Dear researchers, I have been working on exact solution methods of partial differential equations for more than 10 years. I noticed that the researchers working in this field are always people from a certain geography, such as Turkey, Pakistan, Egypt, Saudi Arabia, India, etc. Why a German, English, or Japanese researchers do not study these topics? Do you think that our work really contributes to science and humanity? I am looking forward to your valuable comments.
For my part, I find doing a computer simulation easier. The physics of change (differential) in the universe can be very complex. The history is that the change equations are only approximations. However, there are rules for how the simulation should behave, for example here must be some form of feedback. That is, as one parameter grows, there must be a correcting process to have it shrink. For example, as a predator population grows, the prey population decreases which with time cause a decline in the predators. The same kind of rules (``Sarengeti rules'') seem to apply to physical processes.
In my opinion, exact solutions of differential equations are very important for both pure and applied mathematics.
First, most of special functions (e.g., hypergeometric functions, Bessel functions, elliptic functions, Meijer’s G-function) are solutions of some ODEs and PDEs. If a difficult equation can be solved exactly, new special functions may be found. The same is true even in Functional Differential Equations (FDEs) (e.g., the Ambartsumian equation).
Second, if an exact solution are found, we can do many interesting things. Various examples can be found in Asymptotic Theory, especially in the theory of uniform asymptotics. Once a special function has been thoroughly studied, it can be used like trigonometric and logarithmic functions.
Finally, the structures of solutions of ODEs and PDEs are usually very complicated, exact solutions may sometime provide illuminating examples (e.g. Kummer’s twenty-four solution of the hypergeometric equation).
Thank you for your comments. I agree with you, but many studies in the literature are similar. and I think there is a monopoly by certain groups. In addition, the references section always contains similar studies. The obtained results are not cited by researchers working in other fields, such as physicists and engineers. Therefore, I think that the scope of the studies is narrow.
Dear Sir,
The exact solution methods of partial differential equations are extremely valuable for science and humanity. Partial differential equations (PDEs) are mathematical equations that describe many fundamental physical and natural phenomena, such as fluid flow, heat transfer, electromagnetic fields, optic fiber, quantum mechanics, and many other areas of scientific research.
Moreover, exact solutions of PDEs can enhance our understanding of the underlying physical phenomena and provide insights into the behavior of complex systems. By solving PDEs exactly, scientists and researchers can derive analytical formulas, establish mathematical relationships, and uncover fundamental laws that govern the behavior of physical systems.
The value of exact solutions extends beyond theoretical advancements. They have numerous practical applications in fields such as engineering, physics, chemistry, biology, finance, and many others.
Furthermore, exact solutions often serve as benchmarks for numerical methods. When numerical techniques are used to solve PDEs, they are validated by comparing their results to known exact solutions. This allows researchers to assess the accuracy and reliability of numerical algorithms, ensuring that they provide trustworthy results.
The quest for finding exact solutions of nonlinear PDEs has been significantly advanced by the contributions of mathematicians from the USA, Germany, and Japan. Notable mathematicians in this field include A.M. Wazwaz, L. Akeniyame, Peter J. Olver, J. Bona, David L. Finn, Ernst A. C. Greiner, Rainer Schimming, Hansjörg Kielhöfer, Y. Hirota, Yositaka Sasaki, Tetsu Mizumachi, Hiroshi Matano, and several others.
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