Unstable said:That’s it!
People, especially pure math orientated have to understand that Des (ODE, PDE, DDE,…) are more than just objects to study from a mathematical point of view. Unfortunately, until now most math courses about Des are about theoretical properties. No doubt, this is important, but these guys (most of them are really not able to this) have to show students what one could do with these Des.
I am talking to you as a math PhD.
To start with Des either from a numerical point of view or a modeling point of view, it is much more important in order to understand the power of this tool than just concentrating on mathematical properties.
To understand why the solution of an ODE exists and is unique is important for the examination but it has nothing to do with the real value of an ODE.
As an example, I know money exists and it is important to have, but what you could do with it is a complete different field. The same with Des….
A PDE, or partial differential equation, is a type of differential equation that involves multiple independent variables and their partial derivatives. It is commonly used to describe physical processes or phenomena in fields such as physics, engineering, and mathematics.
The prerequisites for studying PDEs include a strong foundation in calculus, particularly in multivariable and vector calculus. Knowledge of linear algebra and ordinary differential equations is also important. Additionally, familiarity with basic concepts in physics and mathematical analysis can be helpful.
Yes, there are several mathematical techniques and methods that are commonly used in the study of PDEs. These include separation of variables, Fourier series and transforms, and numerical methods such as finite differences and finite element methods. Depending on the specific PDE and its application, other techniques such as Green's functions and complex analysis may also be useful.
Some PDEs can be solved analytically, meaning a closed-form solution can be obtained using mathematical techniques. However, many PDEs do not have analytical solutions and require numerical methods to approximate the solution. The choice of analytical or numerical methods depends on the specific PDE and its complexity.
PDEs have numerous real-world applications in fields such as physics, engineering, and finance. They are commonly used to model and analyze physical processes such as heat transfer, fluid dynamics, and wave propagation. PDEs also have applications in image processing, computer graphics, and option pricing in finance.