solution of the ordinary differential equation... 

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

E. Hopf, A remark on linear elliptic differential equations of the second order, Proc. Amer. Math. Soc., 3 (1952), pp. 791-793. 

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... 

The solution of this differential equation could be expressed in terms of elliptic functions. The transverse-vibration frequency v is determined by the formula analogous to Eqs. (431)-(432b) ... 

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. 

---