Various Types Of Equations Research Articles

Interpolated differential operator (IDO) scheme for solving partial differential equations

We present a numerical scheme applicable to a wide variety of partial differential equations (PDEs) in space and time. The scheme is based on a high accurate interpolation of the profile for the independent variables over a local area and repetitive differential operations regarding PDEs as differential operators. We demonstrate that the scheme is uniformly applicable to hyperbolic, ellipsoidal and parabolic equations. The equations are solved in terms of the primitive independent variables, so that the scheme has flexibility for various types of equations including source terms. We find out that the conservation holds accurate when a Hermite interpolation is used. For compressible fluid problems, the shock interface is found to be sharply described by adding an artificial viscosity term.

Some Considerations on Modeling Heat and Mass Transfer in Porous Media

In this paper some considerations are presented about the equations needed to set up a model of the process of heat and mass transfer in porous media. A clear classification is made of the various types of equations used and of their physical meaning. Special attention is paid to the thermodynamic equilibrium equations and to their derivation since they are too often taken for granted. The importance of the various transport mechanisms (of mass and energy) is analyzed and the consequences that can arise when some term is neglected are indicated.

Pre-asymptotic Expansions

We introduce the concepts of pre-asymptotic schemes and pre-asymptotic expansions to study the divergent series that formally are solutions of various types of equations.

Dynamical effects from equation of state on topographies and geoid anomalies due to internal loading

The effects of mantle compressibility on geoid and topographic signatures caused by internal loading in the mantle are studied. For a Cartesian compressible model the Green's functions of these geophysical signatures are determined by solving a two‐point boundary value problem with depth for each horizontal wave number associated with the density anomaly. Various types of equations of state have been considered, which include both constant and depth‐dependent thermodynamic parameters. Surface topographies are not changed too much by the effects of mantle compressibility, even for long wavelengths. The deformation of bottom boundary is not seriously influenced by mantle compressibility, except for long wavelengths and large viscosity contrasts between the upper and lower mantles. Geoid signals with horizontal wavelengths, exceeding 10,000 km, can be greatly influenced by mantle compressibility. For large viscosity contrasts, differences of up to 100% between incompressible and compressible geoid responses can be found. The admittance function also shows great sensitivity to mantle compressibility at long wavelengths. There are significant differences in the geophysical signatures from the ways in parameterizing the viscosity stratification in the mantle. An exponentially dependent viscosity model produces smaller geoid and surface topographical signals than for a single step function viscosity model.

Methods and Traditions of Babylonian Mathematics, II: An Old Babylonian Catalogue Text with Equations for Squares and Circles

A General Discussion of the Equations and Their Solutions The tablet BM 80209, published as CT 44 39, was described in the of CT 44 as a mathematical text whichdeals with areas; note, in line 4, the difficult term di-ik-9um (see CAD 4, p. 138b). The text is, in fact, a systematically arranged catalogue of mathematical equations of a number of different types, each represented by one or several examples. The various types of equations will be considered below in the order in which they appear on the tablet. For the sake of greater clarity and more exact rendering of the original I have chosen to depart slightly from the standard way of transliterating Babylonian sexagesimal numbers and special signs for fractions. Thus I will write 8 06 40 rather than 8;6,40 or 8,6;40, for example, and 3, 2', 3' rather than 2/3, 1/2, and 1/3; igi-4-gil, igi-5gil, and so on will, however, be written as 1/4, 1/5 and the like.

A modification of Miller's recurrence algorithm

Miller's recurrence algorithm for tabulating the subdominant solution of a second-order difference equation is modified so as to take the asymptotic behaviour of the solution into account. The asymptotic solutions of various types of equations are listed, and a method is given for estimating the error in the tabulated solution.

Dimensional analysis in mathematical biology I. General discussion

Dimensional analysis is discussed from the viewpoint of its basic group properties and shown to be an algebraic Abelian group that is useful for analysis of physical measurements. The application of the method to various types of equations and the formulation of previously unclassified dimensions are discussed. Functional dimensional analysis is applied to the problems of cell size and biomass proliferation; future applications are also noted. A number of dimensionless terms have been formulated for cellular physiochemical phenomena. They apparently represent the first systematic study of biological dimensionless numbers recorded in the literature. A dimensionless proliferation law is suggested. A brief analysis of the physical dimensionality associated with information measures is carried out. Entropy and “information” are shown to be completely different in their dimensional meaning; other informational measures of possible interest in biology are proposed. The dimensional coding and computor analysis of biomathematical equations is suggested.