About one method of numeral decision of differential equalizations in partials using geometric interpolants E.V.Konopatskiy1, O.S.Voronova1, O.A.Shevchuk1, A.A.Bezditnyi2 e.v.konopatskiy@mail.ru 1 Donbas National Academy of Civil Engineering and Architecture, Makeyevka, Donetsk region, Ukraine; 2 Sevastopol branch of «Plekhanov Russian University of Economics», Sevastopol, Russia The article presents a new vision of the process of approximating the solution of differential equations based on the construction of geometric objects of multidimensional space incident to nodal points, called geometric interpolants, which have pre-defined differential characteristics corresponding to the original differential equation. The incidence condition for a geometric interpolant to nodal points is provided by a special way of constructing a tree of a geometric model obtained on the basis of the moving simplex method and using special arcs of algebraic curves obtained on the basis of Bernstein polynomials. A fundamental computational algorithm for solving differential equations based on geometric interpolants of multidimensional space is developed. It includes the choice and analytical description of the geometric interpolant, its coordinate-wise calculation and differentiation, the substitution of the values of the parameters of the nodal points and the solution of the system of linear algebraic equations. The proposed method is used as an example of solving the inhomogeneous heat equation with a linear Laplacian, for approximation of which a 16-point 2- parameter interpolant is used. The accuracy of the approximation was estimated using scientific visualization by superimposing the obtained surface on the surface of the reference solution obtained on the basis of the variable separation method. As a result, an almost complete coincidence of the approximation solution with the reference one was established. Keywords: multidimensional approximation, multidimensional interpolation, geometric interpolant, heat equation, differential equations So, for a one-dimensional geometric interpolant (1- 1. Introduction parameter interpolant) the tree of the geometric model is Traditionally, one of the possible results of the just one line (Fig. 1), passing through the predetermined numerical solution of differential equations (DE) is a points. certain geometric model, the visualization of which allows you to visually evaluate the result. Thus, for most abstract solutions, there is a geometric interpretation. For example, the solution to an ordinary differential equation is a line, and the solution to the inhomogeneous heat equation of the rod is the surface compartment. Those, the result of solving the differential equation is a Fig. 1. 1-parameter interpolant geometric object. Change the causal relationship to the inverse. Then it turns out that in order to solve the In the BN-calculus [6-8], such an interpolant can be differential equation it is necessary to simulate some represented as the following point equation of a one- geometric object that has the required differential parameter set M of points: characteristics. A similar approach was implemented in n [1, 2]. Of course, DE have a wide variety of varieties, and M = ∑ M i pi ( u ), (1) i =1 not for every differential equation there is an exact where M – the current point of the arc of the curve of the solution. Therefore, for the numerical solution of the line passing through the predetermined points; Mi – the differential equation it is enough that the required starting points through which the arc of the curve should differential characteristics are provided at some discrete pass; pi(u) – function of the parameter u; u – is the points (network nodes) that belong to the simulated current parameter, which varies from 0 to 1; n – the geometric object. In this case, the intermediate values of number of starting points of the arc of the curve line; i – the resulting solution will be determined using serial number of the starting point. multidimensional interpolation. Then, to approximate the Moreover, the condition is that the one-parameter set solution of the DE, it is convenient to immediately use belongs to the space of the selected dimension: one of the geometric interpolants. n 2. A bit about geometric interpolant ∑ p ( u ) = 1 . This condition is satisfied using special i =1 i A geometrical interpolant is a parameterized algebraic curves obtained on the basis of the Bernstein geometrical object passing through predetermined points, polynomial [9]. The fulfillment of this condition is whose coordinates correspond to the initial experimental- mandatory for all subsequent interpolants and is not statistical information, or possessing the necessary, given in the article below, since it is calculated in a predetermined, properties. In accordance with the similar way. geometric theory of multidimensional interpolation [3-5], The point equation (1) is a symbolic notation. Having the geometric interpolant is formed by analytically performed the coordinate-wise calculation for two- describing the tree of the geometric model. dimensional space, we obtain a system of the same type parametric equations: Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0)  n dimensional space passing through predetermined points  xM = ∑ xM i pi ( u ); (Fig. 3).  i =1  n y =  M ∑i =1 yM i pi ( u ). Similarly, any point equation for a space of any dimension can be represented as a system of parametric equations. Moreover, the presented system of parametric equations is an analytical description of the projections of Fig. 3. 3-parameter interpolant the arc of a plane curve on the axis of the global coordinate system. The computational algorithm for determining the 3- A two-dimensional geometric interpolant represents a parameter interpolant will include m of 2-parametric two-parameter set of points – the surface of 3- interpolants forming an even more extended tree of the dimensional space passing through predetermined geometric model (Fig. 3): (Fig. 2).  l  M ij = ∑ M ijk pijk ( u );  k =1 ...............................  n  M = M q ( v );  i ∑ j =1 ij ij (3)  ...............................  m  M = ∑ M i ri ( w ),  i =1 where ri(w) – function of the parameter w. Summarizing this approach, we can obtain a Fig. 2. 2-parameter interpolant geometric interpolant of n dimension corresponding to n- parameter set of points or hypersurfaces (n+1)-th space The computational algorithm for determining a 2- passing through the predetermined points. At the same parameter geometric interpolant can be represented as the time, the belonging of the nodal interpolation points to following sequence of point equations, which include m the simulated geometric interpolant is ensured by the of 1-parametric interpolants at the stage of tree formation passage of all points through the guide lines (one- of the geometric model (Fig. 2): dimensional interpolants) at each stage of the formation  n of the model tree: Fig. 1 → Fig. 2 → Fig. 3.  M 1 = ∑ M 1 j p1 j ( u );  j =1 It should be noted that the geometric theory of ............................... multidimensional interpolation was developed and is  n effectively used to model and optimize multifactor  M = M p ( u );  i ∑ ij ij processes and phenomena based on any experimental j =1 statistical information [10-12]. However, in the context of  (2) ............................... the above studies, it is used for a different purpose,  n namely, for solving DE.  M m = ∑ M mj pmj ( u ); To solve the equations of mathematical physics [13],  j =1 the choice of a geometric interpolant depends primarily  m on the dimension of the Laplacian. So, for the numerical  M = M q ( v ),  ∑i =1 i i solution of the inhomogeneous heat equation with linear Laplacian ∂2U/∂x2 served as a two-parameter interpolant where qi(v) – function of the parameter v. U=f(x,t) [2]. Then with a flat Laplacian ∂2U/∂x2 + ∂2U/∂y2 To describe a 2-parameter interpolant (Fig. 2), a 3- the use of a three-parameter interpolant is necessary dimensional Cartesian coordinate system is used (although the proposed equations are also valid for an U=f(x,y,t), and with spatial ∂2U/∂x2 + ∂2U/∂y2 + ∂2U/∂z2 - affine coordinate system). In addition, such a geometric four-parameter interpolant U=f(x,y,z,t). interpolant can exist in a space of higher dimensions. In 3. Analytical description of geometric this case, the point equation will remain unchanged, but interpolants when performing the coordinate-wise calculation of the parametric equations of the system, there will be more, For the analytical description of geometric and their number will directly depend on the dimension interpolants, the point equations of algebraic curves arcs of the space in which the simulated geometric object is passing through the predetermined points obtained on the located. basis of Bernstein polynomials [9] are used. The need to Similarly, a three-parameter interpolant is defined by determine such curves lies in the fact that when modeling a 3-parameter set of points – a hypersurface of 4- multi-factor processes for each separate problem, it is necessary to solve systems of linear algebraic equations (SLAE) in determining the desired equation. To obtain a accordingly, for solving differential equations with a universal approach to modeling multifactor processes [3- Laplacian of any dimension 5], it was necessary to obtain such equations of arcs of Another important feature of the obtained equations algebraic curves into which you can substitute any values of the curve arc is the uniform distribution of the of the points coordinates (both fixed and variable), and parameter values, which was originally laid down in the immediately obtain the desired result. For this, the SLAE method for determining the curve arc passing through solution process was laid down directly at the stage of predetermined points. Moreover, for each specific curve modeling. As a result, we obtained the point coordinate axis having a uniform distribution of the equations of algebraic curves arcs passing through coordinates of the source points, a linear relationship predetermined points, which are the main tool of the between the natural value of the factor belonging to the geometric theory of multidimensional interpolation and projection axis and the current parameter is valid. This approximation. significantly reduces the amount of necessary It should be noted a very important distinguishing calculations when approximating the solution of the feature of the obtained equations. For point equations, the differential equation, allowing us to consider them on a belonging of a geometric object to a space of a specific regular multidimensional network of points. Moreover, dimension is determined by the sum of functions of a the method is universal in nature and without making any parameter (condition to equation (1)), which must be changes, it can be fully used for both regular and equal to 1. The using of Bernstein polynomials made it irregular network of points. possible to ensure that this condition is met regardless of In this way, point equations of arcs of curves of 2–10 the dimension of the space of the global coordinate order, passing through 3–11 points, respectively, were system. The functions of the parameter are determined by obtained. For example: the Newton binomial, which is expanded for the 1. The point equation of an arc of a curve of the 2nd parameter and its complement to 1. By this, it provides order passing through 3 predetermined points: the condition that the arc of the curve belongs to a M= M 1u (1 − 2u ) + 4uuM 2 + M 3u ( 2u − 1) , (4) specific space, regardless of its dimension. In other where u = 1 − u - parameter addition u to 1. words, the obtained parametric equations of the arc of the 2. The point equation of an arc of a third-order curve curve can be used for a space of any dimension and, passing through 4 predetermined points: M= M 1 ( u 3 − 2,5u 2 u + uu 2 ) + M 2 ( 9u 2 u − 4,5uu 2 ) + M 3 ( −4,5u 2 u + 9uu 2 ) + M 4 ( u 2 u − 2,5uu 2 + u 3 ) . (5) 8. We solve the obtained SLAE and determine the 4. General approach to the approximation of necessary values at the nodal points of the the solution of differential equations interpolant. After that, we substitute the result of The main idea of the proposed approximation method calculations in the approximation equation from the is that at the nodes of the selected interpolation network 5th point. of points the condition of the original differential 9. We analyze the result and check its reliability. In the equation is satisfied. For its implementation, the case of insufficiently accurate results, we increase following fundamental computational algorithm was the number of nodal points of the geometric formed: interpolant. 1. Depending on the source differential equation, form Of course, each engineering task is separate in nature a network of points of the required dimension and and has its own characteristics, but this will not affect the density, which will be the basis for creating the tree fundamental approach to solving differential equation. of the geometric model. For example, with a large number of nodal points, it is 2. Select arcs of approximating curves for an analytical possible to use composite approximating curves that will description of a geometric interpolant, thereby form composite geometric objects in multidimensional forming a computational sub algorithm. space. If necessary, they can be docked with the required 3. Perform coordinate-wise calculation and, in the case smoothness order [14-19]. And with an increase in the of using a regular network of points, go from the order of the differential equation, an increase in the order parametric equations system of the geometric of the approximating curve is necessary. Moreover, in interpolant analytical description to its equation in an order to obtain the correct result of solving the explicit form. differential equation, it is necessary that the order of the 4. Enter the coordinates of the points corresponding to approximating curve be greater than the order of the the initial and boundary conditions. original differential equation. 5. To differentiate the obtained equations and substitute It should be noted that the result of the them in the original DE. implementation of the proposed computational algorithm 6. Substitute parameter values at the nodal points, will be a general control solution. It can have an infinite thereby forming a local system of linear algebraic number of particular solutions. A specific solution is equations (SLAE). distinguished from a variety of particular solutions using 7. In the case of using piecewise approximation, we initial and boundary conditions, which, unlike most repeat the first 6 points of the computational methods for solving differential equation, must be laid algorithm several times, thus accumulating local down in the form of input data at the stage of creation SLAEs to form a global SLAE. and analytical description of the geometric interpolant, thereby forming point 4 of the computational algorithm. In other words, the geometric display of the initial and ∂U ∂ 2U boundary conditions are also some geometric objects: = a 2 2 + 2 x + 1, ∂t ∂x points, lines, surfaces, etc. Thus, the desired geometric (6) interpolant must be a carrier of geometric objects 0 < x < 1, t > 0, U ( 0, t ) = 1, corresponding to the initial and boundary conditions. U (1, t )= 2, U ( x, 0 )= x + 1. 5. An example of approximation of the solution To approximate the solution of equation (6), we use a of the heat equation using a two-parameter 16-point 2-parameter interpolant [2]. Using the point interpolant equation of the arc of a third-order curve passing through 4 predetermined points, we obtain the following Consider the use of the proposed method on the computational algorithm for determining a 16-point 2- example of solving the following inhomogeneous heat parameter interpolant, which is determined using the equation: point equation (5) by the following sequence of point equations:  M= 111 M 1111 ( u 3 − 2,5u 2 u + uu 2 ) + M 1112 ( 9u 2 u − 4,5uu 2 ) + M 1113 ( −4,5u 2 u + 9uu 2 ) + M 1114 ( u 2 u − 2,5uu 2 + u 3 ) ,   M= 112 M 1121 ( u 3 − 2,5u 2 u + uu 2 ) + M 1122 ( 9u 2 u − 4,5uu 2 ) + M 1123 ( −4,5u 2 u + 9uu 2 ) + M 1124 ( u 2 u − 2,5uu 2 + u 3 ) ,   M= M 1131 ( u 3 − 2,5u 2 u + uu 2 ) + M 1132 ( 9u 2 u − 4,5uu 2 ) + M 1133 ( −4,5u 2 u + 9uu 2 ) + M 1134 ( u 2 u − 2,5uu 2 + u 3 ) , (7)  113  M= M ( u 3 − 2,5u 2 u + uu 2 ) + M ( 9u 2 u − 4,5uu 2 ) + M ( −4,5u 2 u + 9uu 2 ) + M ( u 2 u − 2,5uu 2 + u 3 ) ,  114 1141 1142 1143 1144  M= M ( v 3 − 2,5v 2 v + vv 2 ) + M ( 9v 2 v − 4,5vv 2 ) + M ( −4,5v 2 v + 9vv 2 ) + M ( v 2 v − 2,5vv 2 + v 3 ) ,  11 111 112 113 114 where u = 1 − u and v = 1 − v . U. Thus, the number of equations in the sequence (7) will Perform coordinate-wise calculation of the sequence triple. Given the special properties of arcs of algebraic of equations (7) for 3-dimensional space. To do this, we curves obtained on the basis of Bernstein polynomials adopt a Cartesian coordinate system with axes: x, t, and and described above, we obtain: t = u;   x = v; (8)  (3 2 2 ) 2 ( 2 ) 2 ( 2 2 ) 2 ( 3 U 11 = U 111 v − 2,5v v + v v + U 112 9v v − 4,5v v + U 113 − 4,5v v + 9v v + U 114 v v − 2,5v v + v , ) 4 5 where U= M1141 U= M1142 U= M1143 U= M1144 2 , U M1121 = and U M1131 = . 3 3 U= 111 U M1111 ( u − 2,5u u + uu ) + U M1112 ( 9u u − 4,5uu ) + U M1113 ( −4,5u 2 u + 9uu 2 ) + U M1114 ( u 2 u − 2,5uu 2 + u 3 ) ; 3 2 2 2 2 U= 112 U M1121 ( u 3 − 2,5u 2 u + uu 2 ) + U M1122 ( 9u 2 u − 4,5uu 2 ) + U M1123 ( −4,5u 2 u + 9uu 2 ) + U M1124 ( u 2 u − 2,5uu 2 + u 3 ) ; U= 113 U M1131 ( u 3 − 2,5u 2 u + uu 2 ) + U M1132 ( 9u 2 u − 4,5uu 2 ) + U M1133 ( −4,5u 2 u + 9uu 2 ) + U M1134 ( u 2 u − 2,5uu 2 + u 3 ) ; U= 114 U M1141 ( u 3 − 2,5u 2 u + uu 2 ) + U M1142 ( 9u 2 u − 4,5uu 2 ) + U M1143 ( −4,5u 2 u + 9uu 2 ) + U M1144 ( u 2 u − 2,5uu 2 + u 3 ) . Using the linear dependence of the first two equations indicate the corresponding coordinates of the points along of system (8), we pass to the explicit equation of the the axis U: UM1111=UM1112=UM1113=UM1114=1, approximating 2-parameter interpolant. Thus, it remains to determine the values of the Further, to ensure the initial and boundary conditions, geometric interpolant at 6 points: M1122, M1123, M1124, it is necessary that the obtained geometric interpolant M1132, M1133 and M1134 that correspond to the following passes through 3 straight lines: U(0,t)=1, U(1,t)=2 and values of the parameters of the nodal points of the U(x,0)=x+1. To ensure these conditions, it suffices to interpolant: 1 1 1 2 1 = xM1122 =; tM1122 ;= xM1123 =; tM1123 ;=xM1124 =; tM1124 1; 3 3 3 3 3 2 1 2 2 2 = xM1132 =; tM1132 ;= xM1133 =; tM1133 ;= xM1134 = ; tM1134 1. 3 3 3 3 3 As a result, we obtain a SLAE of 6 equations with 6 U =1 + x − 0, 608t 3 x 3 + 1, 212t 3 x + 1, 247t 2 x 3 − unknowns: U1122, U1123, U1124, U1132, U1133 and U1134. Solving this SLAE and substituting the obtained values in −2, 634t 2 x − 0, 785tx 3 + 1, 783tx − the equation of an approximating two-parameter −0, 604t 3 x 2 + 1,387t 2 x 2 − 0,998tx 2 . interpolant, taking into account the rounding of the Having checked the result obtained by comparing the coefficients of the equation, we obtain: obtained solution with the solution obtained on the basis of the variable separation method: U ( x, t ) = x + 1 + them on each other (Fig. 4). In this case, the green solution shows the reference solution obtained by the ( 6 ( −1) + 2) 1 − e n +1 ( ) sin (π nx ). method of separation of variables. ∞ ∑ π n π na 2 − (π na ) t + ( ) 2 n =1 For a visual comparison of the obtained results, we will visualize the obtained surfaces and superimpose Fig. 4. Comparison of the results of solving the inhomogeneous heat equation As can be seen from Figure 4, with the help of the 16- higher-dimensional Laplacian. In this case, the point geometric interpolant, it was possible to achieve an computational algorithm does not have fundamental almost complete degree of coincidence with the reference differences. Only increases the dimension of the solution. Moreover, further use of the obtained geometric interpolant and the number of equations of polynomial equation for engineering calculations is more coordinate calculation. Based on this, we consider not a preferable in comparison with the equation obtained by particular, but a general solution of the heat equation, the method of separation of variables. It should be noted given in a general form for a three-dimensional that, if necessary, the number of nodal points of the Laplacian: approximating network can be practically any and can ∂U  ∂ 2U ∂ 2U ∂ 2U  always be increased to achieve the required accuracy of = a 2  2 + 2 + 2  + f ( x, y , z ) . (9) ∂t  ∂x ∂y ∂z  the solution. As a geometric interpolant, we choose a 4-parameter 6. A generalization of the proposed solution of hypersurface belonging to a 5-dimensional space. As an the inhomogeneous heat equation to a example, let us take a curve of the third order passing multidimensional space through 4 forward given points (5) as an approximating arc. Then the computational algorithm for determining Let us consider a generalization of the proposed solution of the inhomogeneous heat equation for a the 4-parameter interpolant takes the following form:  M=1 M 11 ( w3 − 2,5w2 w + ww2 ) + M 12 ( 9 w2 w − 4,5ww2 ) + M 13 ( −4,5w2 w + 9 ww2 ) + M 14 ( w2 w − 2,5ww2 + w3 ) ,   M=2 M 21 ( w3 − 2,5w2 w + ww2 ) + M 22 ( 9 w2 w − 4,5ww2 ) + M 23 ( −4,5w2 w + 9 ww2 ) + M 24 ( w2 w − 2,5ww2 + w3 ) ,   M= M 31 ( w3 − 2,5w2 w + ww2 ) + M 32 ( 9 w2 w − 4,5ww2 ) + M 33 ( −4,5w2 w + 9 ww2 ) + M 34 ( w2 w − 2,5ww2 + w3 ) , (10)  3  M = M ( w3 − 2,5w2 w + ww2 ) + M ( 9 w2 w − 4,5ww2 ) + M ( −4,5w2 w + 9 ww2 ) + M ( w2 w − 2,5ww2 + w3 ) ,  4 41 42 43 44  M= M (ϕ 3 − 2,5ϕ 2ϕ + ϕϕ 2 ) + M ( 9ϕ 2ϕ − 4,5ϕϕ 2 ) + M ( −4,5ϕ 2ϕ + 9ϕϕ 2 ) + M (ϕ 2ϕ − 2,5ϕϕ 2 + ϕ 3 ) ,  1 2 3 4 where w = 1 − w and ϕ = 1 − ϕ . equation with a flat Laplacian ∂2U/∂x2 + ∂2U/∂y2 the It should be noted that the sequence (10) did not number of nodal points will be 64. include 16 2-parameter interpolants Mij, which also must We perform the coordinate-wise calculation of the be determined by analogy with the sequence (7). Thus, sequence of equations (10) for a 5-dimensional space. To the desired geometric interpolant will pass through 256 do this, we adopt a Cartesian coordinate system with nodal points. Accordingly, for the inhomogeneous heat axes: x, y, z, t and U. Given the special properties of the arcs of algebraic curves obtained on the basis of Bernstein polynomials and described above, we obtain:  =t at u + bt ; = x ax v + bx ;  =  y a y w + by ; (11)  =  z a z ϕ + by ; U= U ϕ 3 − 2,5ϕ 2ϕ + ϕϕ 2 + U 9ϕ 2ϕ − 4,5ϕϕ 2 + U −4,5ϕ 2ϕ + 9ϕϕ 2 + U ϕ 2ϕ − 2,5ϕϕ 2 + ϕ 3 ,  1( ) 2( ) 3( ) 4( ) where ax, ay, az, at, bx, by, bz, bt – parameters that are determined depending on the initial and boundary Acknowledgments conditions for solving the differential equation. This work was completed and published with Further, taking into account the linear dependence of financial support from the Russian Foundation for Basic the first 4 equations of system (11), we proceed to the Research, grant 19-07-01024. equation given explicitly U=f(x,y,z,t). We differentiate it in accordance with equation (9) and, substituting the References parameter values at the nodal points of the interpolant one by one, we compose a SLAE, solving which we [1] Konopatskiy E.V. The solution of differential obtain the desired numerical solution of the equations by geometric modeling methods. inhomogeneous heat equation. Proceedings of the 28th International Conference on Similarly, other arcs of curves passing through Computer Graphics and Machine Vision GraphiCon forward given points of a higher order or obtained in 2018. September 24-27, 2018. Tomsk: TPU, 2018. some other way can be used to approximate the solution pp. 322-325. of the differential equations. 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