Using An Accelerating Method With The Trapezoidal And Mid-Point Rules To Evaluate The Double Integrals With Continuous Integrands Numerically

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1 ISSN -50 (Paper) ISSN 5-05 (Olie) Vol.7, No., 017 Usig A Acceleratig Method With The Trapezoidal Ad Mid-Poit Rules To Evaluate The Double Itegrals With Cotiuous Itegrads Numerically Azal Taha Abdul Wahab 1* Rusul Hassa Naser 1. Faculty of Educatio for girls\uiversity of Kufa, Najaf, IRAQ.. Faculty of Educatio for girls\uiversity of Kufa, Najaf, IRAQ. * of the correspodig author: Azalt.almussawy@uokufa.edu.iq Abstract I this research we have used a compoud method which is composed of Trapezoidal ad Mid-Poit Rules to evaluate the approximate values of the double Itegrals with Cotiuous Itegrads because of the resultat approximate values are fast whe approximatig from the true values of itegrals if compared with other Newto -Cotes formulas. [ S.S. Sastry, 00". The symbol of this rule is, i this method is, the umber of subitervals [a,b], equivalet to m ( the umber of subitervals of [c,d] ad (h=ћ), we accelerate the resultat approximate to the true values of itegrals through applyig Aitke's accelerate o the compoud rule to procure a ew method we called it A() where the symbol A idicates to the Aitke s acceleratio method ad the symbol ( y idicates to the Trapezoidal rule o the exteral ad x idicates to the Mid-poit rule o the iteral dimesios). Keywords: double itegrals, Trapezoidal ad Mid-Poit Rules ad Newto-Cotes Itroductio Numerical aalysis topic is characterized by iovatig ew several various methods to reach approximate solutios of certai mathematics problems effectively, sice the method efficiecy depeds o smoothess whe applyig it, therefor the selectio of the appropriate method for approximatig problem solutio gets effected by chage that occurs i computer techology. The importace of double Itegrals i fidig surface area mid-poit positios ad momet of iertia of flat surface ad to fid a volume layig uder a double Itegrals surface, all that urge may researchers to work i double Itegrals field, amog those who shed light o evaluatig Itegrals with Cotiuous Itegrads of the formulas f(x, y)= f 1 (x)f ( y) are Has Schjar ad Jacobse i 1973[Has Schjar ad Jacobse 1973], some studied Itegrals with improper Itegrads but they eglected impropriety as Phillip J. Davis ad Phillip Rabiowitz i 1975[3][Phillip J. Davis ad Phillip Rabiowitz, 1975]. I 010, Akar [Master thesis submitted by Batool Hatam Akar, 010] used a ew method where she itroduced a umerical method to evaluate double Itegrals values through usig Romberg's method o the resultat values of Mid-poit Rules i two dimesios x ad y whe umber of subitervals of the iteral itegrals are equivalet to umber of subitervals of the exteral itegrals, where (h=ћ) we call it RMM where ( MM ) symbolizes to Mid-poit rule method that is applied o both dimesios ad R idicates to acceleratio method ad good results were obtaied with a few umber of used subiterval. I 01, the researcher, Nada Ahmed Muhammed [Master thesis submitted by Muhammed, Nada Ahmed, 01], preseted three ew theorems by usig the Trapezoidal ad Mid-Poit Rules with derivatio of error formula to each method she called it, TT ad MT, ad accelerated reachig to the results by usig Richard's accelerate. Applyig Richard accelerate with the three rules have give good results regardig accuracy whe applied o cotiuous double itegrads. I this research we apply Aitki accelerate o three values resultig from applicatio of to calculate the approximate values of double itegrals with cotiuous itegrals whe the symbol of the umber of subitervals [a, b], is equivalet to m equal to umber of subitervals of [c, a] ad we have obtaied a ew method ad called it A, this rule gives faster results i reachig to aalytical aalysis tha the rule whe applyig o the cotiuous itegrads. - Method To derive a method to evaluate the double Itegrals with Cotiuous Itegrads Numerically, we assume the double Itegrals I it is defied as followig

2 ISSN -50 (Paper) ISSN 5-05 (Olie) Vol.7, No., 017 (1) Where f(x, y) is a Cotiuous Itegrads i each poit of Itegrals area [a, b] [c, d] I geeral, the itegral ca be writte as followig: d b c a, I f x y GG h. () Where GG(h) represets Itegrals value umerically by usig of the formulas ad b a d c h m (We will choose m= because of attaiig better ad faster results whe reachig to the true values. Akkar [Master thesis submitted by Batool Hatam Akar, 010] ad Nassir [Master thesis submitted by Nassir, Rusul Hassa, 011]) The formulas ca be obtaied through applicatio Mid-Poit rule o the iteral dimesio x ad the Trapezoidal rule o the exteral dimesio whe m= ( represets the umber of subitervals [a, b] ad m represets umber of subitervals of [c, d] which meas (h=ћ), where, d c h m, b a h Nada Ahmed Muhammed [Master thesis submitted by Muhammed, Nada Ahmed, 01] (3) Geometric represetatio illustrates itegral area ad two -dimesio itegral subiterval accordig to the rule. Whe we start with placig =m= ito the formula above, that meas we evaluate the approximate values of the 17

3 ISSN -50 (Paper) ISSN 5-05 (Olie) Vol.7, No., 017 double Itegrals I with method ad write that value i the table whe =m=, the we place =m= ad evaluate, also we write that value i the table as it is the approximate values of the double Itegrals, by usig the same method to evaluate the approximate values of the double Itegrals whe =m=. Thus, it is reachable to attai a better value through usig each three values from rule by applyig Aitke s acceleratio method to procure better value for itegral. 3- Method with Aitke's acceleratio method. Aitke's acceleratio method is a process to accelerate the covergece of the approximate values to the true values of itegrals. To apply method with Aitke's acceleratio method, we use the followig rule: h h ((h) (( )) ( ( ))...() h h ( ( )) ( ( )) ( (h)) Where A() is a value i the ew colum ad each of h h ( h ),( ),( ) are values i the colum that lays before the first oe, ad we will apply them i the metioed rule to accelerate obtaiig better values of itegrals istead of cotiuity i icreasig the subitervals, so if it was for istace, represets a value i the rule we will get - value whe applyig Aitke's accelerate o it through use of the equatio (6), the we also use Aike's o - value that leads to get - value. Thus, we cotiue coductig that till we obtai the desirable accuracy. Nassir [Master thesis submitted by Nassir, Rusul Hassa, 011] Sigificat otes cocerig the selected examples: 1- I rule we used the values m= 1,,,,,. = 1,,,,,. Nassir [Master thesis submitted by Nassir, Rusul Hassa, 011] - Numerical values of itegrals are evaluated through MATLAB programmig laguage, which is cosidered oe of the most sigificat programs that presets solutios i mathematic field ad what emaates of it from egieerig specialties that deped basically upo mathematics. 3. Examples:- The itegrad of itegral I l( x y ) 11 defied of each (x, y) [1,] [1,] ad it's aalytical value is which is approximated to te decimal places Table (1) double itegral evaluatio l by usig ( ) I x y A method

4 ISSN -50 (Paper) ISSN 5-05 (Olie) Vol.7, No., 017 We coclude from the table (1) the followig: whe = m = 3, the value i rule is true oly for the five decimal places. Whe usig Aitke's acceleratio method with this rule we have obtaied a value that is idetical with aalytical value which is true for te decimal places with (10 subiterval). For evaluatig the double itegral umerically 1 1 ( x y) I xe 0 0, it is obvious that itegrad is defied to each (x, y) [0,1] [0,1] ad it's aalytical value is which is approximated to te decimal places Table () double itegral calculatio 1 1 ( x y) I xe 0 0 by usig ( ) A method We otice i this table, whe =m=6, the value i rule is true oly for four decimal places. Whe usig Aitke's acceleratio method with this rule we have attaied a value that is idetical with aalytical value, which is true for te decimal places, whe =m=6 ( 1 subiterval) Also the itegrad of itegral 11 I si( ( x y)) 00 is defied of each (x, y) [0,1] [0,1] ad it's aalytical value is which is approximated to ie decimal places

5 ISSN -50 (Paper) ISSN 5-05 (Olie) Vol.7, No., 017 Table (3) double itegral calculatio 11 I si( ( x y)) by usig ( ) 00 A method. We otice i this table whe =m=6, the value i rule is true oly for four decimal places. Whe usig Aitke's acceleratio method with this rule we have obtaied a value that is idetical with aalytical value, which is true for ie decimal places, with (1 subiterval) Also the itegrad of itegral I 1 10 l( x y) x y defied of each (x, y) [0,1] [0,1] ad it's aalytical value which is approximated to te decimal places. The table () illustrates the results Table () illustrates that the umerical value of itegral I 1 10 l( x y) x y by usig A() Method It is obvious from this table that the umerical value I 1 10 l( x y) x y Whe =m=6 rule is true oly for five decimal places by usig A() method ad by applyig Aitke's acceleratio method o the resulted values, we have obtaied a value that is idetical with (the aalytical value), which is true for te decimal places, whe =m=6, that meas ( 1 subiterval) 5- Discussio It is obvious from what have come i all tables metioed above that the applicatio of rule aloe for Cotiuous Itegrads will give true approximate values for four of five decimal places whe m= ( represets the umber of subitervals [a,b] ad m represets the umber of subitervals [a,b] [c,d] that meas (h=ћ), we could accelerate the results through implemetig Aitke's acceleratio method o the resultat values ad obtaiig umerical values idetical to true aalytical values for te decimal places i some examples, ie decimal places i other examples by usig relatively a few subitervals ad i a short time where the time was calculated through MATLAB program laguage. 0

6 ISSN -50 (Paper) ISSN 5-05 (Olie) Vol.7, No., 017 Refereces [1] S.S. Sastry, 00" Itroductory Methods of Numerical Aalysis",pp , chapter 5,. [] Has Schjar ad Jacobse 1973"Computer Programs for Oe-ad Two-Dimesioal Romberg Itegratio of Complex Fuctio ", the Techical Uiversity of Demark Lygby, pp. 1-1 [3] Phillip J. Davis ad Phillip Rabiowitz, 1975" Methods of Numerical Itegratio ", BLASDELL Publishig Compay, pp. 1-, 599, 113, chapter 5. [] Master thesis submitted by Batool Hatam Akar, 010 "Some umerical methods to evaluate sigle, double ad triple itegrals," to uiversity of Kufa. [5] Master thesis submitted by Muhammed, Nada Ahmed, 01" Derivatig compoud methods from Trapezoidal ad Mid-poit, Rules to evaluate the Double Itegrals Numerically ad their error formulas ad improve results by usig acceleratig Methods", to uiversity of Kufa. [6] Master thesis submitted by Nassir, Rusul Hassa, 011 "The compariso betwee Aitke's acceleratio method ad Romberg's i evaluatig itegrals umerically, to uiversity of Kufa. 1