We know applied mathematics is a very rich field that is constantly evolving. The field has always been driven by applications in science and industry and the mineral industry has been a significant source for creative ideas. Mining engineering is responsible for the program design, operation, management, extraction and processing minerals from naturally occurring environment and mathematical formulated solutions are critical for the successful completion of very complex tasks.

There is no need to go very far to find cases of successful applications with mathematical models of use in mining, which produce enormous savings to the industry, not only applicable in planning mining, but in extraction and processing of minerals. Generally, when we refer to the use of applied mathematics in industry, we imply the use of numerical analysis techniques to implement proven mathematical theory.

Very widely used and of great value is the use of optimization problems for which the solution is characterized by the largest or smallest value of a numerical function depending on several parameters. Such a function is often called the objective or goal function. Many kinds of problems can be expressed in optimization, i.e, finding the maximum or the minimum of a goal function. In mining it is frequently used to determine allocation of laborers to work areas and also to calculate optimum load extraction based on expected total output production volume and ore grade requirements.

Many other prototypical solutions involve theory based on non-linear analysis, numerical integration, function evaluation, complex numbers, boundary value solutions, eigenvalue solutions, and differential equations. A few cases to mention: Underground mining operations requiring construction of platforms to carry assorted equipment with specific dimensions; Exploratory drilling tasks for obtaining sample core bits in a formation where several methods of combined drilling are used in somewhat complex geological sections and finding stress concentrations around holes; Use of the fourth-order Runge-Kutta integration methods to resolve polluted reservoirs exacerbated by large-scale earth disturbances and acid drainage and find acceptable levels of concentration of the pollutants as fresh water would replenish the reservoir; Applications in underground mining for tubing diameter variations due to diametric contractions regarding site temperatures; Applications in Cut-and-Fill stoping mining methods problems using rock mechanics techniques including finite element, finite difference and boundary integral methods.

Many mathematical functions used in numerical computation are defined by an integral, a recurrence formula, or a series expansion. While such definitions can be useful to a mathematician, they are quite complicated to implement in a computer. For one, not every mining engineer knows how to evaluate an integral numerically (most do), and then there is the problem of accuracy, and the evaluation of a function as defined mathematically involves complexity. Plus programming expertise is paramount to produce application module solutions that are accurate, useful and timely.

This being the case, the mathematical representation via numerical modeling represents a major challenge. The importance of applied mathematics in the mineral industry is so significant that future generations of mining engineers need to be very knowledgeable in numerical analysis procedures, possess solid command of high level programming languages and feel comfortable with mathematics reasoning in general.