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

help of sequential multiple calculations can the order of the derivative be increased to the desired value. The same applies to integration. II. M aterials and M ethods This method of calculating derivatives reduces the efficiency of using the differentiation operation, for example, in series expansions, because it requires calculating derivatives of a "high" order, and this is time- consuming and involves calculation errors. Therefore, in such calculations, only the first few terms of the decomposition are taken, and the rest are discarded, which increases the calculation error. As for calculating integrals, especially multiplicities greater than 2 , this is an even more difficult task. Thus, the lack of a simple, reliable and accurate method of differentiation and/or integration significantly hinders computational progress in mathematics. The same problem is observed in physics. Many laws of mathematical physics, most often appearing in simple, accessible calculations, are based on the use, mainly, of the 1st, maximum 2nd derivative (for example, current i = dq / dt , force F = m · d 2 x/ dt 2 ) and a single integral, for example, voltage across the capacitor u(t) = 1 / C · ∫ ( ) . It is very rare in everyday physics or mathematics to find a 3rd derivative or a 3-fold integral. This does not happen often. One of the ways to use a 3-fold integral is the Ostrogradsky-Gauss integral to calculate the volume of a body if the surface bounding this body is known. And if you look more broadly, then neither in physics nor in mathematics have the everyday laws of the universe using fractional derivatives and integrals been discovered so far, because their calculation is fraught with great difficulties [1]. At the same time, it is possible that all the diversity of the world exists exactly there, in a fractional dimension, which can be described and studied, precisely with the help of fractional (analog), and not integer (discrete) integrals and differentials. Take, for example, the mechanism of describing multidimensional structures, for example, multidimensional space. Our 3-dimensional space and one-dimensional time are described by discrete (integer) coordinate values, in this case one and three. At the same time, the question of the existence of a space having, not 3, but, say, 2,345 coordinates is of great scientific and practical interest. In other words, the structure of a special "fractional" space, no longer two-dimensional, is a plane (because to describe the plane, you need 2 coordinates, and we have more – 2,345), but also not a three-dimensional volume (where 3 coordinates are needed), i.e. something average between the plane and the volume. It is very difficult to imagine such a structure. In nature, such a space does not seem to exist. It is even more difficult to determine the velocity or acceleration in such a space, i.e. to describe the kinematics of the motion of bodies. If it is possible to define the force in such a space (or to use the already existing classical method of specifying forces), then we can count on success in creating the dynamics of such structures, i.e., in other words, to create the mechanics of multidimensional space. At the same time, our classical 3- dimensional mechanics will turn out to be a special case of a more general mechanics – the mechanics of multidimensional spaces. This can be said about other physical laws of the universe. And whether our idea of the world will change with the emergence of a new, more general, idea of space. So far we don't know much about this, because our concepts are tied to a three-dimensional dimensional space, and all the diversity of the world "lies" in a multidimensional "fractional" world that has not been studied at all. Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 10 Year 2023 © 2023 Global Journals ( ) I Application of Differentialintegral Functions A number of legitimate questions arise: - What kind of space is "located", say, between a plane (2-dimensional space) and a volume (3-dimensional), i.e. a substance with the dimension of space R , where 2<R <3 ? - What kind of physical quantity, which is between speed and acceleration between y <1> and y <2> from the move, i.e. a physical quantity, defined, for example, the fractional derivative of y <1,23> , the order of 1,23 (not 1 or 2)? - Whether Newton's laws are applicable to the so-called fractional space? - How will the definition of force in fractional space change (if it changes)? - Will it be possible to apply the classical laws of mechanics to fractional space, or will it be necessary to create a new, more general, mechanics of the macro and microcosm? - Will the interaction between space and time change if we "replace" the classical concept of space with a fractional one? - Will there be changes in Einstein's theory of relativity and will the concepts of "gravitational, electromagnetic and other interactions" and much, much more remain the same? Answers to these and other questions can be obtained if you have a convenient apparatus for calculating derivatives/integrals of any order/multiplicity, including fractional ones. In other words, it is necessary to create such