Numerical Methods for Evaluation of Double integrals with Continuous Integrands

Transcription

1 Volume 9 No. 0 08, ISSN: (rinted version); ISSN: (on-line version) url: htt:// ijam.eu Numerical Methods for Evaluation of Double integrals with Continuous Integrands Safaa M. Aljassas College of Education for Girls, Mathematics Det., Iraq. Abstract: The main objective of this research is to introduce new numerical methods for calculating double integrations with continuous integrands. We combined between the best methods of finding aroximation values for single integration, based on accelerating Romberg with a Traezoidal rule (RT), the Mid- oint rule (RM), the Simson rule (RS) on the outer dimension Y with the Suggested method (RSu) on the inner dimension X. We call the obtained methods by RT(RSu), RM(RSu) and RS(RSu) resectively. Alying these methods on double integrals gives high accuracy results in short time and with few artial eriods.. Introduction: Numerical integral is one of the mathematics branches that connects between analytical mathematics and comuter, it is very imortant in hysical engineering alications. Moreover, it has been used in dentistry where dentists used some numerical methods to find the surface area of the cross section of the root canal. Because of the imortance of double integrals in calculating surface area, middle centres, the intrinsic limitations of flat surfaces and finding the volume under the surface of double integral, so many researchers worked in the field of double integrals. Furthermore, they imroved the results using different accelerating methods. Usually, by these methods, we get an aroximation solution which means that there is an error, so here is the role of numerical integrals. In 984, Muhammad [7] alied comlex methods such as Romberg (Romberg) method, Gauss (Gauss) method, Romberg (Gauss) method, and Gauss (Romberg) method on many examles of double integrals with continuous integrands and he comared between all these comlex methods to find that Gauss method (Gauss) is the best one in terms of accuracy and velocity of aroach to the values of analytic integrals as well as the number of artial eriods. However, Al-Taey [] used accelerating Romberg with the base oint on the outer dimension Y beside the base of Simson on the internal dimension X to give good results in terms of accuracy and a few artial eriods. In the other side, RMM method, Romberg acceleration method on the values obtained by Mid-oint method on the inner and outer dimensions X, Y when the number of artial eriods on the two dimensions is equal was resented by Egghead [4]. For more information about this,see [] and[9]. 385

2 In this work, we resent three new numerical methods for calculating double integrals with continuous integrands using single integral methods, RT, RM,RS, [7] and R(Su), [8] when the number of divisions on the outer dimension is not equal to the number of divisions on the internal dimension and we get good results in terms of accuracy and velocity of aroach to the values of analytic integrals as well as the number of artial eriods.. Newton-Cotes Formulas The Newton-Coats formulas are the most imortant methods of numerical integrals, such as Traezoidal rules, Mid-oint, Simson. In this section we resent these methods and their correction limits. Let's assume the integral J is defined to be : t J g( x ) dx ( k ) ( k ) R () [5]. s Such that ( k ) is the numerical base to calculate the value of integral J, reresents the tye of the rule, ( k ) is the correction terms for the rule ( k ) and R is the remainder after truncation some terms from ( k ). The general formulas for the above three methods are: w k T ( k ) g( s ) g( t ) g ( s k ) Traezoidal Rule w M ( k ) g( s ( 0.5) k ) Mid-oint Rule w w k S ( k ) g( s ) g ( t ) g( s k ) 4 g( s ( ) k ) 3 Simson's Rule A great imortance for the correction limits is to imrove the value of integral and accelerate the numerical value of integral to the analytical or exact one. Therefore, Fox [5] has found the series of correction terms for each rule of Newton-Coates rules for continuous integrals. Thus the series of correction terms for T ( k, ) Mk ( ) and Sk ( ) resectively are: ( k ) k k k 4 6 T T T T ( k ) k k k 4 6 M M M M ( k ) k k k S S S S Such that S, S, S,, M, M, M,, T, T, T, are constants. 3. Suggested method (Su) The Suggested method that reresented by [8] and others, is one of the single integral methods that derived from the Traezoidal and Mid-oint rules. This method gave good results in terms of 386

3 accuracy and velocity of aroach to the values of analytic integrals better than the mid-traezoid method. The general formula of the Suggested method is : w k Su ( k ) g( s ) g( t ) g( s ( w 0.5) k ) (g( s ( 0.5) k ) g ( s k )) 4 Where its correction terms are: 4 6 Su ( k ) Su k Su k Suk, such that Su, Su, Su, are constants. 4. Romberg Accelerating Werner Werner ( ) reresented this method in 955, in which a triangular arrangement consist from numerical aroximations for the definite integral by alying Richardson's external adjustment reeatedly to Traezoidal rule, Mid-oint or Simson's rule which reresents some of the Newton Coates formulas. The general formula for this accelerating is : L k k L Such that L 4,6,8,0,... (in Simson's rule ) and L,4,6,8,0,... (in Traezoidal rule and Midoint rule), is the value in that we written it in a new column in tables whereas k and k are values in its revious column. Moreover, the first column from the table reresents one of Newton Coates values [3]. 5. Numerical Methods for Calculating double Integrals with Numerically continuous integrands: In this section, we will aly the comound method RT(RSu) to calculate the aroximated value for the integral : J v t u s g x, y dxdy () We can write it as: v J G( y) dy u Such that: G ( y ) g ( x, y ) dx t s For examle, the aroximated value for (3) on outer dimension y by Traezoidal rule is :...( 3) (4) w k J G( u) G( v) G( y) T( k) (5) 387

4 u -v where y u k,,,..., w, w is the number of the interval [ uv, ] divisions, k w and the correction terms on outer dimension Y is ( k ). We shall mention that in Traezoidal and Midoint case, w,, 4,8,... while in Simson case, w,4,8,.... To calculate G ( y ), G ( v ), G ( u ) aroximately using formula (4), we write: t G ( u ) g ( x, u ) dx s t G ( v ) g ( x, v ) dx s T (6) (7) t G ( y ) g ( x, y ) dx, (8) s Alying the suggested method ( on inner dimension X) on (6),(7),(8), we get the following formulas: w k G ( u ) g( s, u ) g( t, u ) g( s ( w 0.5) k, u ) (g( s ( r 0.5) k, u) g ( s rk, u )) k 4 r...(9) w k G ( v ) g( s, v ) g( t, v ) g( s ( w 0.5) k, v ) (g( s ( r 0.5) k, v ) g ( s rk, v )) k 4 r...(0) w k G ( y ) g( s, y ) g( t, y ) g( s ( w 0.5) k, y ) (g( s ( r 0.5) k, y ) g ( s rk, y )) 3k 4 r...() where w is the number of the interval [ st, ] divisions, outer dimension X for (9),(0) and () are 3 k, k, k t s k and the correction terms on w resectively. To imrove the results, we alied Romberg accelerating on (9),(0) and (), then substituted the new results in (5) to get the aroximated value for the integral J in the formula. After substitute the correction terms T ( k ) in (5) and alying Romberg accelerating again, we will calculate the aroximated value for the integral J in (3) which means that we got the aroximated value for the original integral J in (). Remark: With the same above rocess, we can calculate the double integrals using RS(RSu) and RM(RSu). 6. Examles: 6. : The integrand of.5x 0.6 y ye dxdy that defined for (, ) [0,] [0,] 00 x y which analytic value is (aroximates to fourteen decimal) had been shown in Fig.:. 388

5 .50.6 y 0.6 y From (4) we have G ( y ) y e e 5, alying the three methods 4 RT ( RSu), RM ( RSu), RS ( RSu ) with Es 0 for the outer dimension y and Es 0 for the inner dimension x as follows: Alying RT ( RSu), RS ( RSu ) with w w 3, we got a value that equal to the exact value on artial interval, resectively. Moreover by alying RM ( RSu ) with w 6, w 3, we got a value that aroximates to thirteen decimal) on artial interval tables (),() and (3).. All these details shown in Fig. 6. : The integrand of 3 y ln( x y ) dxdy that defined for ( x, y) [,] [,3] which analytic value is (aroximates to fourteen decimal) had been shown in Fig.:. G ( y ) y ( y )ln( y ) ( y )ln( y ), alying the three methods From (4) we have 4 RT ( RSu), RM ( RSu), RS ( RSu ) with Es Es 0 as follows: Alying RT ( RSu), RS ( RSu ) with w 3, w 6, we got a value that equal to the exact value on artial interval, same value on artial interval 0 resectively. Moreover by alying ( ) RM RSu with w w 6, we got the 0. All these details shown in tables (4),(5) and (6). Fig.: 389

6 6.3: The integrand of cosh(0.4x 0.6 y ) dxdy that defined for ( x, y) [, ] [, ] which analytic value is (aroximates to fourteen decimal) had been shown in Fig.:3. G ( y ).5 sinh( y ) sinh( y ), alying the three methods From (4) we have RT ( RSu), RM ( RSu), RS ( RSu ) with Es 4 0 and 0 value that equal to the exact value on artial interval, tables (7),(8) and (9). Es 0 with w w 6, we got a 9 resectively. All these details shown in 6.4 The integral cos( x) sin( y) Fig.:3 4 4 dxdy that defined on,.5,.5, x y value is unknown had been shown in Fig.:4. From (4) we have, RS ( RSu ) with cos( x) sin( y) x y, but its analytic 4 4 dxdy,alying the three methods RT ( RSu), RM ( RSu) x y Es 4 0 and Es analytic value, but alying ( ), ( ) 0 we got that although, this integral with unknown RT RSu RS RSu with w w and w w 6 for 3 RS(RSu), its value will be which fixed. Thus we can say that this fixed value is 0 9 the correct value for the integral aroximates to fourteen decimal on artial interval and resectively. All these details shown in tables (0),() and (). 390

7 Fig.:4 6.5 The integral 33 had been shown in Fig.:5. From (4) we have Es 4 0 and ( xy ) y dxdy that defined onx, y,3,3, but its analytic value is unknown 33 ( xy ) y dxdy, alying the three methods RS ( RSu), RM ( RSu), RT ( RSu ) with Es alying ( ), ( ) 0 we got that although, this integral with unknown analytic value, but RT RSu RS RSu with w 3, w 6 and for RM(RSu), w 64, w 6, its value will be which fixed. Thus we can say that this fixed value is the correct value 0 for the integral aroximates to fourteen decimal on artial interval and resectively. All these details shown in tables (3),(4) and (5). Fig:5 7. Conclusion : In this work, we calculate the aroximated value for the double integrals with continuous integrands using the comound methods which consist of RSu ( on inner dimension) and RT, RM, RS (on outer dimension), and we called them as RT ( RSu), RM ( RSu ) and RS ( RSu ) 39

8 resectively. All tables show that, these comound methods give correct value ( for many decimals) on numbers of artial eriods comared with the exact values of the integrals. In more accuracy, we got exact values for the integrals aroximated to (3-4) decimal on (6-64) artial eriod using these comound methods. Moreover, using these three methods allow us to calculate the value for unknown analytic value integrals when the aroximated value is fixed like was in Examle 6.4 and 6.5. Therefore, the comound methods give results with higher accuracy in calculating the double integrals with continuous integrands on less artial eriods and less time. 8. Tables: 39

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10 9. References: [] Alkaramy, Nada A., Derivation of Comosition Methods for Evaluating Double Integrals and their Error Formulas from Traezoidal and mid- oint Methods and Imroving Results using Accelerating Methods, MS.c dissertation, 0. [] Al-Taey, Alyia.S, Some Numerical Methods for Evaluating Single and Double Integrals with Singularity, MS.c dissertation, 005. [3] Anthony Ralston,"A First Course in Numerical Analysis " McGraw -Hill Book Comany,965 [4] Eghaar, Batool H., Some Numerical Methods For Evaluating Double and Trile Integrals, MS.c dissertation, 00. [5] Fox L.," Romberg Integration for a Class of Singular Integrands ", comut. J.0, , 967 [6] Fox L. And Linda Hayes, " On the Definite Integration of Singular Integrands " SIAM REVIEW.,, , 970. [7] Mohammed, Ali H., Evaluations of Integrations with Singular Integrand,, MS.c dissertation,984. [8] Mohammed Ali H., Alkiffai Ameera N., Khudair Rehab A., Suggested Numerical Method to Evaluate Single Integrals, Journal of Kerbala university, 9, 0-06,

11 [9] Muosa, Safaa M., Imroving the Results of Numerical Calculation the Double Integrals Throughout Using Romberg Accelerating Method with the Mid-oint and Simson s Rules, MS.c dissertation,

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