Differential equation numerical integration

Monte Carlo simulation is an extremely powerful tool available to the scientist/ engineer that can be used to solve multivariable systems, ordinary and partial differential equations, numerical integrations, etc. 

The implementation of the multistep methods is ineffective for general differential equation numerical integration programs because of their complex initialization and variation in the integration step. 

In the work that follows, the experimental data were fitted by minimizing the sum of least squares and the differential equations were integrated numerically. 

The dependent variable y is most frequently the reaction rate independent variables are the concentration or pressure of reaction components, temperature and time. If in some cases the so-called integral data (reactant concentrations or conversion versus time variable) arc to be treated, a differential kinetic equation obtained by the combination of a rate equation with the mass balance equation 1 or 3 for the given type of reactor is used. The differential equation is integrated numerically, and the values obtained arc compared with experimental data. 

The method of characteristics, the distance method of lines (continuous-time discrete-space), and the time method of lines (continuous-space discrete-time) were used to solve the solids stream partial differential equations. Numerical stiffness was not considered a problem for the method of characteristics and time method of lines calculations. For the distance method of lines, a possible numerical stiffness problem was solved by using a simple sifting procedure. A variable-step fifth-order Runge-Kutta-Fehlberg method was used to integrate the differential equations for both the solids and the gas streams. 

Once some idea has been obtained of the type of expression that is required, it is then possible to integrate the rate equations numerically. An iterative method is often convenient the iteration would normally take the exponents in the various concentration terms as fixed and would proceed by substituting trial values of the rate coefficients (estimates of these having been obtained from prior experiments) in the differential equations and integrating in this way, it is possible to find values which give a satisfactory match of calculated and observed data. The question of the criterion of fit is outside the scope of this chapter. 

Using the boundary conditions (equations (5.7) and (5.8)) the boundary values uo and Un+1 can be eliminated. Hence, the method of lines technique reduces the linear parabolic ODE partial differential equation (equation (5.1)) to a linear system of N coupled first order ordinary differential equations (equation (5.5)). Traditionally this linear system of ordinary differential equations is integrated numerically in time.[l] [2] [3] [4] However, since the governing equation (equation (5.5)) is linear, it can be written as a matrix differential equation (see section 2.1.2) ... 

The system of differential equations is integrated using CVODE numerical integration package. CVODE is a solver for stiff and nonstiff ordinary differential equation systems [60]. The fraction of dose absorbed is calculated as the sum of all drug amounts crossing the apical membrane as a function of time, divided by the dose, or by the sum of all doses if multiple dosing is used. 

If all we cared about was hydrogen-like atoms, we would not need to employ expansion methods, but could solve the differential equations numerically, as is done in existing codes for 4-component calculations on atoms [1,2]. However, the electron-electron interaction of many-electron systems gives rise to integrals of the form... 

The change in concentration of reactants is at the centre of interest in photokinetics as well as the determination of these partial photochemical quantum yields. The time laws cannot be integrated in a closed form. Therefore to avoid the problems with solving these differential equations, the integrals are numerically calculated - a procedure named formal integration. This method also turns out to be advantageous in thermal and photochemical examinations. 

When integrating a differential equation numerically, one would expect the suggested step size to be relatively small in a region in which the solution curve displays much variation and to be relatively large where the solution curve straightens out to approach a line with a slope of nearly zero. Unfortunately, this is not always the case. The DDEs that make up the mathematical models of most chemical engineering systems usually represent a collection of fast and slow dynamics. For instance, in a typical distillation tower, the liquid mechanics (e.g., flow, hold-up) is considered as fast dynamics (time constant seconds), compared with the tray composition slow dynamics (time constant minutes). Systems with such a collection of fast and slow ODEs are denoted stiff systems. 

The variables j/, r) and v(x, y, t) describe the scaled concentration distributions of the propagator (activator) and controller (inhibitor), x and y are the scaled 2D space coordinates and r the scaled time. For the numerical solution of these PDFs a straightforward two-dimensional space discretization with a 5- or 9-point approximation of the Laplacian [28] was programmed in C. The differential equations were integrated by the explicit Euler method. 

The partial-differential equations (3) are solved numerically through finite difference approximation for the spatial derivatives and the method of line for time advancement. The model medium is represented by a discretized line with a resolution from 50 up to 200 points. The resulting set of ordinary differential equations is integrated with a stiff ODE solver [85]. Care is taken to vary the spatio-temporal resolution in order to check the reliability of the reported phenomena. 

The calculus of finite differences may be characterized as a two-way street that enables the user to take a differential equation and integrate it numerically by calculating the values of the function at a discrete (finite) number of points. Or, conversely, if a set of finite values is available, such as experimental data, these may be differentiated, or integrated, using the calculus of finite differences. It should be pointed out, however, thatnumerical differentiation is inherently less accurate than numerical integration. 

Gear C W 1966 The numerical integration of ordinary differential equations of various orders ANL 7126... 

The method of Ishida et al [84] includes a minimization in the direction in which the path curves, i.e. along (g/ g -g / gj), where g and g are the gradient at the begiiming and the end of an Euler step. This teclmique, called the stabilized Euler method, perfomis much better than the simple Euler method but may become numerically unstable for very small steps. Several other methods, based on higher-order integrators for differential equations, have been proposed [85, 86]. 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

Another difference is related to the mathematical formulation. Equation (1) is deterministic and does not include explicit stochasticity. In contrast, the equations of motion for a Brownian particle include noise. Nevertheless, similar algorithms are adopted to solve the two differential equations as outlined below. The most common approach is to numerically integrate the above differential equations using small time steps and preset initial values. 

Verlet, L. Computer Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Physical Review 159 (1967) 98-103 Janezic, D., Merzel, F. Split Integration Symplectic Method for Molecular Dynamics Integration. J. Chem. Inf. Comput. Sci. 37 (1997) 1048-1054 McLachlan, R. I. On the Numerical Integration of Ordinary Differential Equations by Symplectic Composition Methods. SIAM J. Sci. Comput. 16 (1995) 151-168... 

W. Gautschi. Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math., 3 381-397, 1961. 

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