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Solution of Equation

The equations are solved by Gaussian elimination as described by Richtmyer and Morton (1967) and by Roache (1972). We perform a standard LU decomposition of the coefficient matrix V on the left-hand side [Pg.73]

Putting U = Y, we then solve successively, by back-substitution, the equations [Pg.73]

This solution allows us to proceed from the 6 to the boundary of the flame, after which a similar procedure is carried out in reverse from to 6 to give the and so complete the cycle of computation for the updating of the values of the dependent variables. The algorithm is [Pg.73]


Let us assume that stress gradient in axial direction is present but smooth. Then we can use a perturbation method and expand the solution of equation (30) in a series. The first term of this expansion will be a solution of the plane strain problem and potential N will be equal to zero. The next terms of the stress components will contain potential N also. [Pg.138]

Using the orbitals, ct)(r), from a solution of equation Al.3.11, the Hartree many-body wavefunction can be constructed and the total energy detemiined from equation Al.3,3. [Pg.90]

All equations of two variables, such as equation (A2.1.12). are necessarily integrable because they can be written in the fonn dy/dx = fix, y), which detennines a unique value of the slope of the line tln-ough any point (x, y). Figure A2.1.4 shows a set of non-intersecting lines in V-Q space representing solutions of equation (A2.1.12F... [Pg.334]

We then write the solution of equation B1.5.7 as a power series expansion hi temis of the strength X of the perturbation ... [Pg.1268]

The amplitude of the response, x 2ca), is given by the steady-state solution of equation B1.5.10 as... [Pg.1269]

Surfaces in polar solvents and particularly in water tend to be charged, tlirough dissociation of surface groups or by adsorjDtion of ions, resulting in a charge density a. Near a flat surface, < ) only depends on the distance x from the surface. The solution of equation (C2.6.6) then is... [Pg.2677]

Though the solution procedure sounds straightforward, if tedious, practice difficulty is encountered immediately because of the implicit nature of the available flux models. As we saw in Chapter 5 even the si lest of these, the dusty gas model, has solutions which are too cumbersc to be written down for more than three components, while the ternary sol tion itself is already very complicated. It is only for binary mixtures therefore, that the explicit formulation and solution of equations (11. Is practicable. In systems with more than two components, we rely on... [Pg.111]

Stewart s argument provides a prescription for constructing a solution of equations (11.61) - (11.63) provided the matrix Is nonsingular for all relevant values of jc, and provided the differential equations (11.64) and (11.65) have solutions consistent with their boundary conditions. It is possible, in principle, to check the nonsingularity of for any... [Pg.143]

Let us first consider the standard Galerkin solution of Equation (2.80) obtained using the previously described steps. [Pg.55]

After the aussembly of elemental equations into a global set and imposition of the boundary conditions the final solution of the original differential equation with respect to various values of upwinding parameter jS can be found. The analytical solution of Equation (2.80) with a = 50 is found as... [Pg.61]

The right-hand side in Equation (6.18) is known and hence its solution yields the error 5x in the original solution. The procedure can be iterated to improve the solution step-by-step. Note that implementation of this algorithm in the context of finite element computations may be very expensive. A significant advantage of the LU decomposition technique now becomes clear, because using this technique [A] can be decomposed only once and stored. Therefore in the solution of Equation (6.18) only the right-hand side needs to be calculated. [Pg.207]

The wave functions ij/ resulting from solution of Equation (1.66) are... [Pg.25]

For an isothermal system the simultaneous solution of equations 30 and 31, subject to the boundary conditions imposed on the column, provides the expressions for the concentration profiles in both phases. If the system is nonisotherm a1, an energy balance is also required and since, in... [Pg.261]

The solution of equation 16 is a decreasing, simple exponential where = k ([A ] + [P ]) + k. The perturbation approach generates small deviations in concentrations that permit use of the linearized differential equation and is another instance of pseudo-first-order behavior. Measurements over a range of [A ] + [T ] allow the kineticist to plot against that quantity and determine / ftom the slope and from the intercept. [Pg.510]

Work in the area of simultaneous heat and mass transfer has centered on the solution of equations such as 1—18 for cases where the stmcture and properties of a soHd phase must also be considered, as in drying (qv) or adsorption (qv), or where a chemical reaction takes place. Drying simulation (45—47) and drying of foods (48,49) have been particularly active subjects. In the adsorption area the separation of multicomponent fluid mixtures is influenced by comparative rates of diffusion and by interface temperatures (50,51). In the area of reactor studies there has been much interest in monolithic and honeycomb catalytic reactions (52,53) (see Exhaust control, industrial). Eor these kinds of appHcations psychrometric charts for systems other than air—water would be useful. The constmction of such has been considered (54). [Pg.106]

The example demonstrates that not all the B-numbers of equation 5 are linearly independent. A set of linearly independent B-numbers is said to be complete if every B-number of Dis a product of powers of the B-numbers of the set. To determine the number of elements in a complete set of B-numbers, it is only necessary to determine the number of linearly independent solutions of equation 13. The solution to the latter is well known and can be found in any text on matrix algebra (see, for example, (39) and (40)). Thus the following theorems can be stated. [Pg.106]

In terms of linear vector space, Buckingham s theorem (Theorem 2) simply states that the null space of the dimensional matrix has a fixed dimension, and Van Driest s rule (Theorem 3) then specifies the nullity of the dimensional matrix. The problem of finding a complete set of B-numbers is equivalent to that of computing a fundamental system of solutions of equation 13 called a complete set of B-vectors. For simplicity, the matrix formed by a complete set of B-vectors will be called a complete B-matrix. It can also be demonstrated that the choice of reference dimensions does not affect the B-numbers (22). [Pg.106]

The numerieal solution of equation 4.35 is suffieient in most eases to provide a reasonable answer for reliability with multiple load applieations for any eom bination of loading stress and strength distribution (Freudenthal et al., 1966). [Pg.185]

Several standard seetion sizes for unequal angles are listed in Table 4.15 (BS 4360,1990). For eaeh standard seetion, first the statistieal variation of the area properties, distanees and angles ean be estimated using Monte Carlo simulation, whieh are then used to determine stresses at points A and B on the seetion from solution of equations 4.108 and 4.111. The stresses found at points A and B for the seetions listed are also shown in Table 4.11. [Pg.239]

Consider the solution of Equation 6-170 for eaeh of the four types of rate expressions to determine the optimum temperature progression at any given fraetional eonversion X. ... [Pg.532]

For a first order reaction (-r ) = kC, and Equation 8-147 is then linear, has constant coefficients, and is homogeneous. The solution of Equation 8-147 subject to the boundary conditions of Danckwerts and Wehner and Wilhelm [23] for species A gives... [Pg.743]

Solution of equation 6.4 is the now familiar exponential distribution which is linear in a semi-logarithmic plot In n L) versus L (Figure 6.11a)... [Pg.168]

Valence orbital Xij is the lowest energy solution of equation 9.23 only if there are no core orbitals with the same angular momentum quantum number. Equation 9.23 can be solved using standard atomic HF codes. Once these solutions are known, it is possible to construct a valence-only HF-like equation that uses an effective potential to ensure that the valence orbital is the lowest energy solution. The equation is written... [Pg.172]

Second solution of Equation 9-129 to determine line for plot... [Pg.398]


See other pages where Solution of Equation is mentioned: [Pg.879]    [Pg.93]    [Pg.228]    [Pg.563]    [Pg.1051]    [Pg.1062]    [Pg.1266]    [Pg.1501]    [Pg.2344]    [Pg.3059]    [Pg.116]    [Pg.142]    [Pg.48]    [Pg.50]    [Pg.62]    [Pg.175]    [Pg.135]    [Pg.106]    [Pg.343]    [Pg.469]    [Pg.716]    [Pg.115]    [Pg.283]    [Pg.288]    [Pg.391]   
See also in sourсe #XX -- [ Pg.419 ]

See also in sourсe #XX -- [ Pg.352 ]




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A Numerical Solution of Ordinary Differential Equations

An Integral Representation for Solutions of the Creeping-Flow Equations due to Ladyzhenskaya

Analytic solution of the Michaelis-Menten kinetic equation

Analytic solution of the chemical master equation

Analytical Solution of Mass Transport Equations

Analytical Solution of the Kiln Equation for Slow Coke

Analytical Solutions of the diffusion equation

Analytical and Numerical Solutions of Balance Equations for Three-Phase Reactors

Approximate Solutions of the PB Equation

Approximate Solutions of the Schrodinger equation

Approximate solution of the Dirac equation

Approximate solutions of equations

Better Approximate Solutions of the Time-Independent Schrodinger Equation

Boundary Layer Solution of the Mass Transfer Equation

Boundary Layer Solution of the Mass Transfer Equation Around a Gas Bubble

CAO Solution of Fock Equations

Chapman—Enskog solutions of the Boltzmann equation

Discrete-time solution of the state vector differential equation

Discretization and Solution of the Poisson Equations

Discretization and Solution of the SCF equations

Dois Approximate Solution of the Smoluchowski Equation

Equation of Condition for Dilute Solutions

Equation of Radiative Transfer Formal Solution

Equation of the solution

Equations of Motion and their Solution

Equilibrium Solutions of Boltzmans Equation

Exact Solutions of Linear Heat and Mass Transfer Equations

Exact Solutions of the Schroedinger Equation

Exact Solutions of the Stokes Equations

Existence and uniqueness of solutions to a nonlinear algebraic equation

Existence of solutions to operator equations and inequalities

Factorization and Solution of the Secular Equation

Finite difference solution of parabolic equations

Full Solution of the Kohn-Sham Equations

Functions and a Solution of Laplaces Equation

Fundamental Solution of Poissons and Laplaces Equations

Fundamental Solutions of the Creeping-Flow Equations

Fundamental Solutions of the Homogeneous Equation

General Solution of the Transport Equation

General comments on the solution of boundary layer equations

General solution of equation

General solution of the Feller equation

General solution of the differential equations

Graphic solution of equations

Graphical Solution of Equations

Graphical Solution of the CSTR Design Equation

Greens Function Solutions of the Wave Equations

High frequency approximations in the solution of an acoustic wave equation

Highly-accurate solutions of the Schrodinger equation

Homogeneous Solutions of Higher Order Constant Coefficient Equations

Iterative Component-Wise Solution of the Nonlinear Equations

Iterative Solution of Nonlinear Algebraic Equations

Logistic Solution of Haldane-Radic Equation

MRM solution of the Euler equation

Method of Solution for Homogeneous Equations

Method of Solution for Inhomogeneous Equations

Numerical Methods for Solution of Partial Differential Equations

Numerical Solution of Algebraic Equations

Numerical Solution of Boltzmann Equation

Numerical Solution of Equations

Numerical Solution of Integral Equations

Numerical Solution of Linear Equations

Numerical Solution of Matrix Equations

Numerical Solution of Nonlinear Equations in One Variable

Numerical Solution of Ordinary Differential Equations

Numerical Solution of Partial Differential Equations

Numerical Solution of Schrodingers Equation

Numerical Solution of Simultaneous Linear Algebraic Equations

Numerical Solution of Stiff Equations

Numerical Solution of the Governing Equations

Numerical Solution of the Model Equations

Numerical Solution of the One-Dimensional Time-Independent Schrodinger Equation

Numerical Solution of the Population Balance Equation

Numerical Solution of the Radial Schrodinger Equation

Numerical Solution of the Time-Dependent Schrodinger Equation

Numerical Solution of the diffusion equation

Numerical Solutions of Differential Equations

Numerical Solutions of Nonlinear Equations

Numerical solution of SCF equations

Numerical solution of the Percus-Yevick equation

Numerical solution of the equations

Numerical solution, of model equations

P orbital solutions of Schrodinger wave equation for

Periodic Solutions of Operator-Differential Equations

Physical Interpretation of the Angular Equation Solutions

Power-Series Solution of Differential Equations

Reduced System of Equations and Solutions

Relaxation Times via General Solution of Blochs Equations

Rotating Wave Solution of the Ginzburg-Landau Equation

SOLUTION OF FICKS SECOND LAW EQUATION

Series solutions of differential equations

Simultaneous Solution of Linear Equations

Simultaneous Solution of Nonlinear Algebraic Equations

Solution of Batch-Mill Equations

Solution of Boltzmann Equation for Hydrogenous Systems

Solution of Complex Equations

Solution of Condensed Phase Equations

Solution of Differential Equations with Laplace Transforms

Solution of Diffusion Equation Near an Interface

Solution of Equations using Optimization

Solution of Example 2.1 Equations

Solution of Gas Phase Equations

Solution of Linear Algebraic Equations

Solution of Linear Equation Systems

Solution of Linear Equations

Solution of Nonlinear Algebraic Equations

Solution of Nonlinear Equations

Solution of Onsager Equations in a Simplified Case

Solution of Ordinary Differential Equations

Solution of Parabolic Partial Differential Equations for Diffusion

Solution of Parabolic Partial Differential Equations for Heat Transfer

Solution of Partial Differential Equations

Solution of Partial Differential Equations Using Finite Differences

Solution of Posissons equation Using a Constant Strain Triangle

Solution of Schrodingers Equation for the Kepler Problem

Solution of Schrodingers equation for

Solution of Simultaneous Algebraic Equations

Solution of Simultaneous Linear Algebraic Equations

Solution of a differential equation

Solution of a diffusion equation

Solution of algebraic equation

Solution of differential equations

Solution of matrix equations

Solution of the Atmospheric Diffusion Equation for an Instantaneous Source

Solution of the CC equations

Solution of the Coagulation Equation

Solution of the Colebrook Equation

Solution of the Condensation Equation

Solution of the Coupled Dirac Radial Equations

Solution of the Dispersion Equation

Solution of the Energy Equation

Solution of the Free-Electron Dirac Equation

Solution of the Governing Equations

Solution of the Harmonic Oscillator Schrodinger Equation

Solution of the Kohn-Sham-Dirac Equations

Solution of the Laplace and Poisson Equations

Solution of the Linearized P-B Equation

Solution of the MESH Equations

Solution of the Model Equations

Solution of the Momentum Equation

Solution of the Multicomponent Diffusion Equations

Solution of the Navier-Stokes Equation

Solution of the R, , and Equations

Solution of the Schrodinger equation

Solution of the Secular Equation

Solution of the Simultaneous Equations

Solution of the Stationarity Equations

Solution of the Steady-State Equations

Solution of the Thermal Boundary-Layer Equation

Solution of the Time-Dependent Schrodinger Equation

Solution of the Transient Gas-Phase Diffusion Problem Equations

Solution of the Transient Gas-Phase Diffusion Problem Equations (11.4) to

Solution of the Transport Equations

Solution of the diffusion equation when Le

Solution of the network equations

Solution of the reaction-diffusion equations

Solution of the spray equation

Solution of the state vector differential equation

Solution of the stationary-value equations

Solution of the transfer equation for

Solutions of Algebraic Equation Systems

Solutions of Eulers equations

Solutions of HF Equations

Solutions of PNP Equations

Solutions of the Boltzmann Equation

Solutions of the Differential Equations for Flow Processes with Variable External Stress and Field

Solutions of the Differential Equations for Flow-Processes

Solutions of the Dirac equation in field-free space

Solutions of the Equations in Relevant Cases

Solutions of the Klein-Gordon Equation

Solutions of the Master Equation

Solutions of the Poisson Equation

Solutions of the Radial Diffusion Equation

Solutions of the Radial Dirac Equation

Solutions of the Spin-Free Modified Dirac Equation

Solutions of the Steady-State Atmospheric Diffusion Equation

Solutions of the diffusion equation

Solutions of the diffusion equation parallel flux

Solutions of the motion equation for various stages

Some Cases for which there is no Solution of the Diffusion Equation

Stepwise Solution of the Combined Dipolar and NOE Equations

Systems of linear equations and their general solutions

Techniques for the numerical solution of partial differential equations

The Constitutive Equation for an Isothermal Solution of Rouse Chains

The Schrodinger equation and some of its solutions

The Solution of Algebraic Equations

The Solution of Population Balance Equations

The Solution of Simultaneous Algebraic Equations

Useful concepts in the solution of mass transport equations

What is the solution of a partial differential equation

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