Numerical Solution of Differential Equations

Many differential equations occur for which no solution can be obtained with pencil and paper. A lot of these occur in the study of chemical reaction rates. With the use of programmable computers, it is now possible to obtain numerical approximations to the solutions of these equations to any desired degree of accuracy. 

This is a method that is extremely simple to understand and implement. However, it is not very accurate and is not used in actual applications. Consider a differential equation for a variable x as a function of time that can be schematically represented 

Like any other formal solution, this cannot be used in practice, since the variable x in the integrand function depends on t in some way that we don t yet know. 

Euler s method assumes that if t is small enough, the integrand function in Eq. (8.110) can be replaced by its value at the beginning of the integration. We replace t by the symbol At and write 

A small value of At is chosen, and this process is repeated until the desired value of t is reached. Let x, be the value of x obtained after carrying out the process i times, and let ti equal i At, the value of t after carrying out the process i times. We write 

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Carnahan, B., and J. O. Wilkes. Numerical Solution of Differential Equations—An Overview in Foundations of Computei-Aided Chemical Fi ocess Design, AIChE, New York (1981). 

An accurate indication is achieved by carrying out the calculations in small time steps, such as At = 0.004 s, and then by calculating the vaporization, humidity change, and corresponding temperature rise at each time step. This is the numerical solution of differential equations (4.326) and (4.328). The results of a calculation of this type are shown in Table 4.12. 

Solution of Sets of Simultaneous Linear Equations 71. Least Squares Curve Fitting 76. Numerical Integration 78. Numerical Solution of Differential Equations 83. 

The successful numerical solution of differential equations requires attention to two issues associated with error control through time step selection. One is accuracy and the other is stability. Accuracy requires a time step that is sufficiently small so that the numerical solution is close to the true solution. Numerical methods usually measure the accuracy in terms of the local truncation error, which depends on the details of the method and the time step. Stability requires a time step that is sufficiently small that numerical errors are damped, and not amplified. A problem is called stiff when the time step required to maintain stability is much smaller than that that would be required to deliver accuracy, if stability were not an issue. Generally speaking, implicit methods have very much better stability properties than explicit methods, and thus are much better suited to solving stiff problems. Since most chemical kinetic problems are stiff, implicit methods are usually the method of choice. 

W.E. Milne. Numerical Solution of Differential Equations. Dover Publications, New York, 1970. 

S.K.Godunov and V.S.Ryabenkiy, Numerical Solutions of Differential Equations, Nauka, Moscow, 1973. 

Remember 2.4 The method for numerical solution of differential equations with complex variables is to separate the equations into real and imaginary parts and to solve them simultaneously. 

Gas flow in a reaction is considered to follow the principles of conservation of momentum, energy, and total mass. These parameters are used to simulate flow behaviour through numerical solutions of differential equations. The general derivation and form of these equations are given in standard references [53-55], and these principles and related equations are briefly introduced below. 

H. A. Deans and L. Lapidus [AJChE J., 6, 656, 663 (I960)] consider the reactor to consist of an assembly of cells, with complete mixing within each cell and with a separate cell associated with each catalyst pellet. The balances then are written as difference equations. Numerical solution of differential equations by machine computation is described in L. Lapidus, Digital Computation for Chemical Engineers, McGraw-Hill Book Company, chap. 4, New York, 1960. 

Levy, H., Baggott, E. A., "Numerical Solutions of Differential Equations, ... 

Mathematica carries out numerical solutions of differential equation for which no exact solution can be written. The solution is given in terms of an interpolating function, which is a table of values of the unknown function for different values of the independent variable. The program finds a numerical value of the function for a specific value of the independent variable by interpolation in this table. The statement NDSolve is used to solve the differential equation, as in the next example ... 

Ixaru39 has derived several exponentially-fitted formulae for numerical derivatives, numerical quadratures, and numerical solution of differential equations. The most theoretical results have been produced in other papers. 

SE7 Mathematically inexact deconvolution. Numerical procedures such as numerical integration, numerical solution of differential equations, and some matrix-vector formulations of linear systems are numerical approximations and as such contain errors. This type of error is largely eliminated in the direct deconvolution method where the deconvolution is based on a mathematical exact deconvolution formula (see above). Similarly, the prescribed input function method ( deconvolution through convolution ) wiU largely eliminate this numerical type of error if the convolution can be done analytically so that numerical convolution is avoided. 

When the EKR system is enhanced by the addition of an acid, these equations should be adequately modified in the same manner as was done for the onedimensional model. Rnally, the new transient values of the concentrations at the time t + At can be calculated using a numerical solution of differential equations Uke Euler s method ... 

Numerical Solution of Differential Equations". Academic Press, New-York, San Fransisco, London, 1979. 

We have seen in Chapter 3 that finite difference equations also arise in Power Series solutions of ODEs by the Method of Frobenius the recurrence relations obtained there are in fact finite-difference equations. In Chapters 7 and 8, we show how finite-difference equations also arise naturally in the numerical solutions of differential equations. 

Rosenbrock, H. H., Some General Implicit Processes for the Numerical Solution of Differential Equation, Comput. J. 5, 329-340 (1963). 

Mass transport in soils and porous mineral materials is usually calculated based on Eqs. 7.24 to 7.28. Further information about parameters used can be found in the GLR recommendation El-10 Mass Transport Models for the Barrier Effect of Liner Layer (DGGt 1997). Mass transport in clayey barrier systems for waste disposal facilities is discussed in great detail in the book (Rowe et al. 1995a). Programs for the numerical solution of differential equation 7.24 for various boundary conditions and selection of parameters are commercially available (Rowe et al. 1994). 

Ralston, Anthony, and Philip Rabinowitz. A First Course in Numerical Analysis. 2d ed. Mineola, N.Y Dover Publications, 2001. A very good introductory book on numeric solution of differential equations. 

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