Arithmetic Mean Derivative Based Midpoint Rule

Transcription

1 Applied Mthemticl Sciences, Vol. 1, 018, no. 13, HIKARI Ltd Arithmetic Men Derivtive Bsed Midpoint Rule Rike Mrjulis 1, M. Imrn, Symsudhuh Numericl Computing Group, Deprtment of Mthemtics University of Riu, Peknbru 893, Indonesi Copyright c 018 Mrjulis, Imrn nd Symsudhuh. This rticle is distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Abstrct In this rticle, we discuss the modifiction of double midpoint rule nd corrected midpoint rule by dding the derivtive evluted t rithmetic men of the nodes to pproximte definite integrl. The proposed rules give increse of precision over the existing rules. Lstly, the effectiveness of the proposed rules is illustrted by numericl exmples nd the results re compred with the existing rules. Mthemtics Subject Clssifiction: 65D30, 65D3 Keywords: Midpoint rule, rithmetic men, derivtive-bsed qudrture 1 Introduction Definite integrl of the form I = f(x)dx (1) is n object of intensive reserch in mthemtics. In some pplictions, we need to clculte the integrl. If the ntiderivtives of f(x) is known, the integrl cn be clculted using Newton-Leibniz rule. If the ntiderivtive function is unvilble or not esy to obtin, the integrl cn be pproximted using numericl integrtion method. 1 Corresponding uthor

2 66 Mrjulis, M. Imrn nd Symsudhuh The most common numericl integrtion process is derived by pproximting f(x) with polynomil interpoltion, known s Newton-Cotes method. The generl form of the formul cn be expressed s f(x)dx = n w i f(x i ) + E n, () i=0 where weights w i nd nodes x i [, b] re to be determined nd E n is the error of the formul. If the end points of the intervl integrtion re included s node points, the formul is nmed by closed Newton-Cotes formul, otherwise it is clled open Newton-Cotes formul [3]. The improvement of closed Newton-Cotes method by including the end of intervl integrtion s new vribles to be determined ws proposed by Deghn et l. [7]. Rmchndrn et l. [11, 13, 1, 1] introduced n improvement of closed Newton-Cotes method by dding the derivtive of the function evluted t the geometric men, heronin men, root men squre, nd hrmonic men. Furthermore Rmchndrn nd Priml [8] modified closed Newton-Cotes method by dding the derivtive of the function evluted t the centroidl men vlue. In [10], Rmchndrn et l. performed comprison between their methods which were proposed in [15, 11, 1]. Deghn et l. [5] nd [6] hve modified the open nd semi-open Newton- Cotes method by the sme strtegy s the improvement of closed Newton- Cotes methods. Menwhile Zfr et l. [16] modified the open Newton-Cotes method by combining the function vlues nd the evlution of derivtive t uniformly spced points. Furthermore Rmchndrn et l. [9] lso modified the midpoint rule [, h. 69] ( ) + b M r (f) = (b )f + (b )3 f (ξ), ξ (, b), (3) by dding the evlution of derivtive in the midpoint, so tht they obtined ( ) ( ) + b (b )3 + b M m (f) = (b )f + f, ξ (, b). () Zfr et l. [16] generlized the corrected midpoint rule ( ) + b (b ( ) ) M ct (f) = (b )f + f (b) f () 7(b )5 f () (ξ), ξ (, b), (5) 5760 by tking the liner combintion of the function nd its derivtive vlues t the node points

3 Arithmetic men derivtive-bsed midpoint rule 67 Formul midpoint cn lso be modified by tking x 0 = + 1 (b ) nd x 1 = + 3 (b ) to obtin double midpoint rule [1, h. 17] M d (f) = (b ) (f(x 0 ) + f(x 1 )) + (b )3 f (ξ), ξ (, b). (6) 96 In section two, we discuss the modifiction of double-midpoint rule (6) nd corrected midpoint rule (5) by dding the derivtive which is evluted t rithmetic men of the nodes. Then it is followed by doing the error nlysis of the proposed rule in section three. The discussion is closed by performing some numericl computtion to see the effectiveness of the proposed rules. Arithmetic Men Derivtive-Bsed Double Midpoint Rule In this section firstly we proposed the modifiction of double midpoint rule by dding the derivtive evluted t rithmetic men in eqution (6). Theorem 1 If f C [, b] then the rithmetic men derivtive-bsed double midpoint rule to pproximte f(x)dx given by [ ( ) ( )] ( ) (b ) 3 + b + 3b (b )3 + b M d (f) = f + f + f, (7) 96 hs the precision three. Proof: We show tht the formul (7) is exct for f(x) = x 3. The exct vlue of x3 dx = 1 (b ). From (7) we obtin M d (f) = (b ) [ (3 ) 3 ( ) ] 3 + b + 3b + = 7 3 b + b 3 + 7b 3 = 8b 8 3 M d (f) = 1 (b ). + 6(b )3 ( + b) b b 3 + b 3 This proves the theorem. Next ) we improve the corrected midpoint rule by tking the vlue of ξ = in eqution (5). ( +b

4 68 Mrjulis, M. Imrn nd Symsudhuh Theorem If f C [, b] then the rithmetic men derivtive-bsed corrected midpoint rule to pproximte f(x)dx given by ( ) + b (b [ ] ) M ct (f) = (b )f + f (b) f () 7 ( ) + b 5760 (b )5 f (), (8) hs the precision five Proof: The eqution (8) is verified by f(x) = x 5 which hs the exct vlue x5 dx = 1 6 (b6 6 ). From (8) we obtin ( ) 5 + b M ct (f) = (b ) + (b ) [ 5b 5 ] 80 (b ) = b 15 b + 15 b + 1b 5 + 3b b 0 b + 0 b 0b 5 + 0b b 35 b + 35 b 8b 5 + 7b 6 96 M ct (f) = 1 6 (b6 6 ). ( ) + b This proves the theorem. We hve shown tht the rithmetic men derivtive-bsed double midpoint rule hs the precision three nd the rithmetic men derivtive-bsed double midpoint rule hs the precision five. 3 Error Anlysis We derive the error term of the rithmetic men derivtive-bsed double midpoint rule nd the rithmetic men derivtive-bsed double midpoint rule using x the reminder of the qudrture formuls for the monomil p+1 nd the exct (p+1)! 1 b vlue of (p+1)! xp+1 dx, where p is the precision of the formul. Theorem 3 If f C [, b] then error term for rithmetic men derivtivebsed double midpoint rule is E Md (f) = (b )5 f () (ξ), (9) where ξ (, b). This rule hs fifth order ccurte.

5 Arithmetic men derivtive-bsed midpoint rule 69 Proof: To show (9), we let f(x) = x! in eqution (3). The exct solution is x! dx = 1 10 (b5 5 ). By pplying the rithmetic men derivtive-bsed double midpoint rule (7), we hve [ ( ) (b ) b M d (f) = + 1 ( ) ] [ ( ) ] + 3b (b )3 1 + b +!! 96 = ( b b 6 b b + 1b 5 ) 61 + ( 5 + b + 3 b b 3 b + b 5 ) 768 M d (f) = ( b + 3 b b b + 9b 5 ). 61 Hence the error form bsed on the Burg [] is E Md (f) = f(x)dx M d (f) = (b )5 f () (ξ), ξ (, b). The precision, the orders nd the error terms for M m [9], M r [], M d [1], nd proposed method M d re shown in Tble 1. Tble 1: Comprison of error terms Rules Precision Order Error terms 1 M r 1 3 (b )3 f (ξ) 1 M m 3 5 (b 190 )5 f () (ξ) 1 M d 1 3 (b 96 )3 f ( ) (ξ) 11 M d 3 5 (b 3070 )5 f () (ξ) Theorem If f C 6 [, b] then error term for rithmetic men derivtivebsed corrected midpoint rule is E Mct (f) = (b )7 f (6) (ξ), (10) where ξ (, b). This rule hs seventh order ccurte.

6 630 Mrjulis, M. Imrn nd Symsudhuh Proof: Let f(x) in eqution (10) is verified by f(x) = x6 6! whose exct solution is x 6 6! dx = (b7 7 ). Next by using rithmetic men derivtive-bsed corrected midpoint rule, we obtin ( ) 6 [ ] (b ) + b (b ) b 5 M ct (f) = ( ) (b ) 5 + b = ( b 9 5 b 5 b b + 9 b 5 + 5b 6 + b 7 ) ( b 5 b + b 5 b 6 + b 7 ) 880 ( b 7 5 b 35 b b + 7 b 5 1b 6 + 7b 7 ) 6080 M ct (f) = b 18 5 b + 30 b b + 18 b 5 6b b Then bsed on the Burg [], the error form is obtined s E Mct (f) = f(x)dx M ct (f) E Mct (f) = (b )7 f (6) (ξ), ξ (, b). The precision, the orders nd the error terms for M ct [16] nd proposed method M ct re shown in Tble. Tble : Comprison of error terms Rules Precision Order Error terms M ct (b 5760 )5 f () (ξ) M ct (b )7 f (6) (ξ) Numericl Exmples In this section, the vlues of 1 0 ex dx, π 0 tn xdx nd 1 (1 + 0 x6 )dx re pproximted by M r, M m, M d, M d, M ct, nd M ct. The results re shown in Tbel 3.

7 Arithmetic men derivtive-bsed midpoint rule 631 Tble 3: Comprison of M r, M m, M d, M ct, nd M ct. 1 0 ex dx π Integrl Method 0 tn x dx 1 0 x6 )dx Error Error Error M r 6.96e e 0 1.7e 01 M m 8.6e e 0.91e 0 M d 1.78e e e 0 M d 5.95e 0.e 0 3.e 0 M ct.03e e e 01 M ct 3.09e 05.9e 0 1.3e 0 From the computtionl results s depicted in Tble 3, it cn be seen tht the error rithmetic men derivtive-bsed double midpoint rule nd the rithmetic men derivtive bsed-corrected midpoint rule is smller thn the comprison rules. Wht we found here is in greement with the error nlysis where the precision of the method is incresed respectively by one nd two precisions. Funding. This reserch is conducted without ny funding support from ny institutions. References [1] H.M. Anti, Numericl Methods for Scientists, Tt McGrw Hill, Delhi, [] K.E. Atkinson, An Introduction to Numericl Anlysis, Second Ed., John Wiley Sons, New York, [3] R. L. Burden nd J. D. Fires, Numericl Anlysis, Ninth Edition, Brooks/Cole, Boston, 010. [] C. O. E. Burg nd E. Degny, Derivtive-bsed midpoint qudrture rule, Applied Mthemtics, (013), [5] M. Deghn, M. Msjed-Jmei nd M. R. Eslhchi, On numericl improvement of open Newton-Cotes qudrtue rules, Applied Mthemtics nd Computtion, 175 (006),

8 63 Mrjulis, M. Imrn nd Symsudhuh [6] M. Deghn, M. Msjed-Jmei nd M.R. Eslhchi, The semi open Newton- Cotes qudrtue rules nd its numericl improvement, Applied Mthemtics nd Computtion, 171 (005), [7] M. Deghn, M. Msjed-Jmei nd M. R. Eslhchi, On Numericl Improvement of Closed Newton-Cotes Qudrtue Rules, Applied Mthemtics nd Computtion, 165 (005), [8] T. Rmchndrn nd R. Priml, Centroidl men derivtive-bsed closed Newton-Cotes qudrture, Interntionl Journl of Science nd Reserch (IJSR), 5 (016), [9] T. Rmchndrn nd R. Priml, Open Newton-Cotes qudrture with midpoint derivtive for integrtion of lgebric functions, Interntionl Journl of Reserch in Engineering nd Technology (IJRET), (015), [10] T. Rmchndrn, D. Udykumr nd R. Priml, Comprison of rithmetic men, geometric men nd hrmonic men derivtive-bsed closed Newton Cotes qudrture, Progress in Nonliner Dynmics nd Chos, (016), [11] T. Rmchndrn, D. Udykumr nd R. Priml, Geometric men derivtive-bsed closed Newton-Cotes qudrture, Interntionl J. of Pure nd Engg. Mthemtics (IJPEM), (016), [1] T. Rmchndrn, D. Udykumr nd R. Priml, Hrmonic men derivtive-bsed closed Newton-Cotes qudrture, IOSR Journl of Mthemtics (IOSR-JM), 1 (016), [13] T. Rmchndrn, D. Udykumr nd R. Priml, Heronin men derivtive-bsed closed Newton-Cotes qudrture, Interntionl Journl of Mthemticl Archive, 7 (016), [1] T. Rmchndrn, D. Udykumr nd R. Priml, Root men squre derivtive-bsed closed Newton-Cotes qudrture, IOSR-Journl of Scientific nd Reserch Publictions, 6 (016), [15] W. Zho nd H. Li, Midpoint derivtive-bsed closed Newton-Cotes qudrture, Abstrct nd Applied Anlysis, 013 (013), Article ID 9507,

9 Arithmetic men derivtive-bsed midpoint rule 633 [16] F. Zfr, S. Slem, nd C.O.E. Burg, New derivtive bsed open Newton- Cotes qudrture rules, Abstrct nd Applied Anlysis, 01 (01), Article ID , Received: My 1, 018; Published: June 5, 018