Some classes of surfaces generated by Nielson and Marshall type operators on the triangle with one curved side

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1 Stud. Univ. Babeş-Bolai Math. 61(2016), No. 3, Some classes of surfaces generated b Nielson and Marshall tpe operators on the triangle with one curved side Teodora Cătinaş Dedicated to Professor Gheorghe Coman on the occasion of his 80th anniversar Abstract. We construct some classes of surfaces which satisf some given conditions, using some Hermite, Nielson and Marshall tpe interpolation operators defined on a triangle with one curved side. Mathematics Subject Classification (2010): 41A05, 41A36, 65D05. Kewords: Blending interpolation, Hermite, Nielson and Marshall interpolation operators, surfaces generation. 1. Introduction In some recent papers, we have introduced and studied some interpolation operators for the functions defined on triangles with curved sides (see, e.g., [8]-[11], [13], [14], [16], [17]). The permit essential boundar conditions to be satisfied eactl and the come as an etension of the interpolation operators on triangles with all straight edges, introduced and studied for eample in [1], [3]-[7], [12], [22]-[28]. We consider here a standard triangle, T, having the vertices V1 = (1, 0), V 2 = (0, 1) and V 3 = (0, 0), two straight sides Γ 1, Γ 2, along the coordinate aes, and the third side Γ 3 (opposite to the verte V 3 ), which is defined b the one-to-one functions f and g, where g is the inverse of the function f, i.e., = f() and =, with f(0) = g(0) = 1. There is no restriction to consider this standard triangle T, since an triangle with one curved side can be obtained from this standard triangle b an affine transformation which preserves the form and order of accurac of the interpolant [4].

2 306 Teodora Cătinaş V 2 1 (,f()) Γ 3 Γ (0,) 0.4 (,) (,) V V 1 (,0) Γ 2 Figure 1: Triangle T. The bending interpolants interpolate on an infinite set of points (segments, curves, etc.), so having such element as a boundar of an object, we ma generate surfaces that contain the given boundar (see, e.g., [2], [18]-[21]). The aim of this paper is to construct some surfaces which satisf some given conditions on the boundar of a domain that can be decomposed in triangles with one curved side. We construct some new surfaces using some Hermite, Nielson and Marshall tpe interpolation operators introduced in [13] and [14]. These operators come as etensions to triangle T, of some interpolation operators for triangles, given, for eample, in [4], [5], [25]. 2. Surfaces generation b Hermite, Nielson and Marshall tpe operators Suppose that F is a real-valued function defined on T, and that it has all partial derivatives needed. We consider three tpes of interpolation operators defined on T : - the Hermite interpolation operators H 1 and H 2 defined b [13]: (H 1 F )(, ) = [2+][ ]2 g 3 () F (0, ) + [ ]2 F (1,0) (0, ) (2.1) + 2 [ 2+3] g 3 () F (, ) + 2 [ ] F (1,0) (, ), (H 2 F )(, ) = [2+f()][ f()]2 f 3 () F (, 0) + [ f()]2 F (0,1) (, 0) + 2 [ 2+3f()] f 3 () F (, f()) + 2 [ f()] F (0,1) (, f()), - the Nielson tpe interpolation operators given b [14]: (N 1 F )(, ) = F (, f()) + (1 f())f (, ), (2.2) (N 2 F )(, ) = F (0, ) + F (, 0) F (0, 0),

3 Classes of surfaces generated b Nielson and Marshall tpe operators the Marshall tpe operators defined b [14]: (Q 1 F )(, ) =F (0, 1) + F (, 0), (2.3) f() ), (Q 2 F )(, ) =F (1, 0) + f()f (0, ( (Q 3 F )(, ) =(f() + (1 f() + )F 1 f()+, ) 1 f()+. For obtaining the first class of surfaces, we consider the boolean sum of the Nielson tpe operators N 1 and N 2, given in (2.2), namel, ((N 1 N 2 )F )(, ) =[F (, f()) F (0, f()) F (, 0) + F (0, 0)] (2.4) + (1 f())[f (, ) F (0, ) F (, 0)] + F (0, ) + F (, 0) f(, and we appl the condition that the roof stas on its support, i.e., F Γ3 = 0. We get S N := F (0, f()) + (1 )F (, 0) + [ f()]f (0, 0) (2.5) + f()f (0, ) + [f() 1]F (, 0). Theorem 2.1. If F Γ3 = 0, then we have the following properties of the operator S N : (S N F )(, 0) = F (, 0). (S N F )(0, ) = F (0, ). (S N F )(, f()) = 0. Proof. The proof follows directl b the epression of S N from (2.5). In the second level of approimation we use the Hermite interpolation operators, given in (2.1), taking into account the condition F Γ3 = 0, i.e., (H1 1 F )(, ) := [2+][ ]2 g 3 () F (0, ) + [ ]2 F (1,0) (0, ) (2.6) + 2 [ ] F (1,0) (, ), (H2 1 F )(, ) := [2+f()][ f()]2 f 3 () F (, 0) + [ f()]2 F (0,1) (, 0) We appl the following approimations: + 2 [ f()] F (0,1) (, f()). F (, 0) (H 1 1 F )(, 0) =(2 + 1)( 1) 2 F (0, 0) + ( 1) 2 F (1,0) (0, 0) (2.7) + 2 ( 1)F (1,0) (1, 0), F (0, ) (H 1 2 F )(0, ) =(2 + 1)( 1) 2 F (0, 0) + ( 1) 2 F (0,1) (1, 0) (2.8) + 2 ( 1)F (0,1) (0, 1), and b (2.5), we obtain the following class of surfaces: (S 1 F )(, ) = (H 1 2 F )(0, f()) + (1 )(H 1 1 F )(, 0) + [ f()]f (0, 0) + f()(h 1 2 F )(0, ) + [f() 1](H 1 1 F )(, 0),

4 308 Teodora Cătinaş i.e., (S 1 F )(, ) = {[2f() + 1][f() 1] 2 F (0, 0) + f()[f() 1] 2 F (0,1) (1, 0) (2.9) + f() 2 [f() 1]F (0,1) (0, 1)} + (1 )[(2 + 1)( 1) 2 F (0, 0) + ( 1) 2 F (1,0) (0, 0) + 2 ( 1)F (1,0) (1, 0)] + [ f()]f (0, 0) + f()[(2 + 1)( 1) 2 F (0, 0) + ( 1) 2 F (0,1) (1, 0) + 2 ( 1)F (0,1) (0, 1)] + [f() 1]{[2 + 1][ 1] 2 F (0, 0) + [ 1] 2 F (1,0) (0, 0) + 2 [ 1]F (1,0) (1, 0)}. For obtaining the second class of surfaces, we consider the boolean sum of the Marshall tpe operators Q 1, Q 2 and Q 3, given in (2.3): ((Q 1 Q 2 Q 3 )F )(, ) = F (, 0) + f()f (0, f() ) + (1 f() + ) (2.10) F ( 1 f()+, 1 f()+ ) F (0, 1) f( [ ] ( / ) 1 f( ) F (1 f( )), 0, and supposing that the roof stas on its support we set the condition F Γ3 = 0, hence we obtain S Q :=F (, 0) + f()f (0, F (0, 1) f( [ 1 f( ) ] F Theorem 2.2. If F Γ3 = 0, then we have (S Q F )(, f()) = 0. f() ) (2.11) ( / ) (1 f( )), 0. Proof. The proof follows directl replacing in (2.11). In the second level we use the Hermite interpolation operators, given in (2.6), and the approimations (2.7) and (2.8), and we obtain the following class of surfaces: (S 2 F )(, ) =(H 1 1 F )(, 0) + f()(h1 2 F )(0, F (0, 1) f( [ ] ( 1 f( ) (H1 1 F ) f() ) / (1 f( )), 0 ),

5 Classes of surfaces generated b Nielson and Marshall tpe operators 309 given below b (S 2 F )(, ) = [2+][ ]2 + 2 [ ] F (1,0) (1, 0) F (0, 0) + [ ] 2 F (1,0) (0, 0) (2.12) + [2+f()][ f()]2 F (0, 0) + [ f()]2 F (0,1) (1, 0) + 2 [ f()] F (0,1) (0, 1) F (0, 1) f( [ ] 1 f( ) {[2m(, ) + 1][m(, ) 1] 2 F (0, 0) (2.13) + m(, )[m(, ) 1] 2 F (1,0) (0, 0) + m 2 (, )[m(, ) 1]F (1,0) (1, 0)}, / where m(, ) denotes (1 f( )). Other classes of surfaces ma be obtained using the conditions F Γ3 = F (0,1) Γ3 = F (1,0) Γ3 = 0. (2.14) We consider the boolean sum of the Nielson tpe operators N 1 and N 2, given in (2.4), taking into account the conditions (2.14), and we get the operator S N given in (2.5). In the second level we use the Hermite interpolation operators, given in (2.1), taking into account the conditions (2.14), so we have (H1 2 F )(, ) := [2+][ ]2 g 3 () F (0, ) + [ ]2 F (1,0) (0, ), (2.15) (H2 2 F )(, ) := [2+f()][ f()]2 f 3 () F (, 0) + [ f()]2 F (0,1) (, 0). Using the following approimations: F (, 0) (H 2 1 F )(, 0) = (2 + 1)( 1) 2 F (0, 0) + ( 1) 2 F (1,0) (0, 0), F (0, ) (H2 2 F )(0, ) = (2 + 1)( 1) 2 F (0, 0) + ( 1) 2 F (0,1) (1, 0), b (2.5), we obtain the following class of surfaces: (S 3 F )(, ) = (H2 2 F )(0, f()) + (1 )(H1 2 F )(, 0) + [ f()]f (0, 0) + f()(h2 2 F )(0, ) + [f() 1](H1 2 F )(, 0),

6 310 Teodora Cătinaş further given as (S 3 F )(, ) = {[2f() + 1][f() 1] 2 F (0, 0) + f()[f() 1] 2 F (0,1) (1, 0)} (2.16) + (1 )[(2 + 1)( 1) 2 F (0, 0) + ( 1) 2 F (1,0) (0, 0)] + [ f()]f (0, 0) + f()[(2 + 1)( 1) 2 F (0, 0) + ( 1) 2 F (0,1) (1, 0)] + [f() 1]{[2 + 1][ 1] 2 F (0, 0) + [ 1] 2 F (1,0) (0, 0)}. Net we consider the boolean sum of the Marshall tpe operators, given in (2.10), taking into account the conditions (2.14), and we get S Q given in (2.11). Further, we appl the Hermite interpolation operators H1 2 and H2 2, given in (2.15), and we get the following class of surfaces: i.e., (S 4 F )(, ) =(H 2 1 F )(, 0) + f()(h2 2 F )(0, F (0, 1) f( [ ] ( 1 f( ) (H1 2 F ) (S 4 F )(, ) = [2+][ ]2 where m(, ) denotes f() ) / (1 f( )), 0 ), F (0, 0) + [ ] 2 F (1,0) (0, 0) (2.17) + [2+f()][ f()]2 F (0, 0) + [ f()]2 F (0,1) (1, 0) F (0, 1) f( [ ] 1 f( ) {[2m(, ) + 1][m(, ) 1] 2 F (0, 0) + m(, )[m(, ) 1] 2 F (1,0) (0, 0)}, / ( 1 f ( )). 3. Numerical eamples Eample 3.1. Consider F : T R, F (, ) = (2 + 2 h 2 ) and f() = 1 2. In Figure 2 we plot the graphs of the surface S 1 F, given in (2.9).

7 Classes of surfaces generated b Nielson and Marshall tpe operators 311 Figure 2: The surface S 1. Eample 3.2. Consider the function f() = 1 2 and F : T R. In Figure 3 we plot the graphs of surface S 2 F, given in (2.12), assigning to the data (F (0, 0), F (0, 1), F (1,0) (0, 0), F (1,0) (1, 0), F (0,1) (1, 0), F (0,1) (0, 1)) the values ( 1/4, 1, 1, 1, 1). Figure 3: The surface S 2. Eample 3.3. Consider the data from Eample 3.1. In Figure 4 we plot the graphs of the surface S 3 F, given in (2.16).

8 312 Teodora Cătinaş Figure 4: The surface S 3. Eample 3.4. Consider same data as in Eample 3.2. In Figure 5 we plot the graphs of the surface S 4 F, given in (2.17). Figure 5: The surface S 4. References [1] Barnhill, R.E., Blending function interpolation: a surve and some new results, Numerishe Methoden der Approimationstheorie, L. Collatz et al. (Eds.), Vol. 30, Birkhauser- Verlag, Basel, 1976, [2] Barnhill, R.E., Representation and approimation of surfaces, Mathematical Software III, J.R. Rice (Ed.), Academic Press, New-York, 1977, [3] Barnhill, R.E., Birkhoff, G., Gordon, W.J., Smooth interpolation in triangles, J. Appro. Theor, 8(1973),

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10 314 Teodora Cătinaş [25] Marshall, J.A., Mitchell, A.R., Blending interpolants in the finite element method, Inter. J. Numer. Meth. Engineering, 12(1978), [26] Nielson, G.M., The side-verte method for interpolation in triangles, J. Appro. Theor, 25(1979), [27] Nielson, G.M., Minimum norm interpolation in triangles, SIAM J. Numer. Anal., 17(1980), no. 1, [28] Nielson, G.M., Thomas, D.H., Wiom, J.A., Interpolation in triangles, Bull. Austral. Math. Soc., 20(1979), no. 1, Teodora Cătinaş Babeş-Bolai Universit Facult of Mathematics and Computer Sciences 1, Kogălniceanu Street RO Cluj-Napoca, Romania