Last Date: 7-2-2013
Dear students, this is to inform that GDB will be started on February 6, 2013 at 00:00 and will be closed on February 7, 2013 at 23:59.
The topic for Graded Moderated Discussion Board is
“Compare the efficiency and characteristics of methods available to you for numerical integration”
Idea Solution
Numerical
integration aims at approximating definite integrals using numerical
techniques. There are many situations where numerical integration is
needed. For example, several well defined functions do not have an
anti-derivative, i.e. their anti-derivative cannot be expressed in terms
of primitive function. A popular example is the function e��x2 whose
anti-derivative does not exist. This function arises in a variety of
applications such as those related to probability and statistics
analyses. Furthermore, many applications in science and engineering are
represented by integral differential equations that require a special
treatment for the integral terms (e.g. expansion, liberalization,
closure ...).
Therefore, numerical integration does not only provide a means for evaluating
integrals numerically, but also grants us the ability to approximate
special functions that are defined in terms of integrals. Without loss
of generality, there are two classes of problems where numerical
integration is needed. In the first class, one wishes to evaluate the
integral of a well defined function. In this case, the integrand can be
evaluated a various points because and numerical integration techniques
help define the optimum number of these points as well as their
locations. The second class of problems for applying numerical
integration is found in differential equations the most common of which
are those that express conservation principles. For example, the
population balance equation, a well known partial differential equation
encountered in process modeling and biological systems, exhibits source
terms that are represented as integrals of the solution variable (e.g.
the number density function). The most common technique for numerical
integration is called quadrature. The recipe for quadrature consists of
three steps
1. Approximate the integrand by an interpolating polynomial using a specified number of points or nodes
2. Substitute the interpolating polynomial into the integral
3. Integrate
