QUALITATIVE AND NUMERICAL ANALYSIS OF LASER-PULSE EQUATION
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Abstract
The study of the equation of the laser pulse, as well as initial and initial-boundary (mixed) problems for this equation is a rather relevant topic for researchers, because of its numerical application. The development of modern technologies makes it possible to widely use laser radiation in biological research (laboratory experiments), medicine (laser surgery, laser therapy) and cosmetology, industry (cutting, welding, painting, etc.), in commercial activities, information transmission systems. It is worth noting the expansion of the use of laser radiation in modern communication technologies. Due to the fact that the laser is able to carry much more information than radio waves, laser technology is now coming out on top, compared to radio engineering. At the same time, laser pulses are studied for the most part by numerical methods. This is because getting an explicit analytic solution is, in some cases, a rather difficult task. However, an explicit analytical solution allows us to explore many qualitative properties, which is quite valuable in terms of the development of the mathematical theory of laser radiation. In the presented work on the use of the Fourier method, we obtain an explicit analytical solution of the first mixed problem for the laser-pulse equation of the fourth-order. One of the effective tools for the study of qualitative properties of solutions of boundary value problems in mathematical physics is also the maximum principle. The problem is that in the classical theory of mathematical physics, the maximum principle is well studied only for parabolic and elliptic equations. The laser pulse equation, which is under consideration of the present work, is an equation of the hyperbolic type, for which the maximum principle does not exist in the classical formulation, in addition, it is a fourth order equation. All these nuances make the task posed in this work relevant both from a mathematical point of view, which allows to increase theoretical developments in this problem, and for further applications of laser technologies.
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References
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