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# Course Introduction

《Introduction to Partial Differential Equation》 |
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- Prerequisite：Multivariable calculus, Linear algebra
- Recommended for： Sophomore
- Introduction：
The main driving force which completely changed the outlook of the scientific-technological aspect of human civilization was the scientific revolution initiated by Newton. The scientific revolution of Newton was the climax of human scientific thought which began immediately after a series of previous great artistic and scientific achievements by the giants since the time of Renaissance. In general, Newton’s scientific revolution, through his invention of Newtonian mechanics, is generally regarded as one of the triumphs of human’s rational thinking in recent human history.
Basically speaking, the main principle of Newtonian mechanics can be realized mathematically as a second-order ordinary differential equation. To a certain extent, the past developments of mathematical tools, which are capable of addressing mathematical problems arising from this second-order O.D.E. in Newtonian mechanics, constituted the whole historical development of Calculus.
To put it in simple words, we can say that Calculus was invented for the sole purpose of solving differential equations. Indeed, being witnesses to the tremendous success of Newtonian mechanics, mathematicians and physicists since the time of Newton began to apply the main ideas of Newtonian Mechanics to all the other branches of Natural Science. In particular, in the studies of the physical phenomena of a vibrating string with fixed end-points, the first partial differential equation, which is known as the wave equation (or the d’Alembert’s equation), was derived for the very first time in the early days of the historical development of P.D.E.’s theory. A partial differential equation is, by definition, an equation satisfied by some of the partial derivatives of the unknown multi-variable function.
Partial differential equations arise naturally in the study of geometry, and in almost all physical laws of Natural Sciences or practical engineering problems. We can even say that, without the developments of the theory of partial differential equations, there could be no development of Science at all in recent human history. Indeed, some of the most famous partial differential equations which appeared in the history of Science are:Laplace equation, Wave equation, Heat equation, Maxwell equation, Schrodinger equation, Navier-Stokes equation and Euler equation, Boltzmann equation, KdV equation.
Basically, one can say that the studies of these equations encompass almost all the mathematical problems arising from Physics and different areas of engineering.
In order to solve these partial differential equations, mathematicians, physicists, and engineers had developed various kinds of mathematical methods, among these the most famous one is the Fourier analysis. As a matter of fact, the subject of Fourier analysis constitutes almost all of the modern mathematical analysis. However, as the development of the theory of partial differential equations became more mature and sophisticated, researchers began to realize the importance of acquiring a better understanding about various kinds of nonlinear phenomena. This naturally motivated the developments of the theory of dynamical systems in recent and modern times, including the recent studies of the theory of Chaos.
Besides standard tools borrowed from mathematical analysis, numerical analysis is another important methodology which is currently used in the studies of solutions to partial differential equations. Through carrying out numerical simulations with the aid of high-speed computers, the use of “Numerical Analysis” in the studies of P.D.E.’s can be regarded as carrying out mathematical experiments. These mathematical experiments through numerical simulations not only help us to gain a deeper understanding about the equation being studied, but even may provide us with clues which can lead to rigorous proofs of mathematical assertions related to the equation being studied.
Finally, but not the least, we should never forget that the sole purpose of a differential equation is to describe natural phenomena. Even priori to our efforts devoted in solving a specific differential equation, the equation itself has already revealed some of the secrets of the Nature to us. In other words, Differential equations themselves naturally speak the language of the Nature. So, it is absolutely true that knowing the basics of Physics will be extremely helpful for the studies of partial differential equations. |