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Computational and Applied Mathematics

Applied Mathematics is concerned  with formulating, analyzing and solving an application problem, whether it arises from physical science or biological science, using mathematical language, theories and techniques.  The hallmark of applied mathematics is mathematical modeling, which may be regarded as the art of abstraction that helps us realize the truth. The power of modeling and abstraction has been clearly evidenced by the enormous advancements and progresses of physical sciences in the past two centuries. As the solutions to all these application problems hinge on the solutions to their mathematical models, analyzing and solving these problems has been critically important for the resolution of many scientific, engineering, and industrial application problems. However, except in some very simple situations, closed form solutions do not exist even for simple, linear, models. As a result, seeking numerical (approximate) solutions becomes the only viable way to find (and to see) the solutions of these mathematical models.

The computations involved in solving a mathematical model,  in particular, a nonlinear model, are often enormous, which are impossible to carry out by hand. On the other hand, the job is perfect for computers because they are good at number-crunching. To realize the potential, a big (and difficult) question which must be answered is how to utilize computers’ superpower of  doing arithmetic calculations to compute solutions of mathematical models. The key to this question is to develop reliable and efficient numerical/computational methods and algorithms which can be efficiently implemented on computers. To develop, analyze and implement these enabling methods and algorithms for all kinds of mathematical problems is a scientific field called Computational Mathematics,  a sub-field of it, that focuses on partial differential equations (PDEs) related mathematical problems, is known as numerical PDEs, which is one of main focuses in Computational  Mathematics.

The centerpieces of Computational Mathematics is “error and speed”. Errors are everywhere; they occur in every step of an algorithm (i.e., approximation error) and in every calculation done by a computer (i.e., round-off error). If a beautiful method or algorithm on paper does not have a mechanism to control its error accumulation when implemented on computers, then it is useless. Such an error-control capability is the watershed to distinguish “good” methods and algorithms  from “bad” ones.  Among “good”  methods and algorithms of the same nature, a faster (in terms of computer execution time) method or algorithm is considered to be a better method or algorithm. Developing  “good” and fast computational methods and algorithms is the heart and soul of Computational Mathematics.

 

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