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Linearized equations

This reduces the calculation at each step to solution of a set of linear equations. The program description and listing are given in Appendix H. [Pg.99]

Using the formulas listed above, it is possible to reach the non-linear equation... [Pg.266]

With constant steps and v = n eq.(5) can be described as linear equation system presented in matrix form ... [Pg.367]

Constant steps are not necessary, but they simplify the matrix g of eq.(6). Eq.(5) and eq.(6) respectively show the relationship between input and output signal for discrete signal processing. It is given by a linear equation system, which can easily be solved. [Pg.367]

In this section we discuss the frequency spectrum of excitations on a liquid surface. Wliile we used linearized equations of hydrodynamics in tire last section to obtain the density fluctuation spectrum in the bulk of a homogeneous fluid, here we use linear fluctuating hydrodynamics to derive an equation of motion for the instantaneous position of the interface. We tlien use this equation to analyse the fluctuations in such an inliomogeneous system, around equilibrium and around a NESS characterized by a small temperature gradient. More details can be found in [9, 10]. [Pg.725]

Detailed derivations of the isothemi can be found in many textbooks and exploit either statistical themio-dynaniic methods [1] or independently consider the kinetics of adsorption and desorption in each layer and set these equal to define the equilibrium coverage as a function of pressure [14]. The most conmion fomi of BET isothemi is written as a linear equation and given by ... [Pg.1874]

Dennis J E and Schnabel R B 1983 Numerical Methods for Unconstrained Optimization and Non-linear Equations (Englewood Cliffs, NJ Prentice-Hall)... [Pg.2355]

Farkas O and Schlegel H B 1998 Methods for geometry optimization In large molecules. I. An O(N ) algorithm for solving systems of linear equations for the transformation of coordinates and forces J. Chem. Phys. 109 7100... [Pg.2357]

The LIN method ( Langevin/Implicit/Normal-Modes ) combines frequent solutions of the linearized equations of motions with anharmonic corrections implemented by implicit integration at a large timestep. Namely, we express the collective position vector of the system as X t) = Xh t) + Z t). (In LN, Z t) is zero). The first part of LIN solves the linearized Langevin equation for the harmonic reference component of the motion, Xh t)- The second part computes the residual component, Z(t), with a large timestep. [Pg.246]

The solution Xh(t) of the linearized equations of motion can be solved by standard NM techniques or, alternatively, by explicit integration. We have experimented with both and found the second approach to be far more efficient and to work equally well. Its handling of the random force discretization is also more straightforward (see below). For completeness, we describe both approaches here. [Pg.247]

C. Lanczos. Solution of systems of linear equations by minimized iterations. J. Res. Nat. Bureau Standards, 49 33-53, 1952. [Pg.431]

To know how linear equations can be used for calculation of enthalpies of gas-phase reactions... [Pg.319]

Equations (8.10) form a set of n independent linear equations for the flux contributions. In principle these can be solved (though in... [Pg.72]

Ire boundary element method of Kashin is similar in spirit to the polarisable continuum model, lut the surface of the cavity is taken to be the molecular surface of the solute [Kashin and lamboodiri 1987 Kashin 1990]. This cavity surface is divided into small boimdary elements, he solute is modelled as a set of atoms with point polarisabilities. The electric field induces 1 dipole proportional to its polarisability. The electric field at an atom has contributions from lipoles on other atoms in the molecule, from polarisation charges on the boundary, and where appropriate) from the charges of electrolytes in the solution. The charge density is issumed to be constant within each boundary element but is not reduced to a single )oint as in the PCM model. A set of linear equations can be set up to describe the electrostatic nteractions within the system. The solutions to these equations give the boundary element harge distribution and the induced dipoles, from which thermodynamic quantities can be letermined. [Pg.614]

Equation (2.45) represents the weighted residual statement of the original differential equation. Theoretically, this equation provides a system of m simultaneous linear equations, with coefficients Q , i = 1,... m, as unknowns, that can be solved to obtain the unknown coefficients in Equation (2.41). Therefore, the required approximation (i.e. the discrete solution) of the field variable becomes detemfined. [Pg.42]

Iterative improvement of the solution of systems of linear equations... [Pg.207]

The least squares derivation for quadratics is the same as it was for linear equations except that one more term is canied through the derivation and, of course, there are three normal equations rather than two. Random deviations from a quadratic are ... [Pg.66]

Fit a linear equation to the following data set and give the uncertainties of the slope and intercept (Fig. 3-2). [Pg.72]

The method of finding uncertainty limits for linear equations can be generalized to higher-order polynomials. The matrix method for finding the minimization... [Pg.76]

A curve fit is then done to create a linear equation, such as... [Pg.244]

Revised material in Section 5 includes an extensive tabulation of binary and ternary azeotropes comprising approximately 850 entries. Over 975 compounds have values listed for viscosity, dielectric constant, dipole moment, and surface tension. Whenever possible, data for viscosity and dielectric constant are provided at two temperatures to permit interpolation for intermediate temperatures and also to permit limited extrapolation of the data. The dipole moments are often listed for different physical states. Values for surface tension can be calculated over a range of temperatures from two constants that can be fitted into a linear equation. Also extensively revised and expanded are the properties of combustible mixtures in air. A table of triple points has been added. [Pg.1287]

These are linear equations which give the symmetry of the strain tensor Sij = Sji- In the general case, the strain tensor is nonlinear,... [Pg.2]

Linear equations for Ford, Shell, and Zahn cups (158). [Pg.182]

Linear equations of the type u = ct — C, where c and C are constants, relate kinematic viscosity to efflux time over limited time ranges. This is based on the fact that, for many viscometers, portions of the viscosity—time curves can be taken as straight lines over moderate time ranges. Linear equations, which are simpler to use in determining and applying correction factors after caUbration, must be appHed carefully as they do not represent the tme viscosity—time relation. Linear equation constants have been given (158) and are used in ASTM D4212. [Pg.182]

Rearrangement yields a set of linear equations with two unknowns ... [Pg.244]


See other pages where Linearized equations is mentioned: [Pg.223]    [Pg.880]    [Pg.880]    [Pg.39]    [Pg.721]    [Pg.723]    [Pg.184]    [Pg.249]    [Pg.422]    [Pg.444]    [Pg.448]    [Pg.204]    [Pg.244]    [Pg.113]    [Pg.134]    [Pg.245]    [Pg.168]    [Pg.168]    [Pg.412]    [Pg.101]    [Pg.510]    [Pg.354]    [Pg.315]    [Pg.243]   
See also in sourсe #XX -- [ Pg.31 ]




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B Some General Consequences of Linearity and the Creeping-Flow Equations

Balance equation, linear

Classification and Characteristics of Linear Equations

Conservation equation linear momentum

Constitutive equation linear regime

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Coupled Linear Differential Equations

Coupled-Cluster Linear Response Equations

Coupling constants, equations, linear

Current-overpotential equation linearization

Difference equation linear

Difference equations linear finite

Differential equation linear integrators

Differential equation, linear, boundary

Differential equation, linear, boundary general solution

Differential equation, linear, boundary particular solutions

Differential equations linear, order

Diffusion equation linearization

Dispersion linearized equation

Equation first-order linear

Equation for Linear Viscoelasticity in Simple Shear

Equation linear form

Equation linearized frequency

Equation second-order linear

Equations linear

Equations linear

Equations linear stability continuity

Equations linear stability momentum

Equilibrium equations linear elastic solid

Ethyl salicylate linear regression equations

Exact Solutions of Linear Heat and Mass Transfer Equations

First order linear ordinary differential equations

First-order differential equations linear, solution

First-order linear homogeneous equations

Fitting Experimental Data to Linear Equations by Regression

General form of steady-state kinetic equation for complex catalytic reactions with multi-route linear mechanisms

Gibbs adsorption equation linear

Governing Equations and Solutions for Linear Elasticity

Higher order linear ordinary differential equations

Homogeneous Linear Differential Equations with Constant Coefficients

Homogeneous Linear Equations with Constant Coefficients

Homogeneous Linear Second-Order Differential Equations

Homogeneous linear differential equation

Homogeneous linear equation

Homogeneous linear equation, solution

Inhomogeneous linear equations

Integral equations linearized

Integrated rate equations linear form

Kinetic equations are non-linear

Kinetic equations, linear forms

Kinetic rate equation, linear

Kriging system of linear equations

Langevin equation linear response theory

Langevin equation, linear

Langmuir equation linear form

Limits Langevin equation, linear

Linear Dependence, Dimensionality, and Gibbs-Duhem Equations

Linear Equations of Higher Order

Linear Fokker-Planck equation

Linear Hansch equation

Linear Higher-Order Differential Equations

Linear Parabolic Partial Differential Equations

Linear constitutive equation

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Linear differential equation

Linear differential equation Exact

Linear diffusion equation

Linear diffusion equations, solution

Linear energy transfer equations

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Linear equations systems

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Linear expansions and eigenvalue equations

Linear first-order differential equations

Linear free energy equations

Linear free energy relationship equations

Linear free energy relationships Hammett equation

Linear inhomogeneous differential equations

Linear integral equations

Linear momentum balance equations

Linear nonhomogeneous simultaneous equations

Linear operator Characteristic equation

Linear operator equations and their solution by iterative methods

Linear ordinary differential equations

Linear parametric equations

Linear partial differential equations

Linear prediction equation

Linear rate equation

Linear reaction equation

Linear regression equations

Linear response equations

Linear test equation

Linear transport equations, liquid phase

Linear using solvation equation

Linear viscoelasticity constitutive equation

Linear wave equations, nonlinear light

Linear-equation method

Linearization of Model Equations

Linearization of rate equation

Linearization, linearized equations

Linearization, linearized equations

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Linearized Boltzmann equation

Linearized Form of the Integrated Michaelis-Menten Equation

Linearized Form of the Michaelis-Menten Equation

Linearized PB equation

Linearized Poisson-Boltzmann equation

Linearized Poisson-Boltzmann equation function

Linearized Poisson-Boltzmann equation, solution

Linearized System Equations

Linearly dependent equations

Linearly independent equations

Linearly independent stoichiometric equation

MATLAB linear equations

Matrices and Systems of Linear Equations

Matrices for solving sets of linear equations

Matrix linear equations

Matrix models linear difference equations

Matrix solution for simultaneous linear equations

Michaelis parameters estimation: linearized rate equations

Multiple linear regression equations

Non-linear Poisson Boltzmann equation

Non-linear equations

Non-linear flux equations in electro-kinetic phenomena

Non-linear transport equations in gaseous medium

Nonhomogeneous Linear Second-Order Differential Equations

Nonlinear phenomena linearized equations

Nonlinear system linear differential equations

Numeric calculation linear equation system

Numerical Solution of Linear Equations

Numerical Solution of Simultaneous Linear Algebraic Equations

Numerical methods linear equations

Ordinary differential equation linearization

Ordinary differential equations linear homogeneous

Oxidation kinetics linear rate equation

Partial differential equation first-order linear

Partial differential equation quasi linear

Partial differential equation second-order linear

Partial differential equations linear second-order hyperbolic

Phase diffusion equation linearized

Phenomenological equations linear

Poisson-Boltzmann equation, linear

Rate equations linearization, 49 Michaelis parameters

Rate equations, linear mechanism

Reduced Linear Equations

Reduced linear equations method

Regression line linear equation

Row Reduction and Systems of Linear Equations

Schrodinger equation linear property

Second Order Linear Constant Coefficient Equation

Similarity solutions linear diffusion equation

Simultaneous Solution of Linear Equations

Simultaneous algebraic equations linear

Simultaneous linear equation

Simultaneous linear equations, solving

Solution Methods for Linear Finite Difference Equations

Solution of Linear Algebraic Equations

Solution of Linear Equation Systems

Solution of Linear Equations

Solution of Simultaneous Linear Algebraic Equations

Solution of the Linearized P-B Equation

Solving Linear and Nonlinear Equations

Solving Sets of Linear Equations

Solving Sets of Simultaneous Linear Equations

Solving Systems of Linear Algebraic Equations

Solving Systems of Linear Equations

Solving linear equations

Solving linear equations (Newtons method)

Static Linear Response of the Schrodinger Equation

Stirred-tank reactor linear equations

System of implicit non-linear equations the Newton-Raphson method

System of linear differential equations

Systems of linear algebraic equations

Systems of linear equations

Systems of linear equations and their general solutions

Systems of non-linear algebraic equations

Systems of non-linear equations

The Linearized Poisson-Boltzmann Equation

Underdetermined system of linear equations

Validity of linear phenomenological equations

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