Global Journal of Researches in Engineering, I: Numerical Methods, Volume 23 Issue 1

Application of Differentialintegral Functions Alexey S. Dorokhov α , Solomashkin Alexey Alekseevich σ , Vyacheslav A. Denisov ρ & Kataev Yuri Vladimirovich Ѡ Summary- The article is devoted to the development and implementation of new mathematical functions, differentialintegral functions that provide differentiation and integration operations not only according to existing algorithms described in textbooks on higher mathematics, but also by substituting a certain parameter k into formulas developed in advance, forming the necessary derivatives and integrals from these formulas. The Purpose of the Research: The expansion of the concept of number, in particular, in classical mechanics, physics, optics and other sciences, including biological and economic, which makes it possible to expand some understanding of the essence of space, time and their derivatives. Materials and Methods: The idea of fractional space, time and its application is given. The usual elementary functions and the Laplace transform were chosen as the object of research. New functions, differentialintegral functions, have been developed for them. A graphical representation of these functions is given, based on the example of the calculation of the sine wave. Examples of calculating these functions for elementary functions are given. Of particular interest is the differentialintegral function, in which the parameter k is a complex number s, s = a + i · b, although in general, the parameter k can be any function of a real or complex argument, as well as the differentialintegral function itself. Research Results: As a result of the research, it is shown how the Laplace transform and Borel's theorem are used to calculate differentialintegral functions. It is shown how to use these functions to carry out differentiation and integration. It is presented how fractional derivatives and fractional integrals should be obtained. Dependencies for their calculation are obtained. Examples of their application for such functions as cos(x), exp(x) and loudness curves in music, Fletcher-Manson or Robinson-Dadson curves are shown. Conclusions: Studies show the possibility of a wide application of differentialintegration functions in modern scientific research. These functions can be used both in office and in specialized programs where calculations of fractional derivatives and fractional integrals are needed. Keywords: differentialintegral functions, derivative, fractional derivative, integral, fractional integral. I. I ntroduction n modern sciences, such as mathematics, physics, astronomy, economics and other sciences, there is little use of differential functions in calculations, because with the help of fractional derivatives and integrals, very few physical, natural, social and other processes are described that use not only the first and second derivatives, single and double integrals, but fractional derivatives and fractional integrals. So in classical mechanics, the first derivative is used as velocity, the second as acceleration, and the third as a jerk. A one-time integral is used to calculate the area under the curve, the mass of an inhomogeneous body, a two–time integral is used to calculate the volume of a cylindrical beam, a three-time integral is used to calculate the volume of the body. They can be found in the equations of mathematical physics, where, in particular, generalized functions and convolutional operations on them are used, and in spectral analysis, and in operational calculus based on integral Fourier and Laplace transformations, and in many other methods where differentiation and integration of functions are used. The basis of all these concepts is the derivative and integral 1 1 And also, definitions of derivatives/integrals based on such concepts as the Riemann-Liouville, Grunwald-Letnikov and Weyl differentialintegrals. . Two mathematical operations that are "opposite" to each other, like addition and subtraction, multiplication and division. Two "reciprocal" functions like sin(x) and arcsin(x) , x 2 and √ , e x and ln(x). Two mathematical operations that logically complement each other, the derivative of the integral does not change the integrable function, as does the integral of the derivative, leaves it unchanged. Let us recall the symbols on graphs and in computer programs. Like any mathematical operation, they have their symbols (designations) on a piece of paper, like ordinary symbols on a computer screen. So, differentiation is denoted as y’ or d/dx , and integration is ∫ ( ) . In this case, a one – time integral is denoted as ∫ ( ) , and a two - time integral is ∬ ( ) ( ) . With the derivative, the situation is more complicated, it has two designations: Author α : Chief Researcher, Doctor of Technical Sciences, Professor. e-mail: dorokhov.vim@yandex.ru Author σ : Leading Specialist. e-mail: littor2013@gmail.com Author ρ : Chief Researcher, Doctor of Technical Sciences. e-mail: va.denisov@mail.ru Author Ѡ : Leading Researcher, Candidate of Technical Sciences. e-mail: ykataev@rgau-msha.ru I © 2023 Global Journals Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 7 Year 2023 ( ) I