On Approximations by Polynomial and Trigonometrical Splines of the Fifth Order

Transcription

1 I. G. Burova T. O. Evdoimova On Approximations by Polynomial and Trigonometrical Splines of te Fift Order I.G. BUROVA St. Petersburg State University Matematics and Mecanics Faculty Universitetsy prospet 8 Peterof St. Petersburg RUSSIA i.g.burova@spbu.ru burovaig@mail.ru T.O. EVDOKIMOVA St. Petersburg State University Matematics and Mecanics Faculty Universitetsy prospet 8 Peterof St. Petersburg RUSSIA t.evdoimova@spbu.ru Abstract: Here we consider several approaces for constructing approximations of a function by te polynomial and te trigonometric splines of te fift order. We compare te approximations to te left te rigt and te middle minimal polynomial splines te approximations to te left te rigt and te middle minimal trigonometrical splines te approximations to te left te middle polynomial integro-differential splines and te approximation to te left te rigt and te middle trigonometrical integro-differential splines. Te quadrature formulas are represented. Te results of some calculations are done. Key Words: Polynomial splines Trigonometrical splines Integro-Differential Splines Interpolation. 1 Introduction Nowadays tere are many different splines for solving different problems [1 10]. B-splines conic splines cubic polynomial and nonpolynomial splines and box spline functions can be used for interpolation or approximation of scattered data simulation of te eart waveform plotting surfaces and etc. Minimal splines are intended for te approximation and interpolation functions. If we now te values of te functions in grid nodes we can construct te approximation on every net interval separately. In te next sections we compare te results of te approximations to te minimal splines see [11] to te polynomial integro-differential splines and to te nonpolynomial integro-differential splines see [1] of te fift order. Polynomial integro-differential splines were first used by Kireev V.I. [13]. In general construction of solutions of delay differential equations is muc more complicated tan te construction of solutions of ordinary differential equations [14 1]. We ave te Caucy problem for a numerical solution on eac interval. Te solution on te interval requires te solution from te previous interval. Some necessary values may be missing among te calculated values but tey may be obtained by interpolating [0]. Interpolation sould use te positions of te discontinuities of te derivatives. Te application splines of te fift order for te delay problem is presented in te last section. We need to use te values of te function only in te given interval so we use te approximations wit te left and te rigt basic splines. Splines of te fift order We consider te grid of equidistant nodes wit te step a = x 0 < x 1 <... < x n = b. Let te function ux be suc tat u C 5 [a b]. We ave to use te interpolation nodes only on te interval [a b]. Terefore we can use polynomial boundaryminimal splines see [11]. Rigt boundary-minimal splines are used on te left side on [a b] and left boundary-minimal splines are used on te rigt side on te interval. We denote by ũx an approximation by te polynomial minimal splines: ũx = ux ω x x [x x +1 ] an approximation by te trigonometric minimal splines: ũx = ux w x x [x x +1 ] an approximation by te polynomial integro- E-ISSN: Volume 1 015

2 I. G. Burova T. O. Evdoimova differential splines: ũx = x +i1 ut dt ω < ii1> x x x x +1 x i an approximation by te trigonometric integrodifferential splines: ũx = x +i1 x i ut dt w < ii1> x x x x +1 were i 1 i are integer numbers ω x w x ω < i +i 1 > x w < i i 1 > x we determine from te system: ũx = ux ux = φ i x i = Here φ i x i = is Cebysev system on [a b] φ i C 5 [x 0 x n ]. In polynomial case we tae φ i x = x i 1 i = ; in trigonometric case we tae φ 1 x = 1 φ x = sinx φ 3 x = cosx φ 4 x = sinx φ 5 x = cosx. 3 Approximation by te middle polynomial integro-differential splines Let us tae an approximation for ux x x x +1 in te form: x ũx = x 1 x + utdt ω < 10> x + x 1 + utdt ω < 1> x+ +1 utdt ω <01> x +3 x+ + utdt ω <1> x + utdt ω <3> x x +1 x + were ω <ss+1> x s = we find from te system 1. Let us tae φ i x = x i 1 i = If we put x = x + t t 0 1 ten we obtain: x + t = 5t t + 10t 0t ω < 1> x +t = 10t4 7 15t + 75t 30t x + t = 30t t 15t 4 75t 60 5 x + t = 13 45t + 10t 4 + 5t 10t ω <3> x + t = 5t t ω < 10> ω <01> ω <1> Let us tae Ũx x a b suc tat Ũx = ũx x x x +1 = 3... n 3. We put f xi x i+1 = sup x x i x i+1 fx f = f Xab = max i sup x x i x i+1 fx. Teorem 1. Let function ux be suc tat u C 5 [a b]. For approximation ux x x x +1 by 3 7 we ave: ũx ux K 1 5 u 5 x x +3 8 R 1 = Ũ u Xx x n K 1 5 u 5 Xx0 x n 9 K 1 = Proof. Inequality 8 follows from te relations 3 7 and Taylor formula wit te remainder term in Lagrange form. Here te next inequalities were used: ω < 1> x 1/0 ω <01> x ω < 10> x 9/0 ω <1> x 9/0 ω <3> x 1/0. Inequality 9 follows from 8. 4 Approximation by middle polynomial minimal splines of Lagrange type Consider te case of te middle minimal splines. Let us tae an approximation for u C 5 [a b] x [x x +1 ] in te form: ũx = ux ω x + ux 1 ω 1 x+ +ux ω x + ux +1 ω +1 x+ 10 +ux + ω + x were supp ω = [x x +3 ] ω +s s = we find from te system 1. Let us tae φ i x = x i 1 i = E-ISSN: Volume 1 015

3 I. G. Burova T. O. Evdoimova If we put x = x + t t [0 1] ten we ave ω x + t = tt 1t t ω 1 x + t = ω x + t = ω +1 x + t = ω + x + t = tt 1t t t 1t t + t tt t + t tt 1t + t Teorem. Let function u be suc tat u C 5 [a b]. For approximation ux x [x x +1 ] by we ave: ũx ux K 5 u 5 [x x + ] 16 R = Ũ u [x x n ] K 5 u 5 [x0 x n] 17 K = Proof. Inequality 16 follows from te inequality: ũx ux [x x +1 ] 1 5! max [x x + ] u5 x max [x x +1 ] x x x x 1 x x x x +1 x x +. Inequality 17 follows from 16. Table 1 sows te actual errors of approximation of te functions. Here R M <P > is te actual error of approximation by te splines 3 7 RM P is te actual error of approximation by te splines on 1 1 wen = 0.1. Calculations were done in Maple Digits=15. Table 1. Actual errors of approximations by te splines 3 7 and by te splines ux R M <P > RM P 1/1 + 5x sinx sin3x x Figure 1 sows te error of approximation of te function 1/1 + 5x by te middle minimal polynomial middle splines Figure 1: Plot of te error of approximation ux = 1/1+5x by te middle polynomial spline Quadrature formula From te approximation by te minimal polynomial middle splines on [x x +1 ] we receive: x+1 x ũtdt = ux ux ux ux ux Now in we can use 18. We ave: x+1 x utdt = x+1 x ũtdt + r were r = 0 if ux = x i 1 i = Left polynomial splines For x [x x +1 ] we tae ũx in te form: ũx = ux 3 ω 3 x + ux ω x+ +ux 1 ω 1 x + ux ω x+ 19 +ux +1 ω +1 x were supp ω = [x 1 x +4 ] ω +s x s = we find from te system 1. Let us tae φ i x = x i 1 i = If we put x = x + t t [0 1] ten we ave: ω +1 x + t = tt + 1t + t ω x + t = t 1t + t tt 1t + t + 3 ω 1 x + t = 4 ω x + t = tt + 3t E-ISSN: Volume 1 015

4 I. G. Burova T. O. Evdoimova ω 3 x + t = tt 1t Let us tae Ũx x a b suc tat Ũx = ũx x x x +1 = n 1. Teorem 3. Let function ux be suc tat u C 5 [a b]. For approximation ux x [x x +1 ] by we ave te estimation: ũx ux K 5 u 5 [x 3 x +1 ] 5 R = Ũ u [x 3 x n ] K 5 u 5 [x0 x n ] 6 were K = Proof. Te inequality 5 follows from te next relation: ũx ux [x x +1 ] 1 5! max [x 3 x +1 ] u5 x max [x x +1 ] x x 3x x x x 1 x x x x +1. Inequality 6 follows from 5. Figure sows te error of approximation of te function 1/1 + 5x by te left polynomial splines Figure : Plot of te error of approximation ux = 1/1 + 5x by te left polynomial spline Quadrature formula From te approximation by te minimal polynomial left splines on [x x +1 ] we receive te next formula: x+1 ũtdt = ux ux x ux ux ux Now in we can use 7. We ave: x+1 x utdt = x+1 x ũtdt + r were r = 0 if ux = x i 1 i = Rigt polynomial splines For x [x x +1 ] we tae ũx in te form: ũx = ux ω x + ux +1 ω +1 x+ +ux + ω + x + ux +3 ω +3 x+ 8 +ux +4 ω +4 x were supp ω = [x 4 x +1 ] ω +s x s = we find from te system 1 for φ i x = x i 1 i = If we put x = x + t t [0 1] ten we ave: ω x + t = ω +1 x + t = ω + x + t = ω +3 x + t = ω +4 x + t = t 4t 3t t tt t 3t tt 1t 3t tt 1t t tt 1t t Let us tae Ũx x a b suc tat Ũx = ũx x x x +1 = n 5. Teorem 4. Let function ux be suc tat u C 5 [a b]. For approximation u x [x x +1 ] by we ave ũx ux K 5 u 5 [x x +4 ] 34 R = Ũ u [x 0 x n 4 ] K 5 u 5 [x0 x n] 35 K = Proof. Inequality 34 follows from te relation: ũx ux [x x +1 ] 1 5! max [x x +4 ] u5 x max [x x +1 ] x x x x +1 x x + x x +3 x x +4. Inequality 35 follows from 34. E-ISSN: Volume 1 015

5 I. G. Burova T. O. Evdoimova Table sows te actual errors of approximation by te left and rigt splines. Here RL P is te actual error of approximation by te splines and RR P is te actual error of approximation by te splines on 1 1 wen = 0.1. Calculations were done in Maple Digits=15. Table. Te actual errors of approximations by te left splines and te actual errors of approximations by te rigt splines ux RL P RR P 1/1 + 5x sinx sin3x x Figure 3 sows te error of approximation of te function 1/1 + 5x by te rigt polynomial splines Figure 3: Plot of te error of approximation ux = 1/1 + 5x by te rigt polynomial splines Quadrature formula From te approximation by te minimal polynomial rigt splines on [x x +1 ] we receive: x+1 x ũtdt = ux ux ux ux ux Now in we can use 36. We ave: x+1 x utdt = x+1 x ũtdt + r were r = 0 if ux = x i 1 i = Middle trigonometrical splines Let us tae an approximation for u C 5 [a b] x [x x +1 ] in te form: ũx = ux w x + ux 1 w 1 x+ +ux w x + ux +1 w +1 x+ 37 +ux + w + x were supp w = [x x +3 ] w +s s = we find from te system 1 were φ 1 x = 1 φ x = sinx φ 3 x = cosx φ 4 x = sinx φ 5 x = cosx. If we put x = x + t x [x x +1 ] t [0 1] ten: w x +t = S 1 = sin t+ sin t S 1 sin/ sin sin3/ sin sin t sin t 38 w 1 x + t = sin / sin sin3/ S = sin t + sin t w x + t = S 3 = sin t + sin t+ w +1 x +t = sin t S sin t 39 S 3 sin sin / 40 sin t S 4 = sin t + sin t+ sin t sin t S 4 sin3/ sin sin / 41 sin t S 5 w + x +t = sin sin3/ sin sin/ 4 S 5 = sin t + sin t+ sin t sin t. It can be sown tat for te polynomial basic splines ω s x 10 and trigonometrical basic splines w x 37 te next relation is fulfilled w x +t = ω x + t + O = Figure 4 sows te error of approximation of te function 1/1 + 5x by te trigonometrical splines Teorem 5. Te error of te approximation by te splines is te next: ũx ux K 5 4u + 5u + u V [x x + ] 43 were x [x x +1 ] K = 0.1. Proof. Te function ux on [x x +1 ] can be written in te form see [1] ux = x 3 x 4u τ+ E-ISSN: Volume 1 015

6 I. G. Burova T. O. Evdoimova Figure 4: Plot of te error of approximation 1/1 + 5x by te trigonometrical splines u τ+u V τ sin 4 x/ τ/dτ +c 1 +c sinx+ c 3 cosx+c 4 sinx+c 5 cosx were c i i = are arbitrary constants. We ave: w x 0.08 w 1 x 0.1 w x 1 w +1 x 1 w + x Using te metod from [1] we obtain Left trigonometrical splines For x [x x +1 ] we tae ũx in te form: ũx = ux 3 w 3 x + ux w x+ +ux 1 w 1 x + ux w x+ 44 +ux +1 w +1 x were supp ω = [x 1 x +4 ] w +s x s = we find from te system 1. Let us tae φ 1 x = 1 φ x = sinx φ 3 x = cosx φ 4 x = sinx φ 5 x = cosx. If we put x = x + t t [0 1] ten we ave: w +1 x + t = T +1 /T 1 45 T +1 = sin t+3 sin t + sin t+ sin t T 1 = sin sin 3 sin sin w x + t = T /T 46 T = sin t+3 sin t + sin t+ sin t T = sin 3 sin sin w 1 x + t = T 1 /T 3 47 T 1 = sin t+3 sin t + sin t sin t T 3 = sin sin T = sin t+3 w x + t = T /T 4 48 sin t+ sin t sin t T 4 = sin sin sin 3 w 3 x + t = T 3 /T 5 49 T 3 = sin t + sin t+ sin t sin t T 5 = sin sin sin 3 sin. Teorem 6. Te error of te approximation by te splines is te next: ũx ux K 5 4u + 5u + u V [x 3 x +1 ] 50 were x [x x +1 ] K = 0.5. Proof is similar to tat done in te proof of Teorem 5. Here te next inequalities were used: w 3 x 0.1 w x 0. w 1 x 0.36 w x 1 w +1 x Rigt trigonometrical splines For x [x x +1 ] we tae ũx in te form: ũx = ux w x + ux +1 w +1 x+ +ux + w + x + ux +3 w +3 x+ 51 +ux +4 w +4 x were supp w = [x 4 x +1 ] w +s s = we find from te system 1. Let us tae φ 1 x = 1 φ x = sinx φ 3 x = cosx φ 4 x = sinx φ 5 x = cosx. If we put x = x + t t [0 1] ten we ave: w x + t = T R /T 0R 5 T R = sin t sin t 3 sin t sin t T 0R = sin sin3/ sin sin/ w +1 x + t = T R +1 /T 1R 53 T+1 R = sin t sin t 3 sin t sin t T 1R = sin3/ sin sin / w + x + t = T R + /T R 54 T+3 R = sin t sin t 3 sin t sin t T R = sin sin / w +3 x + t = T R +3 /T 3R 55 T+3 R = sin t sin t sin t sin t T 3R = sin / sin sin/ w +4 x + t = T R +4 /T 4R 56 T+4 R = sin t 3 sin t sin t sin t T 4R = sin/ sin sin3/ sin. E-ISSN: Volume 1 015

7 I. G. Burova T. O. Evdoimova Teorem 7. Te error of te approximation by te splines is te next: ũx ux K 5 4u + 5u + u V [x x +4 ] were x x x +1 K =. Proof is similar to tat done in te proof of Teorem 5. Here te next inequalities were used: w x 1 w +1 x 1.1 w + x 0.36 w +3 x 0. w +4 x 0.1. Table 3 sows te actual errors of approximation by te left and rigt trigonometrical splines. RL T is te actual error of approximation by te splines RR T is te actual error of approximation by te splines on 1 1 wen = 0.1. Calculations were done in Maple Digits=15. Table 3. Actual errors of approximations by te left splines and by te rigt splines ux RL T RR T 1/1 + 5x sinx sin3x x Table 4 sows te actual and teoretical errors of approximation by te middle trigonometrical splines. RM T is te actual error of approximation by te splines R T M is te teoretical error of approximation by te splines on 1 1 wen = 0.1. Calculations were done in Maple Digits=15. Table 4. Actual errors of approximations by te middle splines and teoretical errors of approximation by te middle splines ux RM T R T M 1/1 + 5x sinx sin3x x Approximation by polynomial integro-differential splines in special form Let us tae for x x x +1 : ũx = J 1 ω <0> x+ +J ω < 11> x + J 3 ω < 1> x+ were J 1 = J 3 = +J 4 ω < 31> x + J 5 ω < 41> x 57 x+ x utdt J = x+1 x utdt J 4 = J 5 = x+1 x+1 x 1 utdt 58 x+1 x 3 utdt 59 x 4 utdt. 60 From ũx = ux u = x i 1 i = we find ω <0> x ω < 11> x ω < 1> x ω < 31> x ω < 41> x. So we ave for x = x + t t 0 1 x + t = 30t + 75t + 36t 3 + 5t x +t = 18t 33t 76t 3 15t x +t= 6 898t 69t + 356t t x +t= 57t 5t t 84t x +t= 55t4 310t 135t + 140t ω <0> ω < 11> ω < 1> ω < 31> ω < 41> Teorem 8. Suppose te function ux be suc tat u C 5 [x 0 x n ] ũx is given by Ten for x x x +1 we ave: ũx ux K 5 5 u 5 x 4 x + 66 K 5 = Proof. We ave from 61 65: ω <0> x 10/384 = 0.315/ ω < 11> x / / ω < 1> x / / ω < 31> x / / ω < 41> x / /. Representing ux by te Taylor formula one obtains 66. Let us tae Ũx x a b suc tat Ũx = ũx x x x +1 = n 3. E-ISSN: Volume 1 015

8 I. G. Burova T. O. Evdoimova Teorem 9. Suppose te ypotesis of te Teorem 3 is fulfilled. Ten: R = Ũ u x 4 x n K 5 u 5 ab 67 K = Proof. Inequality 67 follows from te relation 66. Table 5 sows te errors of approximation of functions by te splines on 1 1 wen = 0.1. Te calculations of te actual error R <P > were done in Maple Digits=15. Table 5. Te errors of approximation of functions by te splines N ux R <P > 1 1/1 + 5x sinx sin3x x Figure 5 sows te errors of approximation of te function 1/1 + 5x by te polynomial integrodifferential splines on 1 1 = utdt utdt + w <1> x + w <3> x x +1 x + 68 were w < 1> x w < 10> x w <01> x w <1> x w <3> x we find from ũx = φ i x φ 1 x = 1 φ x = sinx φ 3 x = cosx φ 4 x = sinx φ 5 x = cosx. So we ave for x = x + t t 0 1: w < 1> x + t = sin 3t 1 sin t+1 + cos + 1 sin t+1 sin 1 t sin sin cos cos w < 10> x + t = sin + sin 4t cos 4 cos sin t 1 cos t sint cos 8 cos 3 + sin cos 7 cos Figure 5: Plot of te error of approximation of te function 1/1 + 5x by te polynomial integrodifferential splines Trigonometric integro-differential splines Let us tae for x x x +1 : + x x 1 ũx = x 1 x utdt w < 10> x + utdt w < 1> x+ +1 utdt w <01> x x+ 1 4 cos sin w <01> x +t = cos3+cos+1+cost cos + 1 cos 1 cost cos cos sin cos 1 1 sin + + sin 71 w <1> x +t = 4 cos sin t+1 cos t 1 sin t 1 cos t+1 + sin + t sin + t cos cos4 cos3 + 4 cos cos 3 1 cos + sin cos sin 7 w <3> x + t = 4 cos cost + cos cos + t cos cos + 1 sin cos+sin cos cos E-ISSN: Volume 1 015

9 I. G. Burova T. O. Evdoimova If we don t now te value of te integrals we can use quadrature formula. For example from trigonometrical splines we obtain +1 utdt = ux J +ux 1 J 1 +ux J 0 + x ux +1 J 1 + ux + J were J = 8 sin cos cos cos 1 sin + sin + cos cos + 1 J 1 = 4 sin cos cos 1 sin cos 3 sin cos + 14 cos 1 J 0 = 4 sin 1 cos 1 sin + cos cos+ + sin 1/ + cos J 1 = 1/ cos 4 cos 3 3 cos +cos+1 8 sin cos 3 3 sin cos sin cos + 3 sin 4 cos cos + cos/4 J = 8 sin cos cos cos 1 cos + cos sin cos sin. It can be easily sown tat between polynomial and trigonometric integro-differential splines te next relation is fulfilled w <ss+1> x + t = ω <ss+1> x + t + O. Table 6 sows te errors of approximation of functions by splines on 1 1 wen = 0.1. Te calculations of te actual error R M <T > were done in Maple Digits=15. Table 6. Te errors of approximation of functions by splines N ux R M <T > 1 1/1 + 5x sinx 0. 3 sin3x x Teorem 10. Te error of te approximation by te splines is te next: ũx ux K 5 4u + 5u + u V x x were x x x +1 K = 0.. Proof is similar to tat done in te proof of Teorem 5. Here te next inequalities were used: w < 1> x 0.05/ w < 10> x 0.45/ w <01> x 1.1/ w <1> 0.45/ w <3> x 0.05/. x 8 Rigt integro-differential trigonometric splines On te left side of [a b] te best approximation gives us te rigt basic splines. In eac x x +1 = n 5 te approximation for ux are presented in te form: + ũx = +1 utdt w <01> x +3 x+ + utdt w <1> x + utdt w <3> x+ x +1 x utdt w <34> x + utdt w <45> x x +3 x were w <ss+1> x s = are determined from te conditions φ 1 x = 1 φ x = sinx φ 3 x = cosx φ 4 x = sinx φ 5 x = cosx. were If we put x = x + t t 0 1 ten we ave w <01> x + t = G <01> 1 G <01> 76 G <01> 1 = 8 sin 3 cos cos + 1 cos 1 1 G <01> = 1 sin cos + 1 cos + 1 cos4 cos 1 cost + + sin sint + 4cos + 1 cos cost + cos4 1 cost w <1> x + t = G <1> 1 G <1> 77 E-ISSN: Volume 1 015

10 I. G. Burova T. O. Evdoimova were G <1> 1 = 4 sin cos + 1cos 1 1 G <1> = sin 4 cos 1 cos + 1 cos cos sint sint sin 8 cos cos+ cos+1 cost + +sin4 + sin + sin cost sin cos4 + cos + 1 cost were w <3> x + t = 1 + cos 4 sin 5 G<3> 1 78 G <3> 1 = cos + cos + cost cos sin 3 sin sint+ sin 1 4 sin4 sin cos sin sint sin3 sin cost were G <34> w <34> x + t = G <34> 1 G <34> 79 1 = 4 sin cos + 1cos 1 1 G <34> = sin 4 cos 1 sint cos cos cos sint cos4 + cos3 + cos sin4 cost+ 4 cos cos+ +cos4+4 cos cos sin cos t were w <45> x + t = G <45> /G <45> G <45> G <45> = 1 1 =8 sin 3 cos cos+1cos 1 sin cos + 1 cos + 1 cos cost + cost 4. Teorem 11. Te error of te approximation by te splines is te next ũx ux K 5 4u + 5u + u V x x were x x x +1 K =. Proof. Te function ux on x x +1 can be written in te form see [1]: ux= 3 4u τ+5u τ+u V τ sin 4 x τ dτ+ x c 1 + c sinx + c 3 cosx + c 4 sinx + c 5 cosx were c i i = are arbitrary constants. Here te next inequalities were used: w <01> x.9/ w <1> x.7/ w <3> x.9/ w <34> x 1.05/ w <45> x 0./. Using te metod from [1] we obtain from Let Ũx x a b be suc tat ũ x Ũx = ũ x x x x +1 = n 5. Teorem 1. For te error of approximation by trigonometric splines we ave te next relation: Ũ u x 0 x n 4 K 5 4u + 5u + u V x0 x n K =. Proof. Te proof follows from 81. Table 7 sows te teoretical R <T R > and te actual R R <T > errors of approximation by te splines tey were found in Maple Digits = 5 wit = 0.1. Table 7. N ux R <T > 1 1 R R <T R > 1+5x sinx sin3x x Trigonometric quadrature Sometimes if te values of te integrals are unnown te next trigonometric quadrature may be useful. From we obtain: x+1 x utdt = ux J 0 + ux +1 J 1 + were: +ux + J + ux +3 J 3 + ux +4 J 4 + r J 0 = Q 01 /Q 0 Q 01 = cos 4 sin cossin + cos + sin + sin + cos Q 0 = 8cos cos cos sin J 1 = Q 11 /Q 1 Q 11 = + sin sin + sin4 cos + sincos cos1 4 cos 4 cos E-ISSN: Volume 1 015

11 I. G. Burova T. O. Evdoimova Q 1 = 4cos cos sin J = Q 1 /Q Q 1 = 3 sin cos + 4 sin cos 3 + sin 4 cos 3 + cos Q = 4cos 3 cos cos + 1 J 3 = Q 31 /Q 3 J 4 = Q 41 /Q 4 Q 31 = sin cos3 + 4 cos + cos cos cos Q 3 = 4cos cos sin Q 41 = sin cos cos cos + sin Q 4 = 8cos cos cos sin. Tis formula suc tat r = 0 if u = 1 sinx cosx sinx cosx. 9 Left integro-differential trigonometric splines On te rigt side of [a b] te best approximation gives us te left basic splines. In eac interval x x +1 = te approximation for ux is presented in te form: + x ũx = x 3 x 4 utdtw < 3 > x 3 + x x 1 utdt w < 4 3> x+ x+ 1 utdtw < 10> x + utdtw < 1> x +1 utdtw <01> x x+ x 8 were w <ss+1> x s = are determined from te conditions from ũx = φ i x φ 1 = 1 φ = sinx φ 3 = cosx φ 4 = sinx φ 5 = cosx. If we put x = x + t t 0 1 ten we ave: w < 4 3> x + t = P < 4 3> were: P < 4 3> /P < 4 3> = 4 sin sin cos + 1 cos 1 P < 4 3> = cos + 1 cos cost + + sin cos cos + t 1 w < 3 > x + t = P < 3 > /P < 3 > 1 84 were: P < 3 > P < 3 > 1 = 4 sin cos + 1cos 1 = sin cost sin cost + sin 4 cos 1 + cos + cos + 1 sin cos t + sin cost + w < 1> x + t = P < 1> were: P < 1> /P < 1> = 4 sin 3 cos 1 P < 1> = 4 cos 3 cos cos t cos cost sin4 + sin cost w < 10> x + t = P < 10> /P < 10> 1 86 were: P < 10> 1 = 4 sin 3 1 cos cos + 1 P < 10> = sin cos + 1 sint + cos cos 1 1 cos t + cos + cos+1 sin sint cost+ cos cos 3 cos t + sin4 cos 1 cos+1 were: P <01> P <01> w <01> x + t = P <01> /P <01> =8 sin 3 cos cos+1cos 1 = cos + 1 cos cost + + cos4 + t + 1 sin cos + 1. Teorem 13. Te error of te approximation by te splines is te next: ũx ux K 5 4u + 5u + u V x 4 x were x x x +1 K =.1. Proof is similar te proof of Teorem 11. Here te next inequalities were used: w < 4 3> x 0./ w < 3 > x 1.05/ w < 1>.7/ w <01> x.9/ w < 10> x.9/. x Let Ũx x a b be suc tat ũ x Ũx = ũ x x x x +1 = n 1. Teorem 14. For te error of approximation by trigonometric splines we ave te next relation: Ũ u x 4 x n K 5 4u + 5u + u V x0 x n K =.1. Proof follows from 88. E-ISSN: Volume 1 015

12 I. G. Burova T. O. Evdoimova Table 8 sows te teoretical R <T L > and actual R L <T > errors of approximation of functions by trigonometric splines calculated in Maple for Digits = 5 and = Table 8. N fx R <T > 1 1 L R <T L > 1+5x sinx sin3x x We can use te next formula if te values of te integrals are unnown. From we obtain: x+1 x utdt ux 3 J 3 + ux J + were: +ux 1 J 1 + ux J 0 + ux +1 J 1 J 3 = Q 31 /Q 3 J = Q 1 /Q Q 31 = sin cos + sin cos cos Q 4 = 8 sin cos cos cos Q 1 = sin cos + sin cos + cos cos3 Q = 4 sin cos cos J 1 = Q 11 /Q 1 J 0 = Q 01 /Q 0 Q 11 = sin cos3+1+sin cos Q 1 = 4 sin 1 cos Q 01 = sin4 cos sin + cos3 + sin 3 cos 3 cos Q 0 = 4 sin cos cos J 1 = Q 11 /Q 1 Q 11 = sin sin sin sin4 + cos + cos Q 1 = 8 sin cos cos cos. 10 Application for solving delay equation We consider te delay differential equations: y = yt 1 for t 1 89 wit constant istory yt = 1 0 t 1. Te solution of equation 89 is suc tat y becomes discontinuous at x = 1 y becomes discontinuous at x = and so on. Here for solving tis problem we apply approximation metods wit te minimal trigonometric splines and te polynomial integro-differential splines. Figures 6 and 7 sow te errors of solution of te delay problem 89 wen = e 07 3e 07 e 07 1e e Figure 6: Plot of te error of solution of te delay problem 89 by trigonometric left and rigt splines. 1e 07 8e 08 6e 08 4e 08 e 08 e e 08 6e 08 8e Figure 7: Plot of te error of solution of te delay problem 89 by polynomial splines Conclusion Polynomial and nonpolynomial minimal and integrodifferential splines are useful for solving different approximation problems. Te results represented in te tables sows tat te constants in te estimations of te errors of approximation represented in te teorems can be diminised. Here tese constants were calculated using te Taylor teorem wit te remainder term in Lagrange form. Furter more we are going to minimize te constants taing te remainder term of te Taylor teorem in anoter form and to find a way for minimizing te constants in te estimations of te errors of approximation for nonpolynomial splines. References: [1] Sarfraz M. Generating outlines of generic sapes by mining feature points. WSEAS Transactions on Systems. Vol pp [] Sala V. Fast interpolation and approximation of scattered multidimensional and dynamic data E-ISSN: Volume 1 015

13 I. G. Burova T. O. Evdoimova using radial basis functions. WSEAS Transactions on Matematics. Vol. 1 Iss. 5 May 013 pp [3] Popoviciu N. Boncut M. On te teorem of curry & scoenberg and te relation between B-Spline and Box-Spline Functions. Recent Researces in Computational Tecniques Non-Linear Systems and Control. Proc. of te 13t WSEAS Int. Conf. on MAMECTIS 11 NO- LASC 11 CONTROL 11 WAMUS pp [4] Kalovretis K. Ganetsos T. Sammas N.Y.A. Taylor I. Andonopoulos J. Development of a computerized ECG analysis model using te cubic spline interpolation metod. Recent Researces in Circuits Systems and Signal Processing - Proc. of te 15t WSEAS Int. Conf. on Circuits Part of te 15t WSEAS CSCC Multiconference Proc. of te 5t Int. Conf. on CSS pp [5] Popoviciu N. A comparison between two box spline algoritms based on inductive metod and geometric metod. Recent Researces in Computational Tecniques Non-Linear Systems and Control. Proc. of te 13t WSEAS Int. Conf. on MAMECTIS 11 NOLASC 11 CON- TROL 11 WAMUS pp [6] Furferi R. Governi L. Palai M. Volpe Y. From unordered point cloud to weigted B- spline - A novel PCA-based metod. Applications of Matematics and Computer Engineering - American Conf. on Applied Matematics AMERICAN-MATH 11 5t WSEAS International Conference on Computer Engineering and Applications CEA pp [7] Caglar H. Yilmaz S. Caglar N. Iseri M. A non-polynomial spline solution of te onedimensional wave equation subect to an integral conservation condition. Proc. of te 9t WSEAS International Conf. on Applied Computer and Applied Computational Science ACA- COS pp [8] Kuragano T. Quintic B-spline curve generation using given points and gradients and modification based on specified radius of curvature Article. WSEAS Transactions on Matematics. Vol. 9 Iss. 010 pp [9] Harus Kalis Oars Lietuvietis. On Finite Difference Approximations for Solving some Problems of Matematical Pysics wit Periodical Boundary Conditions. Advances in Applied and Pure Matematics. Proc. of te 7-t International Conf. on Finite Differences Finite Elements Finite Volumes Boundary Elements Fand-B 14. Gdans. Poland. May pp [10] Fengmin Cen Patricia J.Y.Wong. On periodic discrete spline interpolation: Quintic and biquintic case. Journal of Computational and Applied Matematics pp [11] Burova I.G. Demyanovic Yu.K. Minimal Splines and teirs Applications. Spb.010. in Russian [1] Burova Irina. On Integro- Differential Splines Construction. Advances in Applied and Pure Matematics. Proc. of te 7-t International Conf. on Finite Differences Finite Elements Finite Volumes Boundary Elements F-and-B 14. Gdans. Poland. May pp [13] Kireev V.I. Panteleev A.V. Numerical metods in examples and tass. Moscow p. Russian [14] N.E.Mastorais. An extended Cran-Nicolson metod and its Applications in te Solution of Partial Differential Equations: 1-D and 3-D Conduction Equations. Proceedings of te 10t WSEAS International Conference on APPLIED MATHEMATICS Dallas Texas USA November pp [15] Nicola Guglielmi Ernst Hairer. Numerical approaces for state-dependent neutral delay equations wit discontinuities. Matematics And Computers In Simulation. Vol pp. 1. [16] Kennedy Benamin B. A State-Dependent Delay Equation wit Negative Feedbac and Mildly Unstable Rapidly Oscillating Periodic Solutions. Discrete and Continuous Dynamical System. Series B pp [17] Borges Ricardo; Calsina Angel; Cuadrado Silvi Odo Diemann. Delay equation formulation of a cyclin-structured cell population model Journal of Evolution Equations. Vol. 14 Issue pp [18] Abdellatif Ben Malouf Moamed Ali Hammami. A comment on Exponential stability of nonlinear delay equation wit constant decay rate via perturbed system metod. International Journal of Control Automation and Systems. Vol. 1 Issue pp [19] Bellman R.E. Cooe K.L. On te Computational solution of a class of functional differential equations. J.Mat. Anal. and its Appl. Volume pp [0] Hall G. Watt J.M. Modern Numerical Metods for Ordinary Differential Equations [1] Zang Cengui Agarwal Ravi P. Boner Martin And Li Tongxing. Oscillation of fourt-order delay dynamic equations. Science Cina. January 015 Vol.58 No.1 pp E-ISSN: Volume 1 015