Approximate Solution for the System of Non-linear Volterra Integral Equations of the Second Kind by using Block-by-block Method

Transcription

1 Astralian Jornal of Basic and Applied Sciences, (1): , 008 ISSN Approximate Soltion for the System of Non-linear Volterra Integral Eqations of the Second Kind by sing Block-by-block Method 1 Rostam K. Saeed, and Chinar S. Ahmed 1, University of Salahaddin/Erbil, College of Science, Department of Mathematics-Krdistan Region-Iraq. Abstract: The aim of this paper is for finding the nmerical soltion (sometimes exact) for non-linear system of Volterra integral eqations of the second kind (NSVIEK) by sing block-by-block method. Which avoid the need for special starting procedres, bt ses nmerical qadratre rle. Also some illstrative examples are presented, to elcidate the accracy of this method. Key words: block-by-block, system of non-linear Volterra integral eqation INTRODUCTION A block method is essentially an extrapolation procedre which has advantage of being self-starting and prodces a block of vales at a time (Delves and Mohamed (1985); Delves and Walsh (1974)). Linz (1969) describes two block (block-by-block) method and sed this method to solve Volterra integral eqations of the second kind. Also AL-Asdi (00) sed two and three blocks for solving Hammersetien Volterra integral eqations of the second kind, while Saify (005) sed two, three and for blocks for solving a system of linear Volterra integral eqation of the second kind. In this paper, the approaches of two and three blocks are reformlated and applied to find the nmerical soltion for a system of non-linear VIEK s, in which a block of two and three vales are prodced at each stage, and these vales are obtained sing the two three-point qadratre formla. The reslting system of non-linear eqations from this approaches are solved by modified Newton-Raphson method (mnrm) (Kincaid and Cheney (00). Some Integration Formlas: (Gerald and Wheatley (1984); Kincaid and Cheney (00) In this section we sed some qadratre formlas and qadratic interpolation polynomials sch as trapezoidal, Simpson s 1/3 and Newton-Gregory forward methods. The form of Newton-Gregory forward is: (1) where Here, to derive other integration formlas we sed the emblematic method in terms of the stepping operator E. Ths Corresponding Athor: Rostam K. Saeed, University of Salahaddin/Erbil, College of Science, Department of Mathematics-Krdistan Region-Iraq. rostamkarim64@ni-sci.org 114

2 () Since, E=1+Ä (3) we expand ln(1+ Ä ) as a power series, we get: Sbstitting eqation (3) and (4) into the eqation () to obtain: (4) Using division we get: (5) When n terms are sed, it represents a polynomial of degree n, fitting from x 0 to x n bt the integrated only from x 0 to x 1. First, sing three terms of eqation (4.5) we obtain: (6) Second, sing for terms of eqation (4.5) we obtain: (7) Moreover, the following formla is derived by sing adaptive Simpson s 1/3 rle. (8) where f 1/ can be fond as follows: In the eqation (1), ptting s=1/ and n=, we get: (9) 115

3 Block-by-block Method: The basic interval [a, b] is divided into steps of width h, sch as x j, = jh, j = 0,1,...n and nh = b - a. The approximate soltion of i (x) will be defined at mesh-points x j and denoted by ij; j = 0,1,...,n sch as is an approximation to (x ). ij i j For solving system of non-linear Volterra integral eqations (10) where Rewrite eqation (10) as follows:, (11) where p is some integer and l is. If the vales i0, i1,..., i,pl are known, then the first integral can be approximated by standard qadratre methods. The second integral is estimated by a qadratre rle sing vales of the integrand at have a system of mp non-linear simltaneos eqations. Since the vales of at these points are nknown, we i (1) for, where depend on the qadratre rle sed. For sfficiently small h the system we obtain from eqation (1) has a niqe soltion which can be determined by iteration sch as modified Newton-Raphson method. Ths, a block of p vales of is obtained simltaneosly. Modified Method of Two Blocks: i For this method we take p =, the integration over can be accomplished by Simpson s rle, and the integral over Then eqation (10) becomes: by sing a qadratic interpolation of the integrand at the point (13) and (14) 116

4 where,. Or from eqation (11), eqation (13), (14) can be written as: and Therefore, by eqation (6) the approximate soltion is compted by: (15) (16) It shold be noticed that in the eqation (15) the kernel has to be evalated at the point. Ths, replace the second term in eqation (15) by formla (8) and (9). Then the reslting eqations are: (17) (18) where, 117

5 At each step we constrct m non-linear simltaneos eqations from (17) and (18) to find the nknowns and. Solve the reslting system of non-linear eqations by sing modified Newton-Raphson method. il+1 il+ Algorithm of MBLM: Step (1): Fix Step (): Letting. Step (3): Calclate. Step (4): Using eqation (17) and (18) to find system of eqation for the nknown's il+1 and il+. Step (5): Find the vale of il+1 and il+ by sing mnrm. Step (6): Repeat steps (3)-(4) for l=1,,... Modified Method of Three Blocks: The method of two blocks which presented in section 3.1 can be extended to prodce 3m simltaneos non-linear eqations at each step since in this method we have p = 3. Then throgh the se of eqation (11) eqation (10) can be written as: In practice this method depends on the se of three qadratre formla, Simpson s 1/3 rle, trapezoidal rle and some qadratre interpolation formla. Therefore, the approximate soltion is compted as follows: If l is even (19) (0) (1) If l is odd 118

6 () (3) (4) Setting s=1/, n=3 and sing eqation (1) and (8) the last term in eqations (19), (0), () and (3) becomes: The reslting eqations (19)-(4) become: If l is even, where q=0, 1 (5). (6) (7) 119

7 (8) If l is odd (9) (30) (31) where Therefore, at each step we constrct 3m simltaneosly non-linear eqations from the eqation (6)-(31) which can be solved for the nknown s i,3l+1, i,3l+ and i,3l+3, by sing modified Newton- Raphson method. Algorithm of MBLM3: Step (1): Fix,. Step (): Letting Step (3): If l is even then, calclate. Step (4): Using eqation (6), (7) and (8) to find system of eqations for the nknown's i,3l+1, i,3l+ and i,3l+3. Step (5): Find the vale of, and by sing mnrm. i,3l+1 i,3l+ i,3l+3 10

8 Step (6): If l is odd then, calclate,. Step (7): Using eqation (9), (30) and (31) to find system of eqations for the nknown's and i,3l+3. Step (8): Find the vale of i,3l+1, i,3l+ and i,3l+3 by sing mnrm. Step (9): Repeat steps (3)-(8) for l=1,,... Illstrative Examples: In this section, three examples are presented for demonstrating the methods and a comparison among the soltions obtained by these methods against the exact soltion which has been made depending on the least sqare errors. i,3l+1, i,3l+ Example 1: (Babolian and Biazar (000)) Solve a system of non-linear VIEK s: The exact soltion of this system is: and. After solving this system sing block-by-block methods with h=0.1 in eqation (17), (18) and (6)-(31), we obtain the following nmerical reslts. Table 1: Comparison between the exact soltion x and the nmerical soltion 1(x) and (x) of Example 1 taking h=0.1. 1(x) (x) x Exact soltion MBLM MBLM3 MBLM MBLM L.S.E Table : Shows the least sqare errors for 1(x) and (x) with different vales of h for Example 1. least sqare errors Nmerical soltion of methods h=0.1 h=0.05 h=0.05 1(x) MBLM MBLM X X X 10 (x) MBLM MBLM X X X 10 Example : (Jmaa (005)) Solve a system of non-linear VIEK s: 11

9 The exact soltion of this system is: After solving this system sing block-by-block methods with h=0.1 in eqation (17), (18) and (6)-(31), we obtain the following nmerical soltion. Table 3: Comparison between the exact soltion and the nmerical soltion 1(x) of Example taking h=0.1. x 1(x) Exact soltion MBLM MBLM L.S.E Table 4: Comparison between the exact soltion tan (x) and the nmerical soltion (x) of Example taking h=0.1. (x) x Exact soltion MBLM MBLM L.S.E Table 5: Shows the least sqare errors for 1 (x) and (x) with different vales of h for Example. least sqare errors Nmerical soltion of methods h=0.1 h=0.05 h= (x) MBLM MBLM (x) MBLM MBLM Example 3: (Jmaa (005)) Solve a system of non-linear VIEK s: 1

10 The exact soltion of this system is: After solving this system by block-by-block methods with h=0.1 in eqation (17), (18) and (6)-(31), we obtain the following nmerical soltion. Table 6: Comparison between the exact soltion -1/x and the nmerical soltion 1(x) of Example 3 taking h= (x) x Exact soltion MBLM MBLM L.S.E x Table 7: Comparison between the exact soltion e and the nmerical soltion (x) of Example 3 taking h=0.1. (x) x Exact soltion MBLM MBLM L.S.E Table 8: Shows the least sqare errors for 1 (x) and (x) with different vales of h for Example 3. least sqare errors Nmerical soltion of methods h=0.1 h=0.05 h= (x) MBLM MBLM (x) MBLM MBLM Conclsions: According to the nmerical reslts which obtaining from the illstrative examples we conclde that the method of two blocks is the best bt slower than the method of three blocks. If the fnction, i=1,,..., m are polynomial for sfficiently small h we get a good accracy (exact sometimes) hence by redcing step size length the least sqare error will be redced. REFERENCES AL-Asdi, A. S., 00. The Nmerical Soltion of Hammersetien-Volterra-Second Kind-Integral Eqations, M.Sc. thesis, University of AL-Mstansiriya, Iraq. Babolian, E. and J. Biazar, 000. Soltion of a System of Non-linear Volterra Integral Eqations of the Second Kind, Far East J. Math. Sci. (FJMS), (6):

11 Delves, L.M. and J.L. Mohamed, Comptational Method for Integral Eqations, Cambridge University. Delves, L.M. and J.L. Walsh, Nmerical Soltion of Integral Eqations, Clarendon Press Oxford. Gerald, C.F. and P.O. Wheatley, Applied Nmerical Analysis-Third edition, Addison-Wesley pblishing company, Menlo Park, California. Jmaa, B.F., 005. On Approximate Soltions to a system of Non-linear Volterra Integral Eqations, Ph.D. Thesis, University of Technology, Department of Applied Science, Iraq. Kincaid, D. and W. Cheney, 00. Nmerical Analysis: Mathematics of Scientific Compting, third edition, Wadsworth grop. Brooks/Cole. Linz, P., A Method for Solving Non-linear Volterra Integral Eqations of the Second Kind, Mathematics of Comptation, 3(107): Saify, S.A.A., 005. Nmerical Methods for a System of Linear Volterra Integral Eqations, M.Sc. thesis, University of Technology, Iraq. 14