Systems of algebraic equations

Siep 6 assembly of the elemental stiffness equations into a global system of algebraic equations... 

After the insertion of the boundaiw conditions the solution of the system of algebraic equations in this case gives the required nodal values of 7 (i.e. T2 to 7io) as... 

NUMERICAL SOLUTION OF THE GLOBAL SYSTEMS OF ALGEBRAIC EQUATIONS... 

As mentioned in Chapter 2, the numerical solution of the systems of algebraic equations is based on the general categories of direct or iterative procedures. In the finite element modelling of polymer processing problems the most frequently used methods are the direet methods. 

The Gaussian elimination method provides a systematic approach for implementation of the described forward reduction and back substitution processes for large systems of algebraic equations. 

This technique (also known as the Grout reduction or Cholesky factorization) is based on the transfonnation of the matrix of coefficients in a system of algebraic equations into the product of lower and upper triangular matrices as... 

As mentioned earlier, overall accuracy of finite element computations is directly detennined by the accuracy of the method employed to obtain the numerical solution of the global system of algebraic equations. In practical simulations, therefore, computational errors which are liable to affect the solution of global stiffness equations should be carefully analysed. 

Discretizating by substituting the various finite-difference type approximations for the terms in the integrated equation representing flow processes, which converts the integral equations into a system of algebraic equations. 

In the case of the Cauchy problem with assigned values y and y, we have at our disposal the system of algebraic equations for constants Cj and... 

Following these procedures, we are led to a system of algebraic equations, thereby reducing numerical solution of an initial (linear) differential equation to solving an algebraic system. 

Thus emerged the system of algebraic equations with a tridiagonal matrix. Because of this form, the elimination method may be useful (see Chapter 1, Section 1). 

For (7 0 scheme (4) refers to an implicit two-layer scheme. When the value is sought on the new layer under the natural premise governing system of algebraic equations... 

The books by Gelfand (1967), Samarskii and Nikolaev (1989) cover in full details the general theory of linear difference equations. Sometimes the elimination method available for solving various systems of algebraic equations is referred to, in the foreign literature, as Thomas algorithm and this... 

Usually the finite difference method or the grid method is aimed at numerical solution of various problems in mathematical physics. Under such an approach the solution of partial differential equations amounts to solving systems of algebraic equations. 

In the expression for yp, substitute y x + delx) = y(jc) + dely to derive a system of algebraic equations for dely. Here are the equations presented earlier ... 

The operation of a plant under steady-state conditions is commonly represented by a non-linear system of algebraic equations. It is made up of energy and mass balances and may include thermodynamic relationships and some physical behavior of the system. In this case, data reconciliation is based on the solution of a nonlinear constrained optimization problem. 

The simplest case of this parameter estimation problem results if all state variables jfj(t) and their derivatives xs(t) are measured directly. Then the estimation problem involves only r algebraic equations. On the other hand, if the derivatives are not available by direct measurement, we need to use the integrated forms, which again yield a system of algebraic equations. In a study of a chemical reaction, for example, y might be the conversion and the independent variables might be the time of reaction, temperature, and pressure. In addition to quantitative variables we could also include qualitative variables as the type of catalyst. 

The structure of the present subsection is as follows. First, the governing b.v.p. is formulated and reduced, following the scheme of 4.2, to a system of algebraic equations. Then, two important limit cases are discussed counterion selectivity near equilibrium and selectivity at high concentration polarization. Finally, we present and discuss the results of a numerical solution of the above algebraic system for the intermediate range of deviations from equilibrium. 

This problem is described mathematically as an ordinary-differential-equation boundary-value problem. After discretization (Eq. 4.27) a system of algebraic equations must be solved with the unknowns being the velocities at each of the nodes. Boundary conditions are also needed to complete the system of equations. The most straightforward boundary-condition imposition is to simply specify the values of velocity at both walls. However, other conditions may be appropriate, depending on the particular problem at hand. In some cases a balance equation may be required to describe the behavior at the boundary. 

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