Boundary collocation method

For the electrophoresis of a spherical particle in a circular cylindrical pore, the results obtained from the boundary collocation method show that the presence of the pore wall always reduces the electrophoretic velocity for the entire range of the separation parameter [42]. However, the net wall effect is quite weak, even for the very small gap width between the particle and wall. 

In one of the first articles on this subject [8], the general analytical solution of Eq. (3) was derived. This general solution is easy to find, but it contains infinite series and (integration) constants that depend on the boundary conditions. Those were determined for the central cells of square and triangular arrays, using the boundary collocation method [8]. More recent publications on this subject are based mostly on complete numerical solution using finite-element methods. 

On the other hand, through the use of the boundary collocation technique, the diffusiophoretic motion of a colloidal sphere with a thin but polarized diffuse layer in the direction perpendicular to a plane wall was examined. The wall effect on diffusiophoresis was found to be a complicated function of the properties of the particle and ions. The diffusiophoretic motions of a spherical particle with a thin polarized diffuse layer parallel and normal to two plane walls at an arbitrary positiOTi between them and alrnig the axis of a circular cylindrical pore were also investigated by using the boundary collocation method [9]. Numerical results of the wall corrections to Eqs. 5 and 11 for the particle velocity were presented for various values of the relative separation distances and other relevant parameters. 

When the polarization effect of solute species in the diffuse layer surrounding the particle is considered, the particle-interaction and boundary effects on diffusiophoresis can be quite different from those on electrophoresis, due to the fact that the particle size and some other unique factors are involved in each transport mechanism. Through the use of a boundary collocation method, the quasisteady axis3mmetric diffusiophoresis of a chain of colloidal spheres with thin but polarized diffuse layers was examined and numerical results of the diffusiophoretic velocity of the particles were presented for various cases. The diffusiophoretic velocities of two spherical particles with thin polarized diffuse layers but with arbitrary surface properties, arbitrary sizes, and arbitrary orientation with respect to the imposed solute concentration gradi-... 

Parabolic Equations m One Dimension By combining the techniques apphed to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerldn finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

Boundary value methods provide a description of the solution either by providing values at specific locations or by an expansion in a series of functions. Thus, the key issues are the method of representing the solution, the number of points or terms in the series, and how the approximation converges to the exact answer, i.e., how the error changes with the number of points or number of terms in the series. These issues are discussed for each of the methods finite difference, orthogonal collocation, and Galerkin finite element methods. 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements. One simply combines the methods for ordinary differential equations (see Ordinary Differential Equations—Boundary Value Problems ) with the methods for initial-value problems (see Numerical Solution of Ordinary Differential Equations as Initial Value Problems ). Fast Fourier transforms can also be used on regular grids (see Fast Fourier Transform ). 

The orthogonal collocation method has several important differences from other reduction procedures. Jn collocation, it is only necessary to evaluate the residual at the collocation points. The orthogonal collocation scheme developed by Villadsen and Stewart (1967) for boundary value problems has the further advantage that the collocation points are picked optimally and automatically so that the error decreases quickly as the number of terms increases. The trial functions are taken as a series of orthogonal polynomials which satisfy the boundary conditions and the roots of the polynomials are taken as the collocation points. A major simplification that arises with this method is that the solution can be derived in terms of its value at the collocation points instead of in terms of the coefficients in the trial functions and that at these points the solution is exact. 

The differential equations (7.138) and (7.139) for the top-fired furnace, however, describe a boundary value problem with the boundary conditions (7.140) and (7.141) that can be solved via MATLAB s built-in BVP solver bvp4c that uses the collocation method, or via our modified BVP solver bvp4cf singhouseqr. m which can deal with singular Jacobian search matrices referred to in Chapter 5. On the other hand, the differential equation (7.142) is a simple first-order IVP. 

The aim of this part of the book is to present the main and current numerical techniques that are used in polymer processesing. This chapter presents basic principles, such as error, interpolation and numerical integration, that serve as a foundation to numerical techniques, such as finite differences, finite elements, boundary elements, and radial basis functions collocation methods. 

In Chapter 11 of this book we will use the thin spline radial function to develop the radial basis functions collocation method (RBFCM). A well known property of radial interpolation is that it renders a convenient way to calculate derivatives of the interpolated function. This is an advantage over other interpolation functions and it is used in other methods such us the dual reciprocity boundary elements [43], collocation techniques [24], RBFCM, etc. For an interpolated function u,... 

Another method employed to study the boundary effects on electrophoresis of a single particle is the boundary collocation technique. Investigations have been conducted for the particle migration along the axis of a circular orifice, normal to a conducting circular disk [41], and along the... 

Electrophoretic interactions between spherical particles with infinitely thin double layers can also be examined using the boundary collocation technique [16,54]. This method enables one not only to calculate the interactions among more than two particles, but also to deal with the case of particles in contact, for which the bispherical coordinate solution becomes singular. Analogous to the result for a pair of spheres, no interaction arises among the particles in electrophoresis as long as all the particles have an equal zeta potential. This important result is also confirmed by a potential-flow reasoning [10,55]. 

When the gap width between two particles becomes very small, numerical calculations involved in both the bispherical coordinate method and the boundary collocation technique are computationally intensive because the number of terms in the series required to be retained to achieve a desired accuracy increases tremendously. To solve this near-contact motion more effectively and accurately, Loewenberg and Davis [43] developed a lubrication solution for the electrophoretic motion of two spherical particles in near contact along their line of centers with the assumption of infinitely thin ion cloud. The axisymmetric motion of the two particles in near contact can be approximated as the pairwise motion of the spheres in point contact plus a deviation stemming from their relative motion caused by the contact force. The lubrication results agree very well with those obtained from the collocation method. It is shown that near contact electrophoretic interparticle... 

An orthogonal collocation method for elliptic partial differential equations is presented and used to solve the equations resulting from a two-phase two-dimensional description of a packed bed. Comparisons are made between the computational results and experimental results obtained from earlier work. Some qualitative discrimination between rival correlations for the two-phase model parameters is possible on the basis of these comparisons. The validity of the numerical method is shown by applying it to a one-phase packed-bed model for which an analytical solution is available problems arising from a discontinuity in the wall boundary condition and from the semi-infinite domain of the differential operator are discussed. 

The collocation method is based on the assumption that the solution of the PDE system can be approximated by polynomials of order n. In this method the whole space domain (the column), including the boundary conditions, is approximated by one polynomial using n + 2 collocation points. Polynomial coefficients are determined by the condition that the differential equation must be satisfied at the collocation points. This approach allows the spatial derivatives to be described by the known derivatives of the polynomials and transforms the PDE into an ODE system. 

In the historical survey of the spectral methods given by Canute et al [22], it was assumed that Lanczos [101] was the first to reveal that a proper choice of trial functions and distribution of collocation points is crucial to the accuracy of the solution of ordinary differential equations. Villadsen and Stewart [203] developed this method for boundary value problems. The earliest applications of the spectral collocation method to partial differential equations were made for spatially periodic problems by Kreiss and Oliger [94] and Orszag [139]. However, at that time Kreiss and Oliger [94] termed the novel spectral method for the Fourier method while Orszag [139] termed it a pseudospectral method. 

Non-isothermal ejfectiveness factor The non-isothermal effectiveness factor can be obtained numerically only by integrating the two points boundary value differential equations using different numerical techniques, the most efficient of these techniques is the orthogonal collocation method. 

The pellet mass and heat balances are described by second order differential equations of the two point boundary value type. For this case the reaction is neither too fast nor highly exothermic and therefore the concentration and temperature gradients inside the pellet are not very steep. Therefore the orthogonal collocation method with one internal collocation point was found sufficient to transform the differential equation into a set of algebraic equations which were solved numerically using the bisectional method (Rice,... 

The non-linear two-point boundary value differential equation of the catalyst pellet is best solved using the orthogonal collocation method as described by Elnashaie et al. (1988a). The bulk phase initial value differential equations should be solved using standard integration routines with automatic step size to ensure accuracy (e.g. DGEAR-IMSL library). 

The orthogonal collocation technique is a simple numerical method which is easy to program for a computer and which converges rapidly. Therefore it is useful for the solution of many types of second order boundary value problems. This method in its simplest form as presented in this section was developed by Villadsen and Stewart (1967) as a modification of the collocation methods. In collocation methods, trial function expansion coefficients are typically determined by variational principles or by weighted residual methods (Finlayson, 1972). The orthogonal collocation method has the advantage of ease of computation. This method is based on the choice of suitable trial series to represent the solution. The coefficients of trial series are determined by making the residual equation vanish at a set of points called collocation points , in the solution domain. 

The behaviour of the system is described by equations 7.26-7.30. The non-linear coupled two-point boundary value differential equations are difficult to solve as a part of the maximization procedure due to the excessive computational effort involved. The solid phase equations will therefore be recast into an equivalent set of non-linear algebraic equations using the orthogonal collocation method (Villad-sen and Michelsen, 1978). The application of this method to this problem is explained in Appendix C. 

Boundary value problems are encountered so frequently in modelling of engineering problems that they deserve special treatment because of their importance. To handle such problems, we have devoted this chapter exclusively to the methods of weighted residual, with special emphasis on orthogonal collocation. The one-point collocation method is often used as the first step to quickly assess the behavior of the system. Other methods can also be used to treat boundary value problems, such as the finite difference method. This technique is considered in Chapter 12, where we use this method to solve boundary value problems and partial differential equations. 

In Chapters 7 and 8, we presented numerical methods for solving ODEs of initial and boundary value type. The method of orthogonal collocation discussed in Chapter 8 can be also used to solve PDEs. For elliptic PDEs with two spatial domains, the orthogonal collocation is applied on both domains to yield a set of algebraic equations, and for parabolic PDEs the collocation method is applied on the spatial domain (domains if there are more than one) resulting in a set of coupled ODEs of initial value type. This set can then be handled by the methods provided in Chapter 7. 

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