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

Figure 2: Notation of derivatives and integrals for a parabolay(x) = x 2 At the same time, all derivatives, including fractional ones, having a negative index, are located on the numerical axis on the right, and all integrals with a positive index - on the contrary, on the left. It was possible to arrange the designations differently, change the plus to minus, but the essence would not change at the same time. There are many types of symbols, binding to the numeric axis requires clarification. To bring these notations in line with the numerical axis "K", the 4th line contains universal notations for derivatives of any order and integrals of any multiplicity, using angle brackets. The angle brackets denote the order of the derivative or the multiplicity of the integral, for example, y <0> = y(x) is the function under study, and y <-1> = ∫ ( ) is its integral, multiplicity 1. So y <2> = d 2 /dx 2 = y" is the second derivative, and y <-0,46> is the integral, multiplicity 0,46 . For example, a certain derivative of the order of 1,35 is denoted as y <1,35> . In other words, if there is a positive number in the angle brackets, it means it is some kind of derivative, and if it is negative, it means it is an integral. And it is easy to read, and it is located correctly on the numeric axis: negative values of the k index are on the left, and positive values are on the right. This form of writing integrals and derivatives is very convenient, for example, for their designation on graphs or diagrams. Figure 2 shows an example of the notation of derivatives and integrals for the parabola y(x) = x 2 . In addition to notation on graphs, this method can be used for programmers writing programs in various programming languages, for example, ... int main () { float y, u, z; int n3; ... z= y (4) <1.5>; u=n3 <-0,25>; … where y <1,5> is the derivative of the function y(4) of order 1,5 and n3 <-0.25> is the integral of multiplicity 0,25 of the function n3 . In Figure 2, the integral of multiplicity – 0,46 and the derivative of the order of 1,35 are shown for x > 0 . It should be borne in mind that when calculating a derivative of a "high" order, say, 123 orders – y <123> , previously it was necessary to perform 122 differentiation operations beforehand. This is due to the fact that the definition of the derivative/integral implies an increase in the order of the derivative/integral by only 1. It is impossible, using the existing definition of the derivative, to immediately calculate a high-order derivative from it. Only with the y 1 = y < - 1> 1,35 2,0 1,0 0,0 - 0,46 - 1,0 k x x x x x x y 2 = y < - 0,46 > y 3 = y <0> y 4 = y < 1> y 5 = y < 1,35 > y 6 = y < 2 > y < - 1> = x 3 / 3 y < - 0,46> = 0,62 x 2,46 y <0> = x 2 y <1> = 2 x y <1,35> = 2,22 x 0,65 y <2> = 2 © 2023 Global Journals Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 9 Year 2023 ( ) I Application of Differentialintegral Functions