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

y' and d/dx . Figure 1 shows (as one of the options) the currently existing designations of differentials and integrals, widely used in the literature. Figure 1: Notation of integrals and derivatives As can be seen from Figure 1, all the variety of these notations has one property common to all: they try to reflect in various ways, either with the help of numbers or graphically, the order of derivatives or the multiplicity of the integral. In order to unify the record of derivatives and integrals, consider them relative to a certain numerical axis "K" (Figure 2), where the value of the parameter k corresponds to the multiplicity of the integral or the order of the derivative. So, in this scenario of notation, k = -1 corresponds to the designation of a single integral ∫ ( ) from the 2nd line and the designation of the same integral f 1 *y from the 3rd row, and for k = 1 - we have the designation of the first derivative y' from the 1st row and the designation of the same first derivative d /dx from the 2nd row. The third line contains the notation of differentials and integrals based on convolutional operations of generalized functions: y (k) = f -k * y , where k >0 , a value unequal to an integer is called a fractional derivative of order k . An expression of the form: y (k) = f k * y is called a primitive of order k , i.e. an integral of multiplicity k [1]. < - 1> < - 0,46> <0> <+1> <+1,35> <+2> y y y y y y ---|------------------x------------------|------------------------------|---------------x--------------------|--------> K- 1 - 0,46 0 + 1 + 1,35 + 2 --- --- y y ' --- y '' a b xy ⌠  ⌡ d 2 x yd d 2 --- y x yd d --- f 1 * y f 0,46 * y f 0 * y f -1 * y f -1,35 * y f -2 * y (1) (2) (3) (4) Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 8 Year 2023 © 2023 Global Journals ( ) I Application of Differentialintegral Functions