Integro-differential splines and quadratic formulae

Transcription

1 Inegro-differenial splines and quadraic formulae I.G. BUROVA, O. V. RODNIKOVA S. Peersburg Sae Universiy 7/9 Universiesaya nab., S.Persburg, 9934 Russia Absrac: This wor is devoed o he furher invesigaion of splines of he fifh order approximaion. Here we presen some new formulae which are useful for he approximaion of funcions wih one or wo variables. For each grid inerval (or elemenary recangular) we consruc he approximaion separaely. Here we consruc he basic one-dimensional polynomial splines of he fifh order approximaion when he values of he funcion and he values of is firs derivaive are nown in each poin of inerpolaion. Someimes i is imporan ha he inegrals of he funcion over he inervals are equal o he inegrals of he approximaion of he funcion over he inervals. In ha case he approximaion has some physical parallel. For his aim we use quadraic formulae here wih he sixh order of approximaion insead of he value of inegral. The one-dimensional case can be exended o muliple dimensions hrough he use of ensor produc spline consrucs. Numerical examples are represened. Key Words: Polynomial splines, Inegro-Differenial Splines, Inerpolaion. Inroducion The idea of he spline inerpolaion was born in England a he end of he 9h cenury when Briish engineers designed he firs railroad racs. The spline inerpolaion was hen considered as a more appropriae alernaive o polynomial inerpolaion. Now here are a variey of differen ypes of splines ha are used for solving differen mahemaical, mechanical, physical and engineering problems. This mehod of approximaion using polynomial splines is widely used for he inerpolaion and approximaion of discree daa. A lo of research has been devoed o he applicaion of various splines wih differen properies for approximaion and esimaion of daa. Special aenion is given o mehods of consrucing images [ ]. As is well nown, he one-dimensional case can be exended o muliple dimensions hrough he use of ensor produc spline consrucs [ 3]. Suppose ha n is a naural number, while a, b are real numbers, h = (b a)/n. Le us build he grid of inerpolaion nodes x = a + h, =,,..., n. 2 Lef polynomial splines of one variable Le he funcion u(x) be such ha u C 5 ([a, b]). Suppose ha we now u(x ), u (x ), =,,..., n. The nex quadraic formula is well nown: x+ x u(x)dx = V (u) + O(h 6 ), V (u) = (x + x ) (7u(x ) + 7u(x + )+ 3 6u(x )) (x + x ) 2 (x + ) u (x )). 6 We denoe by ũ(x) an approximaion of he funcion u(x) on he inerval [x, x + ] [a, b]: ũ(x) = u(x )ω, (x) + u(x + )ω +, (x)+ +u (x )ω, (x) + u (x + )ω +, (x)+ +V (u) ω <> (x). () The basic splines ω, (x), ω +, (x), ω, (x), ω +, (x), ω <> (x), we obain from he sysem: ũ(x) u(x), u(x) = x i, i =, 2, 3, 4, 5. (2) Suppose ha supp ω,α = [x, x + ], α =,, supp ω <> = [x, x + ]. I is easy o see ha ω,, ω,, ω <> C (R ). We have for x = x +h, [, ] he nex formulae: ω, (x + h)=(2 + )( ) 2, (3) ω +, (x + h)= (/8) 2 ( ), (4) ISSN: X 384 Volume, 26

2 ω, (x + h)=(/4)h(5 + 4)( ) 2, (5) ω +, (x + h)=(/8)h 2 (5 + 3)( ), (6) ω <> (x + h)=(5/6) 2 ( ) 2 /h. (7) Figures, 2, 3 show he graphics of he basic funcions ω, (x), ω +, (x), ω, (x), ω +, (x), ω <> (x), when h =. Figure 3 (righ) shows he error of approximaion of he Runge funcion u(x) = /( + 25x 2 ) wih he polynomial splines, h =., x [, ] Figure : Plos of he basic funcions: ω, (x) (lef), ω +, (x) (righ) Figure 2: Plos of he basic funcions: ω, (x), when h = (lef), ω +, (x), when h = (righ) K =.225. ũ (x) u (x) [x,x + ] K 2 h 4 u (5) [x,x + ], (9) K 2 =.994. ũ(x) u(x) [a+h,b] K h 5 u (5) [a,b]. () Proof. Inequaliy (8) follows from Taylor s heorem and he inequaliies: ω, (x), ω +, (x), ω, (x).26h, ω +, (x).98h, (x).586/h. Inequaliy () follows from (8). We have he nex expressions for derivaives of basic funcions: ω, (x + h) = 6( )/h, ω +, (x + h) = (3/4)( )/h, ω <> ω <> (x + h) = (5/8)( )/h 2, ω, (x + h) = (3/2) (9/2) , ω +, (x + h) = (3/4) (3/4) 2 + (5/2) 3. Inequaliy (9) follows from Taylor s heorem and he inequaliies: ω, (x).5/h, ω +, (x).626/h, ω, (x), ω +, (x), ω <> (x).8/h 2. Theorem 2. Le funcion u(x) be such ha u C 5 ([a, b]). We have: x+ x (ũ(x) u(x)).8h 6 u (5) [x,x + ] The proof is obvious and i is similar o wha i was in Theorem Figure 3: Plos of he basic funcions: ω <> (x), when h = (lef), and he error of approximaion of he Runge funcion wih he polynomial splines, h =., x [, ] (righ) Le us ae Ũ(x), x [a, b], such ha Ũ(x) = u(x), x [x, x + ]. Le u [a,b] = max u(x). [a,b] Theorem. Le funcion u(x) be such ha u C 5 ([a, b]). For approximaion u(x), x [x, x + ] by (), (3) (7) we have: 3 Compering wih he Hermi inerpolaion Here we shall compare he approximaion ha has been consruced above and he Hermie approximaion: ũ H (x) = u(x )ω, (x) + u(x )ω, (x) +u(x + )ω +, (x) + u (x )ω, (x)+ u (x + )ω +, (x), x [x, x + ]. The basic splines ω, (x), ω, (x), ω +, (x), ω, (x), ω +, (x) we obain from he sysem: ũ(x) u(x) [x,x + ] K h 5 u (5) [x,x + ], (8) ũ H (x) u(x), u(x) = x i, i =, 2, 3, 4, 5. () ISSN: X 385 Volume, 26

3 Suppose ha supp ω, = [x, x +2 ], supp ω, = [x, x + ]. I is easy o see ha ω,, ω s,, C (R ), =,, +, s =, +. We have for x = x + h, [, ], he nex formulae: ω, (x + h)=( + ) 2 ( + ) 2, (2) ω +, (x + h)= (/4) 2 ( + )(5 7), (3) ω, (x + h)=h( + )( + ) 2, (4) ω +, (x + h)=(/2)h 2 ( + )( + ), (5) ω, (x + h)=(/4) 2 ( + ) 2. (6) Table shows he errors R I = max ũ u, x [a,b] R II = max x [a,b] ũh u when [a, b] = [, ], h =.. Calculaions were done in Maple, Digis=5. Table. u(x) R I R II x 4.. /(( + 25x 2 ).47e 2.53e 2 sin(5x) cos(5x).293e e 4 4 Abou approximaions wih wo variables Suppose ha n, m are naural numbers, while a, b, c, d are real numbers, h x = (b a)/n, h y = (d c)/m. Le us build he grid of inerpolaion nodes x = a + h x, =,,..., n, y = c + h y, =,,..., m. On every line parallel o axis y, we can consruc he approximaion in he form: ũ(y) = u(y )ω, (y) + u(y + )ω +, (y)+ u (y )ω, (y) + u (y + )ω +, (y)+ +V ω <> (y), y [y, y + ]. (7) Now he formulae for ω, (y), ω +, (y), ω, (y), ω +, (y), ω <> (y) are similar o he previous ones. If (x, y) Ω, hen we ge he nex expression using he ensor produc: ũ(x, y) = i= p= i= p= u(x +i, y +p )ω +i, (x)ω +p, (y)+ u y(x +i, y +p )ω +i, (x)ω +p, (y)+ i= V +i, (x)ω +i, (x)ω <> (y)+ + i= i= V,+i ω <> (x)ω +i, (y)+ S,+i ω <> (x)ω +i, (y)+ W, ω <> (y)ω <> (x)+ u x(x, y +i )dω, (x)ω +i, (y)+ i= u xy(x, y +i )dω, (x)ω +i, (y)+ i= i= P +i, ω +i, (x)ω <> (y), (8) V +i, = (y + y ) (7u(x +i, y )+ 3 7u(x +i, y + ) + 6u(x +i, y )) (y + y ) 2 6 y(x +i, y + ) u y(x +i, y )), V,+i = (x + x ) (7u(x, y +i )+ 3 7u(x +, y +i ) + 6u(x, y +i )) (x + x ) 2 6 x(x +, y +i ) u x(x, y +i )), S,+i = (x + x ) (7u 3 y(x, y +i )+ 7u y(x +, y +i ) + 6u y (x, y +i )) (x + x ) 2 6 xy(x +, y +i ) u xy(x, y +i )), P +i, = (y + y ) (7u 3 x(x +i, y )+ 7u x(x +i, y + ) + 6u x(x +i, y )) (y + y ) 2 6 yx(x +i, y + ) u yx(x +i, y )), W = (y + y ) (7G(x, y )+ 3 7G(x, y + ) + 6G(x, y )) (y + y ) 2 (G y(x, y + ) G y(x, y )), 6 G(x, y) = (x + x ) (7u(x, y)+ 3 ISSN: X 386 Volume, 26

4 7u(x +, y) + 6u(x, y)) (x + x ) 2 6 x(x +, y) u x(x, y)). Graphs 4, 5 shows approximaions and he errors of approximaions ũ(x, y) u(x, y) by (8), (3) (7), (2) (6) of funcions u (x, y) = sin(3x 3y) cos(3x 3y), u 2 (x, y) = (x y) 2 (x + y) 2, when [a, b] = [, ], [c, d] = [, ], h x = h y = h =.2. Calculaions were done in Maple, Digis= h h h h, [, ], ω, (x +h)= 5h 4 4 3h 2 3 3h h, [, ],, / [, ], ω <> 5 6h 2 ( ) 2, (x + h)= [, ], / [, ]. Figure 6 shows he plos of he basic splines ω,, ω,. The plo of he basic spline ω <> is shown in Figure 3. The consrucion of he nonpolynomial splines wih he same properies and heir applicaion for he solving of differen problems will be regarded in furher papers. Figure 4: Plos of he funcions: ũ(x, y) = sin(3x 3y) cos(3x 3y) (lef) and ũ(x, y) u(x, y) (righ) when h =.2, [, ] [, ] e 5 e 5 5e 6 5e 6 e 5.5e 5.5 Figure 5: Plos of he funcions: ũ(x, y) = (x y) 2 (x + y) 2 (lef) and ũ(x, y) u(x, y) (righ) when h =.2, [, ] [, ] 5 Conclusion Basic spines can be applied for solving various mahemaical problems. We can obain he formulae of our basic splines in he following way. In he inerval [x, x ] we obain basic splines from he sysem: ũ(x) u(x), u(x) = x i, i =, 2, 3, 4, 5, ũ(x) = u(x )ω, (x) + u(x )ω, (x)+ u (x )ω, (x) + u (x )ω, (x)+v ω <> (x). If we ae he basic splines wih he same numbers from [x, x ] and [x, x + ] hen we have: , [, ], ω, (x +h)= , [, ],, / [, ],.5.5 Figure 6: Plos of he basic funcions: ω, (h + h) (lef), and ω, (h + h), when h = (righ) References: [] Safa, S., On he rivariae polynomial inerpolaion, WSEAS Transacions on Mahemaics. Vol., Iss. 8, 22, pp [2] Sala, V., Fas inerpolaion and approximaion of scaered mulidimensional and dynamic daa using radial basis funcions, WSEAS Transacions on Mahemaics. Vol. 2, Iss. 5, 23, pp [3] Sarfraz, M., Al-Dabbous, N., Curve represenaion for oulines of planar images using mulilevel coordinae search, WSEAS Transacions on Compuers. Vol. 2, Iss. 2, 23, pp [4] Sarfraz, M., Generaing oulines of generic shapes by mining feaure poins, WSEAS Transacions on Sysems, Vol. 3, 24, pp [5] Zamani, M., A new, robus and applied model for approximaion of huge daa, WSEAS Transacions on Mahemaics, Vol. 2, Iss. 6, 23, pp [6] C. K. Chui. Mulivariae Splines. Sociey for Indusrial and Applied Mahemaics (SIAM), Pensylvania, USA, 988. ISSN: X 387 Volume, 26

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