Finding Error Formulas for SuSu Method to. Calculate Double Integrals with Continuous and. Improper \ Improper Derivatives Integrands.

Transcription

1 International Journal of Mathematical Analsis Vol. 11, 17, no. 17, HIKARI Ltd, Finding Error Formulas for SuSu Method to Calculate Double Integrals with Continuous and Improper \ Improper Derivatives Integrands Numericall Ameera N. Alkiffai Dep. of Mathematics, College of Education for girls, Universit of Kufa, Iraq Rehab A. Shaaban Dep. of Mathematics, College of Education for girls, Universit of Kufa, Iraq Safaa M. Aljassas Mathematics Dep., College of Education for girls, Universit of Kufa, Iraq Copright 17 Ameera N. Alkiffai, Rehab A. Shaaban and Safaa M. Aljassas. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited. Abstract The main goal for this research is to find the general formula for the correction terms of the suggested method, SuSu, when the integrand is continuous and the derivatives are improper or it is just improper in one point\ more of integration region. This research includes theorems about finding the error formulas for these kinds of double integrals from lower\upper ends or both of them at same time. Three examples are discussed to illustrate different cases of double integrals in addition tables had been attached to show all the details. Kewords: Double integrals, continuous integrand, improper derivatives, Romberg s acceleration

2 8 Ameera N. Alkiffai et al. 1. Introduction Double integrals are important for finding surface area, evaluating intermediate centres and moments of inertia for plane surfaces as well as computing the volume under the double integral surface. As an example for this, the resulting 1 cos around the polar axis. In volume from the rotation of heart curve addition, the importance of double integrals as Frank Eers in [] explained, lies in evaluating the area of the piece of ball x z 36 inside the clinder z 6. Man researchers in the double integrals field, Schjear-Jacobsen [7] in 1973, highlighted the computation of double integrals with continues f x, f x f whereas, Davies and integrands of the expression 1 Rabinowitz [9] in 1975, worked with integrals that including improper integrands. In 9, Dheaa [3] presented four numerical methods composed of Romberg's acceleration with the midpoint method and Romberg's acceleration with Simpson method, RM(RS), RS(RS), RS(RM) and RM(RM), to calculate double integrals values that have continues integrands and improper or with improper derivatives. In 11, [6] discussed three numerical methods composed of Romberg's acceleration with two formulation of Newton-Coats (Simpson and Midpoint), RSS, RMS, and RSM when the number of partial periods on two axis X and Y is equal, to calculate double integrals values that have integrands with improper derivatives or onl improper. RM(RM) and RSS had been tested to be the best methods in terms of accurac and speed to approach to the real value of integrations with a few partial periods. For more information in this area, see [1, 6]. In our research, we have derived the general form for the correction terms of the suggested method, SuSu, that introduced in [5] in case that the integrand is continuous with improper derivative or onl improper in one point or more of integration region.. Finding the error formula using the suggested method SuSu to calculate the double integral with continuous integrands and improper derivative from the lower end Theorem (: Let the function f(x,) be continuous and differentiable at each point of the integration region except at the point ( x, ) ( x, ), the approximate value of the double integral SuSu method: n xn J f ( x, ) dxd can be evaluated using the following x

3 Finding error formulas for SuSu method 83 x n n h SuSu f( x, ) dxd f ( x, ) f ( x, n) f ( xn, ) f ( xn, n) ( f ( x, ( n.5) h) 16 x f ( x, ( n.5) h) f ( x ( n.5) h, ) f ( x ( n.5) h, ) f ( x ( n.5) h, ( n.5) h)) n n1 j1 n f ( x, ) f ( x, ( j.5) h) f ( x, ) f ( x, ( j.5) h) f ( x ( n.5) h, ) j n j n j f ( x ( n.5) h, ( j.5) h) f ( x, ) f ( x, ) f ( x ( j.5) h, ) f ( x ( j.5) h, ) j j n n f ( x, ( n.5) h) f ( x ( j.5) h, ( n.5) h) f ( x, ) f ( x, ( j.5) h) n1 j i j i i1 f ( x ( i.5) h, ) f ( x ( i.5) h, ( j.5) h))...( A) j And the error formula is: h h 3 3 h 4 4 ( Dx D ) ( Dx Dx D DxD D ) ( Dx D ) f ( x1, h h h Su Su Su 1 Where Su, Su, Su, are constants that depend on the partial derivatives of the function. Proof: Suppose that the function f x, is defined at each point of the integration region [ x, x n] [, n] and it is not improper and partial derivatives of the function are not defined at the point x,. This means that the Talor s series of two-variable functions exists at each point of the integration region except at x,, [7]. We can write the double integral J b: x x n1 x n1 x x n n r1 s1 1 n n J f x, dxd f x, dxd f x, dxd f x, dxd f x, dxd...( x x r1 x s1 x x r s 1 1 For the first integral on the partial integration region x, x 1, 1 Talor s series for f x, about x,, 1 1, use

4 84 Ameera N. Alkiffai et al. ( x x ) ( ) f ( x, ) 1 ( x x ) D ( ) D D ( x x )( ) D D D!! x 1 x 1 1 x Dx Dx D Dx D D ( x x ) ( x x ) ( ) ( x x )( ) ( ) ( x x ) 3!!! 3! 4! ( x x ( 3 ( x x ( ( x x ( 3 ( Dx Dx D Dx D DxD 3!!! 3! 4! ( x x 5 ( x x ( 4 ( x x ( 3 ( x x ( D Dx Dx D Dx D 5! 4! 3!!!3! ( x x ( 4 ( 5 Dx D Dx D D f ( x 1, 4! 5! B integrate equation () on ( x, x (, we get: 1 x1 x h h h h h h h h f ( x, ) dxd h D D D D D D D D D D D x x x x x x h h h h h h h h h h 3 h 4 h 5 Dx D Dx D D f ( x 1, Substituting the points: D Dx Dx D Dx D Dx D D Dx Dx D Dx D h h h h h h ( x, ),( x,,( x, ),,( x 1, ),( x, ),( x 1, ),( x 1, ),( x, ) in equation () and adding the result to equation (3) to obtain: 1 x1 x h h f ( x, ) dxd [ f ( x, ) f ( x 1, ) f ( x, f ( x 1, ( f ( x, ) f ( x, ) f ( x, ) f ( x, ) ( f ( x, ))] h h h h h h h 3 3 h 4 4 ( Dx D ) ( Dx Dx D Dx D D ) ( Dx D ) f ( x 1, ) 1...(4) For the other three integrals in equation (, the derivative of the function is continuous, so we can calculate their values and add them to equation (4) to get an equation consists of (A) plus the error formula (. 3. Finding the error formula using the suggested method SuSu to calculate the double integral with continuous integrands and improper derivative from the upper end Theorem (): Let the function f(x,) be continuous and differentiable at each point of the inte-

5 Finding error formulas for SuSu method 85 gration region except at the point ( x, ) ( x, ), the approximate value of the double integral n xn x n J f ( x, ) dxd can be evaluated using the following SuSu method (A). So the error formula is: n h h 3 3 h 4 4 ( Dx D ) ( Dx Dx D DxD D ) ( Dx D ) f ( xn 1, n h h h... H 4 6 Su Su Su Where Su, Su, Su, are constants that depend on the partial derivatives of the function. Proof: Suppose that the function f x, is defined at each point of the integration region [ x, x n] [, n] and it is not improper and the partial derivatives of the function are not defined at the point x n, n. Write the double integral J b: x x n x n x x n n n1 n1 s1 n n r1 n n,,,,, J f x dxd f x dxd f x dxd f x dxd f x dxd x x s x r x x s n1 n1 r n1 n1 Using the same steps in proof Theorem 1, we get an equation consists of (A) plus the error formula ( H ) and the proof is complete. 4. Finding the error formula using the suggested method SuSu to calculate the double integral with continuous integrands and improper derivative from the upper\lower ends Theorem (3): Let the function f(x,) be continuous and differentiable at each point of the ( x integration region except at the point n, n),( x, ), the approximate value of the double integral n xn J f ( x, ) dxd can be evaluated using the following x SuSu method (A). So the error formula is: (5)

6 86 Ameera N. Alkiffai et al h h 3 3 h 4 4 ( Dx D ) ( Dx Dx D DxD D ) ( Dx D ) ( 1, f x h h 3 3 h 4 4 ( Dx D ) ( Dx Dx D DxD D ) ( Dx D ) f ( xn 1, n Suh Suh Suh...H3 Where Su, Su, Su, are constants that depend on the partial derivatives of the function. Proof: We can rewrite the above integral as: n xn 1 x1 n1 s1 x1 n1 s1 n xi1 n s1 xn J f ( x, ) dxd f ( x, ) dxd f ( x, ) dxd f ( x, ) dxd f ( x, ) dxd x 1 1 x s s r s s x s x s xn1 n xn f ( x, ) dxd...(6) n1 xn1 Similar to the steps in proofs of Theorems 1 and, we can evaluate the first and the last integrals in equation (6) respectivel. For the other three integrals, the integrand is continuous derivatives on their integration region, so it is not difficult to calculate them and then add the result to equation consists of (A) plus ( H ) together with (6) to obtain an equation consists of (A) plus the error formula H 3. Thus the theorem is established. 5. Integrals with improper integrands in one or both ends of the integration n xn Suppose that J f ( x, ) dxd, f(x,) is continuous on the integration region x x x but it is not defined at the points x, and,, n n x or on one of them. Thus we cannot appl an of the above three theorems, so to evaluate the improper integral, we will ignore the value of the function on impropriet point as Phillip and Rabinowitz suggested in [9]. 6. Examples 6.1: The integral sin ( ).5.5 x dxd that its function had been shown in Fig:1, has analtic value which approximates to fourteen decimal digits. n n

7 Finding error formulas for SuSu method 87 Fig. (: Geometric shape for 1 sin ( ) x dxd within and on integration region Here, the integrand for this example is continuous in integration region but the partial derivatives are improper at the point (x,) = (1,. From Theorem, the suitable correct terms are E ( h) h a h a h h a h a h h a h (7) SuSu Su 1 Su 3 4 Su 5 where ai, Su, Su,, i 1,,3, are constants. Appling SuSu method, we obtained six correct decimal digits at n=18. Moreover, when we used Romberg s acceleration to improve these results with the above correction terms in (7), we 16 got fourteen correct decimal digits at n=18 with partial periods (which is equal to the analtic value), compared with Musa in [6] when got a correct value for thirteen correct decimal digits at n=56 with the same partial periods. Moreover, Nada in [1] got a correct value for fourteen correct decimal digits at 18 n=56 with partial periods using RTT method as same as Eghaar in [4] using RMM method. Table 1 shows all these details. 6.: The integral J 1 1 dxd that its function had been shown in Fig: (1 x ) 1, has analtic value which approximates to fourteen decimal digits. Fig. (): Geometric shape for 1 (1 x )

8 88 Ameera N. Alkiffai et al. In this example, the integrand is continuous in integration region except at the point (x,) = (,-, so it is improper such that the impropriet is proportional. From Theorem 1, the suitable correct terms are 4 6 E ( h) ch h h Su (8) SuSu Su Su h where c, Su, Su, Su, are constants. Appling SuSu method on axes, we obtained two correct decimal digits at n=18 taken with Phillip- Rabinowitz in [9] suggestion (ignoring the function value at impropriet point). Moreover, when we used Romberg s acceleration to improve these results with the above correction terms in (8), we got an exact value that approximates to four decimal digits at n=18 compared with Musa in [6] when got a correct value for fourteen 14 correct decimal digits at same n but with. Moreover, Nada in [1] got a correct 16 value for thirteen correct decimal digits at n=18 with partial periods using RTT method likewise Eghaar in [4]. Table shows all these details. 6.3: The analtic value of the integral function had been shown in Fig x 4 x dxd is unknown. Its Fig. (3): Geometric shape for x 4 x For this example, the integrand is continuous but it is improper at the points (x,) = (,) (x,) = (.5,.5), so b Theorem 3, the correction terms are E ( h) h t h t h h t h t h h t h (9) SuSu Su 1 Su 3 4 Su 5 Where ti, Su, Su, Su,, i 1,,3, are constants. Table 3 shows the results using SuSu method, It is clear that, although the analtic value of the integral is unknown, but this value is fixed horizontall for four columns at n=56 and for six columns at n=51. Thus we can consider that it is the exact value at least for 18 four decimal digits which is with partial periods. Moreover, Table 5 shows that Musa in [6] got a correct value for fourteen correct 18 decimal digits at n=51 with, while Nada in [1] also got same correct value at

9 Finding error formulas for SuSu method 89 same n but with partial periods using RTT method likewise Eghaar in [4] using RMM method. 7. Conclusion We conclude that the values of double integrals using SuSu method give the correct values of several decimal digits compared with the exact values of the integrals using a number of partial periods without using a method of teasing. Moreover the tables show that the results will be better with a few relativel partial periods as well as the values are correct for several decimal digits which are between thirteen and fourteen correct decimal digits, when we use the Romberg s acceleration with the SuSu method accompaning with the correction terms. In addition, the Romberg acceleration without ignore the impropriet will pla an importance role to improve results in terms of accurac and speed of approach to the real value of integrations with a few partial periods. Therefore we can use SuSu method to evaluate the double integral whether the integrand is continuous or improper. References [1] A. H. Mohammed, A. N. Alkiffai, R. A. Khudair, Suggested Numerical Method to Evaluate Single Integrals, Journal of Kerbala Universit, 9 (1, 1-6. [] A. M. Dheaa, About Single, Double and Triple Integrals, MSc Dissertation, 9. [3] B. H. Eghaar, Some Numerical Methods for Evaluating Double and Triple Integrals, MSc Dissertation, 1. [4] F. J. R. Ares, Schaum s Outline Series: Theor and Problems of Calculus, McGraw-Hill book-compan, 197. [5] H. S. Jacobsen, Computer Programs for One- and Two-Dimensional Romberg Integration of Complex Function, Lautrupvang, Ballerup, Denmark, 1973, 1-1. [6] J. D. Phillip, P. Rabinowitz, Methods of Numerical Integration, Dover Publications, Inc., New York, [7] N. A. Alkaram, Derivation of Composition Methods for Evaluating Double Integrals and their Error Formulas from Trapezoidal and Mid-point Methods and Improving Results using Accelerating Methods, MSc Dissertation, 1.

10 81 Ameera N. Alkiffai et al. [8] S. M. Muosa, Improving the Results of Numerical Calculation the Double Integrals Throughout Using Romberg Accelerating Method with the Midpoint and Simpson s Rules, MSc Dissertation, 11. [9] S. S. Sastr, Introductor Methods of Numerical Analsis, PHI Learning Pvt. Ltd., New Delhi, 8. Received: Jul 8, 17; Published: August 3, 17

11 Finding error formulas for SuSu method 811 Tables

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