Math 745/845                                 Foundations of Applied Mathematics

Instructor: Professor Marianna Shubov

    The main focus of this introductory course is the theory and solution techniques for several partial differential equations (and the corresponding boundary–value problems) that are of primary importance in science and engineering. The approach will be mathematically rigorous, however, the emphasis will be on solution algorithms, which are in the cornerstone of the methods of applied mathematics.

    The laws of nature that govern the fundamental physical phenomena have mathematical formulation in the form of differential equations. Most of those equations are the partial differential equations (PDEs). The entire world of vibrations and wave propagation (including solid elastic structures and electromagnetic phenomena) is governed by hyperbolic PDEs and systems of such PDEs (equations of vibrating elastic strings, rods, beams, thin shells, and the Maxwell equations of electromagnetic theory). The phenomena related to diffusion, heat conduction, chemical reactions, and other processes irreversible in time are governed by parabolic PDEs. The stationary solutions of both the hyperbolic and parabolic PDEs are described by elliptic PDEs.

    Anyone, who wishes to understand the theory and the solution techniques for the aforementioned equations, should have a strong background in the theory of just three most fundamental equations. These equations are:

1) the wave equation (a hyperbolic type PDE);
2) the heat (or diffusion) equation (a parabolic type PDE);
3) the Laplace (or the Poisson) equation (an elliptic type PDE).

    The first two equations are considered for n=1,2, or 3 spatial dimensions, and the last one is considered for n=2,3 (so, in fact, the above list contains 8 equations).
    The course will provide the physical motivation and systematic solution algorithms including the following topics:

In modern Applied Mathematics, the basic approach to a particular problem involves 3 stages:

    The content of this course is an essential part of a background of any scientist or engineer who encounters in his/her research work the above steps.
    The history of science shows us that even the most abstract mathematical theories eventually become essential in science. Recall, for example, that the complex numbers were known already in 16th - 17th  centuries. Who at that time could imagine that such an object as √-1 could have any practical significance? Today Complex Analysis is among the main tools of science. Moreover, Quantum Mechanics just cannot be formulated without complex numbers. As another example, recall that systematic study of vectors started in 19th century, and soon after that the great physicist Lord Kelvin complained that vectors ”has never been of the slightest use to any creature.” Can anyone imagine a modern physicist or engineer who does know vectors? Evidently, progress in abstract mathematical research will always be important for those who work on practical problems. However, currently, students interested in applications should first concentrate on mathematical methods whose practical significance has already been firmly established. An essential part of those methods is a subject of this course.