OPEN NEWTON - COTES QUADRATURE WITH MIDPOINT DERIVATIVE FOR INTEGRATION OF ALGEBRAIC FUNCTIONS

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1 IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 9-6 pissn: -78 OPEN NEWTON - COTES QUADRATURE WITH MIDPOINT DERIVATIVE FOR INTEGRATION OF ALGEBRAIC FUNCTIONS T. Rmchndrn R.Priml Assistnt Professor Deprtment of Mthemtics M.V.Muthih Government Arts College for Women Tmil Ndu Indi Reserch Scholr PG &Reserch Deprtment of Mthemtics Government Arts College (Autonomous) Tmil Ndu Indi Astrct Mny methods re ville for pproximting the integrl to the desired precision in Numericl integrtion. A new set of numericl integrtion formul of Open Newton-Cotes Qudrture with Midpoint Derivtive type is suggested which is the modified form of Open Newton-Cotes Qudrture. This new midpoint derivtive sed formul increse the two order of precision thn the clssicl Open Newton-Cotes formul nd lso gives more ccurcy thn the existing formul. Further the error terms re otined nd compred with the existing methods. Finlly the effectiveness of the proposed lgorithm is illustrted y mens of numericl exmple. Key Words: Numericl Integrtion Open Newton-Cotes formul Midpoint Derivtive Numericl Exmples *** INTRODUCTION Numericl integrtion is the process of computing the vlue of definite integrl from set of numericl vlues of the integrnd. The process of evlution of integrtion of function of single vrile is sometimes clled Mechnicl Qudrture. The computtion of doule integrl of function of two independent vriles is clled Mechnicl Cuture. There re mny methods re ville for numericl integrtion []. Consider the definite integrl I f where the function f(x) is continuous in the closed intervl [] so tht the integrl I(f) exists. An efficient formul is developed for computing pproximte vlue of the integrl using only vlues of the integrnd f(x) t points x []. To pproximte the integrl I(f)we integrte exctly piecewise polynomil pproximtions of f(x) on the intervl []. Generlly qudrture rule hs the form n w i f x i () i Where there (n) distinct points x < x < < x n nd (n) weights w w... w n within the intervl []. The error of pproximtion is given s E n f n w i f x i () i. Definition An integrtion method of the form ( ) is sid to e of order P if it produces exct results (En[f] ) for ll polynomils of degree less thn or equl to P []. In Open Newton-cotes rule the end points of the intervl is excluded in the function evlutioni.e w i f x i () i x Volume: Issue: Oct- n x n for given n distinct points x < x <... < xn nd n weights w w... wn over the intervl ( ) with xi ( i ) h i...n nd h n [ 7 ]. the Open Newton cotes rules re given s follows If n ; where ξ. If n ; f f ( ) (ξ) () f ( ) (ξ) (6) 6

2 IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 9-6 pissn: -78 where ξ. If n ; where ξ. If n ; where ξ. f f f f ξ (7) f f 9 f f f ξ (8) In the closed Newton-cotes qudrture the endpoints re included; wheres the open Newton- cotes qudrture only the interior points re included. The corrected open Newton-cotes qudrture hs higher precision thn the clssicl qudrture rule. There re so mny works hs een done on the numericl improvement of Newtoncotes formuls. Dehghn et l. presented n improvement of open semi-open closed first nd second kind [ 6 8 9] Cheyshev Newton- cotes qudrture rules. In the recent yers Clrence O.E Burg nd his compnions introduced new fmily of derivtive sed rules [ 7]. In Weijing Zho nd Hongxing Li [] introduced new fmily of closed Newton-cotes qudrture with Midpoint derivtive rules. In this pper new fmily of open Newton-cotes qudrture rule is descried which uses the derivtive v l u e t the Midpoint with their error terms. Also some numericl exmples re given with their results nd comprison. The result shows tht the new formuls give etter solution thn the clssicl ones..open NEWTON - COTES QUADRATURE WITH MIDPOINT DERIVATIVE A new Open Newton - Cotes Qudrture rules with Midpoint Derivtive is explined elow which g i v e s higher precision thn the clssicl Newton-Cotes Qudrture rules.. Theorem Open Newton - Cotes Qudrture with Midpoint - Derivtive for (n) is Volume: Issue: Oct- f ( ) The precision of this method is. (9) Since the rule (9) hs the degree of precision. Now we verify tht the rule (9) is exct for f (x) x x. When f x x x dx ; n When f x x x dx ; n 6.. Qudrture with Midpoint - Derivtive is.. Theorem Open Newton - Cotes Qudrture with Midpoint - Derivtive for (n) is f ( ) 6 f The precision of this method is. () Since the rule () hs the degree of precision.now we verify tht the rule () is exct for f (x) x x. When f x x x dx ; n ( ) 6.

3 IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 9-6 pissn: -78 When f x x x dx ; n 6( ) 6. Qudrture with Midpoint - Derivtive is.. Theorem Open Newton - Cotes Qudrture with Midpoint - Derivtive for (n) is f f f f The precision of this method is. () Since the rule () hs the degree of precision.now we verify tht the rule () is exct for f (x) x x. When f x x x dx ; n 6. When f x x x dx ; n Qudrture with Midpoint - Derivtive is. f f 9 The precision of this method is. f f f () Since the rule () hs the degree of precision.now we verify tht the rule () is exct for f (x) x x. When f x x x dx ; n 8. When f x x x dx ; n Qudrture with Midpoint - Derivtive is.. THE ERROR TERMS OF OPEN NEWTON-COTES WITH MIDPOINT DERIVATIVE QUADRATURE The Error terms of Open Newton-Cotes Qudrture with Midpoint derivtive re given elow. The Error terms re the difference etween the exct vlue nd the qudrture rule.. Theorem The Error term of Open Newton - Cotes Qudrture with Midpoint - Derivtive for (n) is. Theorem Open Newton - Cotes Qudrture with Midpoint - Derivtive for (n) is Volume: Issue: Oct- f ( ) ( ) 9 f() (ξ) ()

4 IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 9-6 pissn: -78 where ξ ( ).This is fifth order ccurte with the error term is E f 9 f ξ. Let f x x! f! x dx ;!.6!.6 ( ) 9. Therefore the Error term is. Theorem E f 9 f ξ. The Error term of Open Newton - Cotes Qudrture with Midpoint - Derivtive for (n) is ( ) 6 f f 9( ) 888 f () (ξ) ( ) where ξ ( ).This is fifth order ccurte with the error term is 9 E f f ξ. 888 Let f x x! f! f x dx ; Therefore the Error term is. Theorem E f ( ) f ξ. 888 The Error term of Open Newton - Cotes Qudrture with Midpoint - Derivtive for (n) is f f f f. ( )7 888 f(6) (ξ) () where ξ ( ).This is seventh order ccurte with the error term is E f Let f x x6 f 7. ( 7 7 ) 7. ( )7 888 f(6) (ξ). f x 6 dx f f 7 7 ; Therefore the Error term is E f ( )7 888 ( )7 888 f(6) (ξ). Volume: Issue: Oct-

5 IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 9-6 pissn: -78. Theorem The Error term of Open Newton - Cotes Qudrture with Midpoint - Derivtive for (n) is f f 9 8( ) f f f f (6) (ξ) (6) where ξ ( ).This is seventh order ccurte with the error term is 8( )7 E f 68. f (6) (ξ).. Tle : Exct vlue of e x dx n vlue App. vlue Error App. vlue Error n n n n Tle : Exct vlue of n vlue dx x App. vlue Error App. vlue Error Let f x x6 x 6 f f f 7. ( 7 7 ) 7. dx f 7 7 ; 9 f Therefore the Error term is E f 8( ) ( ) NUMERICAL RESULTS f (6) (ξ). An pproximte vlue of the following exmples using the open Newton- cotes qudrture with Midpoint Derivtive rules re determined nd presented. To demonstrte the ccurcy of the results we evlute the exmples nd the Comprison of results is shown in Tles.. n n n n Tle : Exct vlue of n vlue x dx. App. vlue Error App. vlue Error n n n n Tle : Exct vlue of e x dx n vlue App. vlue Error App. vlue Error n n n n From the results presented in Tles - it is o served tht the Open Newton-Cotes q u d r t u r e with midpoint derivtive g i v e s more ccurcy thn the clssicl ones. Volume: Issue: Oct-

6 IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 9-6 pissn: -78. CLUSION In this work we pplied the Open Ne wton-cotes Q u d r t u r e with Midpoint derivtive over finite intervl []. The Error terms gives two orders of precision thn the S t n d r d m e t h o d s.a numericl exmple is given to clrify the proposed Algorithm. 6. REFERENCES [] K.E.Atkinson An Introduction to Numericl Anlysis John wiley nd Sons New York NY USA Second Edition 989. [] Clrence O.E.Burg Derivtive-sed closed Newtoncotes numericl qudrture Applied Mthemtics nd Computtions vol.8 pp [] Clrence O.E.Burg nd Ezechiel Degny Derivtivesed midpoint qudrture rule Applied Mthemtics nd Computtions vol. pp. 8-. [] M.Dehghn M.Msjed-Jmei nd M.R.Eslhchi On numericl improvementof closed Newton-Cotes qudrture rules Applied Mthemtics nd Computtions vol.6 pp. -6. [] M.Dehghn M.Msjed-Jmei nd M.R.Eslhchi The semi-open Newton-Cotes qudrture rule nd its numericl improvement Applied Mthemtics nd Computtions vol.7 pp. 9-. [6] M.Dehghn M.Msjed-Jmei nd M.R.Eslhchi On numericl improvement of open Newton-Cotes qudrture rules Applied Mthemtics nd Computtions vol.7 pp [7] Fiz ZfrSir Sleem nd Clrence O.E.Burg New Derivive sed open Newton- cotes qudrture rules Astrct nd Applied AnlysisVolume Article ID 98 6 pges. [8] S.M.Hshemiprst M.R.EslhchiM.Dehghn M.Msjed-Jmei On numericl improvement of the first kind Cheyshev-Newton-Cotes qudrture rules Applied Mthemtics nd Computtions vol.7 pp [9]S.M.HshemiprstM.Msjed-Jmei M.R.EslhchiM.Dehghn On numericl improvement of the second kind Cheyshev-Newton-Cotes qudrture rules(open type) Applied Mthemtics nd Computtionsvol.8 pp []M.K.JinS.R.K.Iyengr nd R.K.Jin Numericl methods for Scientific nd Computtion New Age Interntionl (P) limited Fifth Edition 7. [] Weijing Zho nd Hongxing Midpoint Derivtive-Bsed Closed Newton- Cotes Qudrture Astrct nd Applied Anlysis vol. Article ID 97 pges Volume: Issue: Oct-