Investigation of Fundamental and Principles of Mathematical Modeling of Well-Posed Physics and Engineering Problems

One of the scientific methods of solving physics and engineering problem is to analyze natural phenomenon and then modeling them mathematically. In fact mathematical model for a physics or engineering problem is the statement of reality which governs the physics and engineering problem between unknown and data. Generally these modelings are in the form of initial value and boundary value differential equation with some initial or boundary conditions. In this paper we first set forth three questions in this context as follows: 1. Can we give a mathematical algorithm and mathematical model for each physics and engineering problem? 2. Can each mathematical modeling and problem of physics and engineering be solved? 3. Is each prediction and solution which is resulted from physics and engineering applicable? In the second part of the paper, the above questions will be answered with the help of scientific, philosophical and historical background based on the attitude and idea of ancient Greek scientists and philosophers. The contemporary mathematician and physicists will be given. Next, in order to have correct answer to the above questions by physicists and engineers, some well-posed problems will be discussed. Finally, with the aid of mathematician’s general methods, some ill-posed problems will be investigated to have a better understanding of well-posed boundary value differential equation. The extend boundary condition of the problems which appear in physics and engineering will be discussed. Eventually, to promote the skill of mathematical modeling of physics and engineering for physics and engineering students, some proposals will be given.