By Karlos J. Kachiashvili

The consequences mentioned during this ebook are attention-grabbing and necessary for a variety of experts and scientists operating within the box of utilized arithmetic, and within the modelling and tracking of pollutants of average waters, ecology, hydrology, strength engineering and construction of alternative constructions of water gadgets. Their value and functional worth are submitted within the pleasant shape for comprehension and are prepared for direct software for the answer of sensible projects. benefits of the elaborated tools and algorithms are proven not just via theoretical decisions and calculations, but additionally throughout the demonstration of result of specific calculus and modelling.

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3. 41) will be executed automatically. , N ) are constant coefficients. , N ) are additional constant coefficients, and the symbol  marks the ascending differences. The values   j , x j ,  j ,  j are determined by the condition of smoothness of the curve in nodes and the boundary conditions for tangent vector to the curve in the point [ x 0 , y 0 ] . 3). 3. Trigonometrical Spline-Interpolation. , N  1 by means of the following 28 Karlos J. Kachiashvili and D. Y. Melikdzhanian  y  cos  j ;  x   sin  j ;  is unit vector tangent to the curve in the point [ x j , y j ] ;  j  2 / 4 represents slope of this vector to a positive direction of the abscissa axis; a x  ( x j 1  x j ) / 2; a y  ( y j 1  y j ) / 2; a  a x2  a y2 ; 2 a is a position vector conducted from the point [ x j , y j ] to the point [ x j 1 , y j 1 ] ; cos  ( x a x   y a y ) / a; sin   ( x a y   y a x ) / a;  is the angle, on which the vector  should be turned to make it parallel to vector a ;  j  sin  / a; R  1 /  j is the radius of the arc of the circumference coincided with a part of the curve between its points [ x j , y j ] and [ x j 1 , y j 1 ]; at line segment; at  j  0 this arc degenerates into a straight    j  0 the center of this arc [ x j , y j ] is to the left of the curve, and at  j  0 - on the right; therefore, at  j  0 the arc is drawn from the point [ x j , y j ] in [ x j 1 , y j 1 ] counter-clockwise, and at  j  0 - clockwise;  j 1   j  2 ;   2 /  , at   0; j j  j  j 1    j  2 , at  j  0.

34). 29). 29). 33) is changed by condition 46 Karlos J. Kachiashvili and D. Y. 28). Theoretical substantiation of such splitting exists [216]. 35) may be solved by different methods. We have constructed effective finite-difference schemes for their solution [217, 218]. In case of classical boundary conditions, the method of scrolling is used for solving the received tree-point equations [219]. In case of non-classical boundary conditions, the scrolling formulae of special type are derived [30, 221].

Interpolating spline-curve for the sinusoid in case of infrequent nodes. 32 Karlos J. Kachiashvili and D. Y. Melikdzhanian After we compare two offered methods of construction of splines: one - by means of trigonometrical functions and another - by means of integrals of fractional rational functions, it is possible to say, that advantage of the first method is the relative simplicity and reliability of calculations, its imperfection – more sensitivity to “rarefaction” of nodes. The description of offered algorithms contains everything required for their implementation in form of computer programs.

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