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

Taking into account expressions (14) and (16), we can conclude that the operations of differentiation/integration of the original can be replaced by algebraic actions (multiplication/division by s) on their images [3]. Thanks to this replacement, this method has found the widest application in integral and differential calculus [4]. However, the case is of particular interest when the function is represented as L [f (t)] = F(s)/(s -k ) (13) that is, the image is divided by (s-k). In this case, depending on k , we get fractional derivatives/integrals. For k> 0 , fractional derivatives of the order k are formed, and for k <0 , fractional integrals of the same multiplicity are formed. [ ( )] = F(s) − = 1/(Г(− )) (14) SL (x, k) L [f (t)] (15) Let's consider some examples of the use of differential integral functions in solving approximation problems. Suppose must be approximated by a power series ряд _cos(x) in a neighborhood of the point x0, the function cos(x), and choose the polynomial coefficients a 0 ...a 5 so as to minimize the mean square error of approximation of this polynomial are: _cos(x) = a 0 + a 1 ∙x + a 2 ∙x 2 + a 3 ∙x 3 + a 4 ∙x 4 + a 5 ∙x 5 (16) and at the selected point is known for its derivatives and differentials, as an integer and the fraction. To do this, we fulfill the approximation conditions according to which the value of the polynomial _cos(x) and its fractional derivatives (for simplicity of calculation, only six (5) derivatives are use d 6 6 To approximate in this case, it is to decompose into a power series using differential integral functions in the vicinity of the point x 0 , bearing in mind that these points are the values of the function f (x) = cos (x). . To increase the accuracy, you can use more, for example, several dozen derivatives, the computer allows it. Instead of derivatives, its integrals can also be used in the same way) in the vicinity of a given point x0, from the domain of the polynomial definition, should equal the corresponding values of the desired function cos(x) and its fractional derivatives (and integrals). 2 points are selected as points – x = 3 and x = 15 . The fractional derivatives/integrals for the elements of the polynomial are defined as ( , , ) ≔ Г( +1)∙ − Г( +1− ) (17) where x -is the matrix of diagnostic information; n - is the exponent of the polynomial; k - is a parameter that sets the multiplicity of the integral or the order of derivatives. Further, solving a linear algebraic equation of the form: (18) we obtain the solution of this equation in the form of the desired coefficients a 0 ...a 5 (Application A Figure A.1). The solution was made in the MathCad program, the calculation listing is given for the point x = 3 and additionally for x = 15. Another example. In addition to the approximation at a point, using the differential integral functions, it is possible to approximate on a given segment. Examples of this approximation are given below. Let it be necessary to approximate, for simplicity, the known functions cos (x) and the exponent exp(x) , as well as cos(x) on the plot 4 <x <6 , as well as volume curves, according to the type of Fleicher-Manson or Robinson- Dadson curves. For ease of calculation, we approximate 6 points for 2 cos (x) functions, 4 (four) points for the exponent exp(x) and 23 for volume curves. For a sine wave, the desired points will be of two types. In the first case, these are the points -5, -4, -2, 1, 3, 5. In the second case, this is -5, -3, -1, 1, 3, 5. We will approximate the sinusoid with a polynomial (17). Exponent – exponent. These expressions (18) and (19) define fractional derivatives/integrals of order k , and are the differential functions of the desired function f(t). Examples of these functions are shown in Table 1. a =A1 -1 · B1 © 2023 Global Journals Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 13 Year 2023 ( ) I Application of Differentialintegral Functions