Approximate Solution of Real Definite Integrals Over a Circle in Adaptive Environment

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1 Global Jornal o Pre an Applie Mathematics. SSN 9-68 Volme Nmber () pp. - Research nia Pblications Approximate Soltion o Real Deinite ntegrals Oer a Circle in Aaptie Enironment Kmini Meher Samya Ranjan Jena an Arjn Kmar Pal Dept o Mathematics School o Applie Sciences KT Uniersity Bhbaneswar- Oisha nia. Abstract n this note a mixe qaratre rle with an aaptie scheme is implemente oer the circlar srace in the Cartesian two imensional space. The oble mathematical transormations which transorm the circlar srace to a stanar sqare space. The mixe qaratre rle has been teste in aaptie scheme taking nmerical tests an it is on to be more eectie than that o Boole s rle. Keywors: Mixe qaratre rle Degree o precision Error bon Aaptie qaratre scheme Circlar region. MSC : 6D 6D. NTRODUCTON The applications o mixe qaratre rle or the approximation o real integrals xxan x y c yx hae been se by seeral athors [].The c e symmetric Gassian qaratre ormla or integrating arbitrary nctions o two ariables oer the srace o a triangle was propose by []. The symmetric integration ormla with higher orer precision p to egree ten was gien by [6]. An alternatie integration ormla or trianglar inite elements was propose by []. Lastly the mixe qaratre on real einite integrals with inite element methos has been sggeste by [8].P. Dash an S.R. Jena [9] ha implemente the mixe qaratre on real einite integrals oer triangles an spheres with inite element methos respectiely. The generalise Gassian qaratre ormla or integrating

2 Kmini Meher Samya Ranjan Jena an Arjn Kmar Pal arbitrary nctions o two ariables oer the srace o a circle was propose by Shiaram []. n this paper we aopt an application o mixe qaratre with aaptie scheme oer circlar srace x y: a x a a x y a x in the Cartesian two imensional x y space. The mathematical transormation rom x y space to space maps the stanar circle in x yspace to a stanar -sqare space :. Then the another transormation transorms the -sqare space to the space :.Here taking the aantage o the act that Boole s rle an Lobatto or-point rle are o same precision (i.e. precision ) a mixe qaratre rle o higher precision (i.e. precision ) has been obtaine by taking the linear combination o these rles. The mixe qaratre rle so orme has been teste on ierent einite integrals giing better reslts than Boole s rle in aaptie scheme. For a real integrable nction g an interal l m an a prescribe tolerance it is esire to compte an approximation B to the integral = g(x)x so that l B.The basic principle or aaptie qaratre is the aitie property o a einite integral o the orm Q R S with the aaptie integration schemes m l []. Q = g(x)x Where r is any point between l an m. r l R = g(x)x S = m r m g(x)x n aaptie integration the points at which the integran is ealate are so chosen in sch a way that epens on the natre o the integran. The iea is that i we can approximate each o the two integrals R an S within a speciie tolerance then the sm Q gies s the esire reslt. not we can recrsiely apply the aitie property to each o the interals l r an r m.aaptie sbiision o corse has geometrical appeal. t seems intitie that points shol be concentrate in regions where the integran is baly behae. The whole interal rles can take no irect accont o this. This paper is esigne as ollows. Sec- contains ntroction. n sec- we iscss abot the integrals o an arbitrary nction oer a qarter circlar region. n sec- we will constrct a mixe qaratre rle o egree o precision seen by taking two constitent rles Boole s rle an Lobatto or-point rle each o egree o precision ie. The error analysis an error bon are introce in sec-. The nmerical eriication o or propose rle is experimente on some sitable real integrals in sec- an inally the conclsion ollows in sec-6.

3 Approximate Soltion o Real Deinite ntegrals Oer a Circle in Aaptie Enironment. CONSTRUCTON OF NTEGRALS OVER A QUARTER CRCULAR REGON The qarter circlar region is transorme to a nit sqare. Fig- (circlar region) Fig- (sqare region) The nmerical integration o an arbitrary nction oer a qarter circlar region is C a x yxy x x y y x. a x The integral o eqn. can be transorme into an integral oer the srace o the sqare by sbstittion x a y a The eterminant o the Jacobian an ierential area are x y J x y x y Area = xy x x y x y a y Now eqn. becomes a a a a. The integral o eqn. can be rther transorme into an integral oer the stanar -sqare: by monomial transormation

4 Kmini Meher Samya Ranjan Jena an Arjn Kmar Pal. Then the eterminant o the Jacobian an the ierential area are. Now sing eqn. an eqn. in eqn. a a a Where a is the rais o the circle.. CONSTRUCTON OF MXED QUADRATURE RULE OF DEGREE OF PRECSON SEVEN Here the mixe qaratre rle o egree o precision seen is orme by taking the linear conex combination o two constitent rles i.e Boole s rle an Lobatto or point rle each o egree o precision ie. For approximate ealation o real einite integral y x y x. We choose Boole s rle ) ( ) ( B R.

5 Approximate Soltion o Real Deinite ntegrals Oer a Circle in Aaptie Enironment Lobatto- or point rle R L 6 where each rle o eqn. an eqn. is o egree o precision ie. Expaning each term o eqn. an eqn. sing Maclarin s series where RB EB. R E L L R B ! 6! ! R L ! 6! 8... We can write eqn. sing Maclarin s expansion ! 8 6!... 6 Error associate with Boole s rle an Lobatto or-point rle respectiely are 6 6 9!

6 6 Kmini Meher Samya Ranjan Jena an Arjn Kmar Pal E B E R L 6! R 6! B 8 98! L ! 98! ! Now mltiplying in eqn. an sbtracting eqn. rom eqn. we get R E. BL BL where R R R. BL B L This is the reqire mixe qaratre rle. The error associate with the rle E E E BL B L R BL is. ERROR ANALYSS The error analysis o the sai mixe qaratre rle can be represente by the ollowing Theorems. Theorem-. Let x y be siciently ierentiable nction in the close interal. The bon o trncate error E BL associate with the rle R BL is gien by ! E BL Proo- From eqn. R E BL BL where R R R E BL B L E E BL B L

7 Approximate Soltion o Real Deinite ntegrals Oer a Circle in Aaptie Enironment 6 8! Hence Theorem-. E BL 8 8 The bons or the trncate error E R 6M 6! BL BL E BL or an M x y E B 6! 6 6! EB EL ! 6 x x 6! Proo: We hae 6 6 x y is max where E L 6 6 E BL E BL 6 6! y x y 6M 6! Where M x y max x y 6 y x y Which gies only the trncational error bon as are nknown points in an it is obios that the error in approximation will be less i the points are close to each other. 6 Corollary-. The error bon or the trncational error []. E BL is gien by E BL 8M 6! when

8 8 Kmini Meher Samya Ranjan Jena an Arjn Kmar Pal. NUMERCAL VERFCATON Here there is a comparison o mixe qaratre rle with Boole s rle or approximation o some real oble einite integrals in aaptie rotine. The integrals ner consierations are x y x y y x e sinx y x y log x y x x y.. x y x x x y e y x x y Table- x y x. ( Comparison o mixe qaratre rle with Boole s rle in approximation o some real einite integrals oer the qarter circle in aaptie qaratre metho with stopping criterion ) Exact ale o the integrals Boole s rle R B by aaptie qaratre metho No o nterals or R B Mixe qaratre rle R BL by aaptie qaratre metho No o nterals or R BL Maximm amissible absolte error = = = = =

9 Approximate Soltion o Real Deinite ntegrals Oer a Circle in Aaptie Enironment Exact ale RB() RBL() Exact ale RB() RBL() Exact ale RB() RBL() Exact ale RB() RBL() Exact ale RB() RBL() Comparison o Exact ale with R B an R BL o an.

10 Kmini Meher Samya Ranjan Jena an Arjn Kmar Pal 6. CONCLUSON The eectieness o mixe qaratre rle is obios rom aboe examples in Table- an rom the graphical representations in aaptie scheme. The mixe qaratre rle R BL reces the nmber o steps reqire to approximate an integral in aaptie qaratre metho in comparison to its constitent Boole s rle. This work may be extene to any circlar region instea o qarter circle or any arbitrary rais or the real einite integrals with the application o mixe qaratre rle with aaptie qaratre scheme. REFERENCES []. A.K. Tripathy R.B. Dash etal () A mixe qaratre rle blening Lobatto an Gass-Legenre three-point rle or approximate ealation o real einite integrals ˮ nt-j-compting science an Mathematics6()66-. []. S.R. Jena S.C. Mishra (6) Mixe qaratre rle or oble integrals o Simpson's r an Gass-Legenre two-point rle in two ariables ˮ Global jornal o Science rontier researchol-6isse-online SSN: 9-66 an Print SSN: []. P.C. Hammer an A.H. Stro (98) Nmerical ealation o mltiple integrals Math. Tables other Ais Comptation -8. [] G.R. Cowper (9) Gassian qaratre ormlae or triangles nt. J.Nm.Math.Engg-8. []. J.N. Lyness an D. Jespersen (9) Moerate Degree Symmetric Qaratre Rles or the Triangle J.nst.Math.Applic9-. [6]. F.G. Lannoy (9) Trianglar inite elements an nmerical integration Compter Strct 6. []. C.T. Rey an D.J. Shippy (98) Alternatie integration ormlae or trianglar inite elements nt.j.nm.math.engg -9. [8]. S.R. Jena an R.B. Dash (9) Mixe qaratre o real einite integral oer triangles Paciic-Asian Jornal o Mathematicsol--. [9]. P. Dash an S.R. Jena() Mixe qaratre oer sphere" Global Jornal o Pre an Applie Mathematicsol--. []. K.T. Shiaram () Generalise Gassian qaratre oer a sphere nt. Jornal o Scientiic an Engg. Research -. []. Walter G Walter G Aaptie qaratre Reisite ˮ BT Nmerical Mathematics () pp.8-9.

11 Approximate Soltion o Real Deinite ntegrals Oer a Circle in Aaptie Enironment []. Olier J. A obly aaptie Clenshaw-Crtis qaratre metho ˮ Compting centre Uniersity o Essex Wienhoe park Colchester Essex. 9() pp.-. []. P. Patra D. Das etal Approximation o Singlar ntegrals by a Mixe Qaratre o Anti-Gass an Steensen s Qaratre Rles in the Aaptie Enironment ˮ Aances in Theoretical an Applie Mathematics SSN 9- Volme Nmber (6) pp. 9-9 []. S. Conte an C. De. Boor (98) Elementary Nmerical Analysis Tata Mac- Graw Hill.

12 Kmini Meher Samya Ranjan Jena an Arjn Kmar Pal