Ordinary differential equation explicit solution

As discussed in the introduction to this chapter, the solution of ordinary differential equations (ODEs) on a digital computer involves numerical integration. We will present several of the simplest and most popular numerical-integration algorithms. In Sec, 4.4.1 we will discuss explicit methods and in Sec. 4.4.2 we will briefly describe implicit algorithms. The differences between the two types and their advantages and disadvantages will be discussed. 

Temperature profiles can be determined from the transient heat conduction equation or, in integral models, by assuming some functional form of the temperature profile a priori. With the former, numerical solution of partial differential equations is required. With the latter, the problem is reduced to a set of coupled ordinary differential equations, but numerical solution is still required. The following equations embody a simple heat transfer limited pyrolysis model for a noncharring polymer that is opaque to thermal radiation and has a density that does not depend on temperature. For simplicity, surface regression (which gives rise to convective terms) is not explicitly included. 

Solutions for the integration of ODEs such as those given in Equation 7.5 are not always readily available. For nonspecialists, it is difficult to determine whether there is an explicit solution at all. MATLAB s symbolic toolbox provides a very convenient means of producing the results and also of testing for explicit solutions of ordinary differential equations, e.g., for the reaction 2A — B, as seen in MATLAB Example 7.2. (Note that MATLAB s symbolic toolbox demands lowercase characters for species names.)... 

A few examples will be demonstrated in the following section. Each feature was incorporated in the software because it has been found necessary or useful in some modelling project. All the modelling tasks that previously required tailor made solutions in each project can now be solved in a unified manner in the ModEst environment. ModEst is able to deal with explicit algebraic, implicit algebraic (systems of nonlinear equations) and ordinary differential equations. As the standard way to handle PDE systems, the Numerical Method of Lines, which transforms a PDE system to a number of ODE components is used. In addition, any model with a solver provided may be dealt with as an algebraic system. The basic numerical tools are contained in the well tested public domain software (Bias, Linpack, Eispack, LSODE). 

The modern methods for numerical solution of the initial-value problem for systems of ordinary differential equations (ODE) suppose usually the explicit dependence of the derivative of the solution [7]... 

In ref. 145 the authors develop two families of explicit and implicit BDF methods (Backward Differentiation Methods), for the accurate integration of differential equations with solutions any linear combinations of exponential of matrices, products of the exponentials by polynomials and products of those matrices by ordinary polynomials. More specifically, the authors study the numerical solution of the problem ... 

This requires the numerical solution of explicit ordinary differential equations or differential-algebraic equations, see Chs. 4, 5, 6. 

Differential-algebraic equations (DAEs) differ in main aspects from explicit or regularly implicit ordinary differential equations. This concerns theory, e.g. solvability and representation of the solution, as well as numerical aspects, e.g. convergence behavior under discretization. Both aspects depend essentially on the index of the DAE. Thus, we first define the index. 

This is an explicit expression for the solution components X2- Differentiation leads together with Eq. (2.6.5a) to an ODE, the underlying ordinary differential equation (UODE)... 

There are explicit and implicit numerical methods for the finite-difference version of Eqs. 9.27 and 9.28. There are also approximate numerical methods in which the radial derivatives are replaced by the functions obtained by differentiating assumed trial functions for the radial profiles, essentially transforming the equations into ordinary differential equations in z. The cell model (Hlavacek and Votruba, 1977) can also be used for the solution. 

Solution of a set of nonlinear ordinary differential equations by the explicit Euler method. 

In most applications of the theory to date, the solution of the Redfield equation has required first the explicit calculation of the Redfield tensor elements [Eq. (11)] given these, Eq. (10) could be solved as an ordinary set of linear differential equations with constant coefficients, either by explicit time stepping [41, 42] or by diagonalization of the Redfield tensor [37,38]. Since there are such tensor elements for an A -state subsystem, the number of these quantities can become quite large. Because of this, until recently most applications of Redfield theory have been limited to small systems of two to four states, or else assumptions, such as the secular approximation, have been used to neglect large classes of tensor elements. 

Numerical integration techniques are necessary in modeling and simulation of batch and bio processing. In this chapter we described error and stability criteria for numerical techniques. Various numerical techniques for solution of stiff and non-stiff problems are discussed. These methods include one-step and multi-step explicit methods for non-stiff and implicit methods for stiff systems, and orthogonal collocation method for ordinary as well as partial differential equations. These methods are an integral part of some of the packages like MATLAB. However, it is important to know the theory so that appropriate method for simulation can be chosen. 

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