Numerical methods trapezoidal rule

Several related rorles or algorithms for numerical integration (rectangular mle, trapezoidal rule, etc.) are described in applied mathematics books, but we shall rely on Simpson s mle. This method can be shown to be superior to the simpler rules for well-behaved functions that occur commonly in chemistry, both functions for which the analytical form is not known and those that exist in analytical form but are not integrable. 

Semi-open formulas are used when the problem exists at only one limit. At the closed end of the integration, the weights from the standard closed-type formulas are used and at the open end, the weights from open formulas are used. (Weights for closed and open formulas of various orders of error may be found in standard numerical methods texts.) Given a closed extended trapezoidal rule of one order higher than the preceding formula, i.e.. 

Two simple numerical methods are often used to determine the area under the curve that equals the desired integral. They involve the use of the trapezoidal rule and Simpson s rule. 

Integration is accomplished numerically with the Simpson or trapezoidal rules. This method is applied in Example 11.2. 

AUC can be obtained by a numerical integration method such as the trapezoidal rule. A recent recommendation by the EDA has been the use of early exposure as a measure of rate of systemic exposure. This can be calculated as a partial AUC, where the area can be truncated at the population median of the values. Measurement of early exposure may be useful when rapid onset of action is desirable (e.g., an analgesic effect) or if a slow input is required to achieve efficacy or safety. The FDA has recently proposed a shift away from the focus on rate and extent of absorption to the measurement of systemic exposure, which can be determined as total, peak, or early exposure (if needed). This is based on the understanding that these measures better reflect the rate and extent of absorption. - ... 

Equations 3.122 and 3.124 for the gas phase serve only to compute distributions of Y and along bed height, which is necessary to calculate q and Wd. They can easily be integrated numerically, e.g., by the Euler method, at each time step. The integrals in Equations 3.121 and 3.123 can be numerically calculated, e.g., by the trapezoidal rule. This allows Equations 3.121 and 3.123 to be solved by any ODE solver. The model has been solved for a sample case and the results are shown in Eigure 3.20. 

Chapter 7 Numerical Solution Methods (Initial Value Problems) the Trapezoidal rule-, that is,... 

The discretized distance measure is derived by evaluation of the integral in the functional distance measure. This is accomplished by applying numerical integration methods, e g. rectangle rule, trapezoidal rule or Simpson rule. Using the rectangle rule and taking the left-hand sum of the functional distance measure in Equation (5) leads to the discretized... 

A numerical method (such as the trapezoidal rule) is used for evaluating the integral term in the design Equation 3.117. Choose n values of x,i2, x, .., /Xan at... 

ILLUSTRATIVE EXAMPLE 21.18 Many chemical reactor applications require solving differential equations. Some of these equations can be solved numerically or analytically by integrating the describing differential equation. Two simple numerical integration methods that are commonly employed are the trapezoidal rule and Simpson s rule. 

First, we demonstrate numerical problems of the various formulations when using Euler s method and the trapezoidal rule. 

The simplest method for numerical integration is the trapezoidal rule, which is based on joining interval end points with a chord to form a trapezoid, whose area is an approximation to the dehnite integral over the interval. This is depicted in Figure 4.1 with hve nonequally spaced subintervals. 

Unless otherwise stated, all numerical integrations for the remainder of the book will be computed by the trapezoidal rule. It is accurate enough for most engineering applications, does not require equally spaced data, and is easy to implement in Excel and VBA. For information on more complex and accurate numerical integration methods, Google it ... 

There are many more sophisticated methods of quadrature (another name for numerical integration). For reasons already given, the trapezoidal rule is sufficient for most engineering applications. In some specialized cases. Gauss integration (see Section 4.2.3) is the most advantageous method. 

---