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

© 2023 Global Journals Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 11 Year 2023 ( ) I Application of Differentialintegral Functions a calculation algorithm, simple and convenient, especially for novice researchers, where instead of calculating integrals/differentials, it would be possible to use the usual substitution of numbers, in which the desired order or multiplicity could be set without performing calculations, but simply substitute the desired parameter into the desired formula and get a ready derivative/integral without their calculations, i.e. immediately. Such a tool, which could be called, for example, functions - SL(x, k) , would greatly simplify the process of calculating derivatives and integrals and significantly expand the boundaries of our knowledge. First, we introduce the concepts of a differential integral function based on the definition of a differential integral. The differential integral function SL (x, k) is an ordinary function of several arguments, where, separated by commas, its arguments (in this c as e one – x ) and the parameter k , the order of future derivatives and/or the multiplicity of integrals are indicated 2 For example, for a parabola y(x) = x 2 , such a differentialintegral function SL(x, k) will have the form . 3 This is the differential integral function of a parabola, the usual function of 2 argume nts , argument x and parameter k . It represents a whole set of integrals and derivatives of any order and multiplicity . (1) where, x is the argument of the function, k is a parameter that specifies the order of the derivative or the multiplicity of the integral. 4 For example, for a parabola, we substitute k = 0 into it. Then, for k = 0y (x, k) = x 2 , ( Г (3 - k) = 2) (the main, mother function). How to use it? You need to set the parameter k and get the desired derivative or integral. 5 2 Here SL(x, k) is another form of writing a power differential function, different from writing the formy <k> . 3 Here and further calculations are performed in the MathCad program, so it uses a dot in its formulas instead of a comma. 4 As the latter, there may be the differentialintegral functions themselves. In this case, the parameter k can also be a complex value. 5 G(x) - gamma function. the function (parabola) does not change. When k = 1y (x, k) = 2x and the parabola is transformed into its 1st derivative - y <1> . When k = -1 y (x, k) = x 3 /3 and the function becomes its one-time integral – y <-1> , and for k = -2 y (x, k) = x 4 /12 - double - y <-2> . No calculations, just substitution. Fractional derivatives and integrals are of particular interest, because there is no simple and reliable way to calculate them, except for the method indicated above [2]. In this case, the method of obtaining is the same. To calculate them, it is enough to substitute the necessary value of the derivative instead of the parameter k, for example, k = 0.123 and the parabola becomes its derivative of the order 0.123 – y <0.123>: (2) If it is necessary to obtain an integral of multiplicity 3,45 - y <-3,45 > , it is enough to substitute k = -3,45 into the differential function (1) and the parabola becomes its integral of multiplicity 3,45 - y <-3,45> : (3) This method of calculating fractional derivatives is no different from the method of obtaining integer (discrete) derivatives – the same substitution. There is no difference between an integer or fractional derivative/integral. Simple substitution to get a given result. Consider another example: y(x)=sin(x). For a sine wave, the differentialintegral function SL(x,k) will have the following form: (4) This is a sine wave whose phase shift depends on the order of its derivative/multiplicity of its integral. At k = 0 , the sine wave does not change, at k = 1 , and becomes cos(x) , i.e. its the first derivative is y <1> , and at k = -1 it becomes -cos(x) , i.e. its integral is y <-1>. . At -1<k <1 , the function occupies an intermediate position between - cos(x) and cos(x), including sin(x) at k = 0 . The differential integral function for the sine wave (4) is a graphical representation of the differential integral function, namely, the parameter k represents a part of the right angle for unit orts. At k = 1 , the function SL(x,1) becomes the 1st derivative, such a unit ort is perpendicular to the abscissa axis, and at k = var it is a fractional derivative of k order and the angle k (in values from 0 to 1 or in % of 90 degrees) it is only a part of the right angle. For the exponent y(x) = e x , the differential integral function SL(x, k) does not depend on k and all its derivatives and integrals are equal to each other and equal to the exponent itself. ( , ) ≔ 2 ∙ 2− Г(3− ) ( , ) ≔ 2 ∙ 2−0,123 Г(3−0,123) , 3 → 1,12 ( , ) ≔ 2 ∙ 2+3,45 Г(3+3,45) , 3 → 7,6060 −3 5,4 ( , ) ∶= sin ( + ∙ 2 )