Mathematical methods ordinary differential equations

In the following section, we only consider the integration of the equation of linear motion Eq. (20) the procedure for the equation of rotational motion, Eq. (21), will be completely analogous. Mathematically, Eq. (20) represents an initial-value ordinary differential equation. The evolution of particle positions and velocities can be traced by using any kind of method for ordinary differential equations. The simplest method is the first-order integrating scheme, which calculates the values at a time t + 5t from the initial values at time t (which are indicated by the superscript 0 ) via ... 

The mathematical method that will be used to characterize the system response is the stroboscopic map that was used by Kai Tomita (1979) and Kevrekidis et al. (1984,1986). If a point in the phase plane is used as an initial condition for the integration of the ordinary differential equations (odes) (4), a trajectory will be traced out. After integrating for one forcing period (from t = 0 to t = T), the trajectory will arrive at a new point in the phase palne. This new point is defined as the stroboscopic map of the original point and integration for... 

The concept of "mathematical chemistry had been already used by M.V. Lomonosov [1] and later on in the 19th century by Du Bois-Reymond, but for a long time it became inapplicable, apparently due to the lack of a distinct field for its application. As a rule, it was, and has remained, preferable to speak about the application of mathematical methods in chemistry rather than about "mathematical chemistry . To our mind, it is now quite correct to treat mathematical chemistry as a specific field of investigation. Its equations are primarily those of chemical kinetics, i.e. ordinary differential equations with a specific polynomial content. We treat these equations relative to heterogeneous catalytic systems. 

The mathematical models of the reacting polydispersed particles usually have stiff ordinary differential equations. Stiffness arises from the effect of particle sizes on the thermal transients of the particles and from the strong temperature dependence of the reactions like combustion and devolatilization. The computation time for the numerical solution using commercially available stiff ODE solvers may take excessive time for some systems. A model that uses K discrete size cuts and N gas-solid reactions will have K(N + 1) differential equations. As an alternative to the numerical solution of these equations an iterative finite difference method was developed and tested on the pyrolysis model of polydispersed coal particles in a transport reactor. The resulting 160 differential equations were solved in less than 30 seconds on a CDC Cyber 73. This is compared to more than 10 hours on the same machine using a commercially available stiff solver which is based on Gear s method. 

To develope a model for this mathematical problem we can either simplify the differential species mass balance equation (1.39) appropriately or combine the transient shell species mass balance written for the thin layer Az with Pick s law for binary diffusion. The resulting partial differential equation is called Pick s second law. A simple way to obtain a solution for this differential equation is to adopt the method of combination of variables. It is then necessary to define a new independent variable that enable us to transform the partial differential equation into an ordinary differential equation. 

The simplicity of equilibrium-based theories results from the fact that the coupled hrst-order partial differential equations (material balances) can be recast as two ordinary differential equations (but these are still coupled). The mathematical technique employed is called the method of characteristics. The results are... 

The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

Ascher, U.M. and Petzold, L.R. (1998) Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM Society for Industrial and Applied Mathematics. 

In Chapters 2 and 3, various analytical techniques were given for solving ordinary differential equations. In this chapter, we develop an approximate solution technique called the perturbation method. This method is particularly useful for model equations that contain a small parameter, and the equation is analytically solvable when that small parameter is set to zero. We begin with a brief introduction into the technique. Following this, we teach the technique using a number of examples, from algebraic relations to differential equations. It is in the class of nonlinear differential equations that the perturbation method finds the most fruitful application, since numerical solutions to such equations are often mathematically intractable. 

The continuity condition on the first derivative is used to access jS which cannot be obtained analytically. At this stage it is necessary to solve the problem using a mathematical solver. The method for the solution of the ordinary differential equation is given by Levine et al. [3]. 

The mathematics of thermodynamics is, in fact, extremely simple, apart from a few special casts, and consists mainly of the methods of partial differentiation and of ordinary differential equations of simple form. The conceptual aspect of thermodynamics is, in contrast, extraordinary abstract and it is here that the real difficulties arise. It has long been customary to try to avoid these difficulties by means of spurious analogies. It has, however, become clear that this method makes a deeper understanding leading to mastery of the subject more difficult. The characteristic properties of this field must be accepted and, on the one hand, basic concepts must be developed from concrete experience while, on the other, the mathematical structure must be analyzed. These consideration determine the way in which tins book is written. 

Here we will review a few methods for solving the first-order ordinary differential equations. Following each method are examples demonstrating the application of that method. Also, the notion of translating prose into mathematical symbolism is introduced as Problem Setup in Section 2.4. 

The mathematical equations used for each numerical int ration method [7, 8] are summarized in this section, where y stands for a nonlinear ordinary differential equation and h for the step size. Modified Euler integration formula ... 

The final two-dimensional mathematical model thus consists of one partial parabolic differential mass balance equation (3.12) with boundary and initial conditions in (3.14) for each of the j reactions and one partial parabolic differential heat transfer equation (3.15) with boundary conditions in (3.17), (3.18) and initial conditions in (3.20). Simultaneously the pressure drop ordinary differential equation (3.7) and the differential equations for the temperature and pressure in each of the surrounding channels in (3.22) must be integrated. Catalyst effectiveness factors in the catalyst bed must be available in all axial and radial integration points using the methods in Section 3.4. 

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