Marcelo Epstein - Partial Differential Equations: Mathematical Techniques for Engineers

Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, AB, Canada

Marcelo Epstein

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It is not an accident that one of the inventors of Calculus, Sir Isaac Newton (16421727), was also the creator of modern science and, in particular, of Mechanics. When we compare Keplers (15711630) laws of planetary motion with Newtons , we observe a clear transition from merely descriptive laws, that apply to a small number of phenomena, to structural and explanatory laws encompassing almost universal situations, as suggested in Fig.. This feat was achieved by Newton, and later perfected by others, in formulating general physical laws in the small (differentials) and obtaining the description of any particular global phenomenon by means of a process of integration (quadrature).

In other words, Newton was the first to propose that a physical law could be formulated in terms of a system of ordinary differential equations . Knowledge of the initial conditions (position and velocity of each particle at a given time) is necessary and sufficient to predict the behaviour of the system for at least some interval of time. From this primordial example, scientists went on to look for differential equations that unlock, as it were, the secrets of Nature. When the phenomena under study involve a continuous extension in space and time one is in the presence of a field theory , such as is the case of Solid and Fluid Mechanics, Heat Transfer and Electromagnetism. These phenomena can be described in terms of equations involving the fields and their partial derivatives with respect to the space and time variables, thus leading to the formulation of systems of partial differential equations . As we shall see in this course, and as you may know from having encountered them in applications, the analysis of these systems is not a mere generalization of the analysis of their ordinary counterparts. The theory of PDEs is a vast field of mathematics that uses the tools of various mathematical disciplines. Some of the specialized treatises are beyond the comprehension of non-specialists. Nevertheless, as with so many other mathematical areas, it is possible for engineers like us to understand the fundamental ideas at a reasonable level and to apply the results to practical situations. In fact, most of the typical differential equations themselves have their origin in engineering problems.