Fixed point method and its improvement for the system of Volterra-Fredholm integral equations of the second kind

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1 MATEMATIKA, 217, Volume 33, Number 2, c Penerbt UTM Press. All rghts reserved Fxed pont method and ts mprovement for the system of Volterra-Fredholm ntegral equatons of the second knd 1 Talaat I. Hasan, 2 Shaharuddn Salleh, 3 Nejmaddn A. Sulaman 1,3 Department of Mathematcs, Salahaddn Unversty 442 SUH, Erbl, Kurdstan Regon, Iraq 2 Department of Mathematcal Scences, Unversty Technolog Malaysa 8131 UTM Skuda, Johor, Malaysa e-mal (correspondng author): ss@utm.my Abstract In ths paper, we consder the system of Volterra-Fredholm ntegral equatons of the second knd (SVFI-2). We proposed fxed pont method (FPM) to solve SVFI-2 and mproved fxed pont method (IFPM) for solvng the problem. In addton, a few theorems and two new algorthms are ntroduced. They are supported by numercal examples and smulatons usng Matlab. The results are reasonably good when compared wth the exact solutons. Keywords Fxed pont method, mproved fxed pont method, contracton mappng,and system Volterra-Fredholm ntegral equatons of the second knd. AMS mathematcs subject classfcaton 45D5, 34A12 1 Introducton Integral equatons have been one of the prncpal nstruments n many dfferent felds of scence lke appled mathematcs physcs, bology and engneerng [1, 2]. Also ntegral equatons are encountered n numerous applcatons n varous areas [3]. In addton, they arse naturally n applcatons, n many felds of mathematcs, scence and technology [4]. They have studed both at the theoretcal and practcallevels [5]. Soluton of ntegral equatons by numercal methods have grown wdely grown n the last 25 years. The methods of ntegral equaton are strongly used for treatng many problems n mathematcal physcs [6]. Integral equatons have many advantages wtnessed by the ncreasng frequency of the ntegral equatons n the lterature and n many areas because some problems have ther mathematcal representaton appear drectly [7]. The Volterra- Fredholm ntegral equatons appear from parabolc boundary value problems [6]. Integral equaton focused on the numercal method of soluton [8]. Jerr [9] was dscussed fxed pont teraton method to solve lnear Volterra ntegral equaton. Sulaman [1] used fxed pont method (FPM) to solve lnear Fredholm ntegral equaton of the second knd. Hasan [11] solved a certan system of Fredholm ntegral equaton of the frst knd. In addton, Waz waz [12] solved a system of lnear Volterra ntegral equatons by FPM. The proposed method FPM and IFPM are used for obtanng the approxmate soluton of system Volterra-Fredholm ntegral equatons of the second knd. Illustratve examples wll be ncluded to demonstrate the valdty and applcablty of the presented technques to hghlght the sgnfcaton of the FPM and mprove fxed pont method (IFPM).

2 192 Talaat I. Hasan, Shaharuddn Salleh and Nejmaddn A. Sulaman 2 Defntons Defnton 1[9] A functon f : R p R p s sad to be Lpschtz on B R p wth Lpschtz constant L > f f(x) f(y) L x y, for all x, y B for some norm on R p. Defnton 2 [13] A map T s called contracton mappng on the nterval [a, b] f t satsfy the followng condtons: 1. U C[a, b] T(U) C[a, b]. 2. T Lp[a, b], wth Lpchtz constant, < L < 1. Defnton 3 [12, 13] Let V and W be vector spaces over a feld F. A functon T : V W s called a lnear transformaton f t satsfes the followng condtons: 1. A, B V, T(A + B) = T(A) + T(B). 2. A V and r F, T(rA) = rt(a). Defnton 4 [6, 14] The ntegral equaton y(s) = f(s) + λ s a b k(s, t)y(t)dt + λ g(s, t)y(t)dt, s called a lnear Volterra-Fredholm ntegral equaton of the second knd, where the functons k(s, t) and g(s, t) are called kernels of ntegral equaton, < λ < 1 and < λ < 1 such that f(s), k(s, t) and g(s, t) are known functons on R = {(s, t), a t < s b} and y(s) s unknown functon. Defnton 5 [12, 15] A kernel k(s, t) s sad to be a symmetrc kernel f k(s, t) = k(t, s) for all s, t R. 3 System of lnear Volterra-Fredholm ntegral equatons of the second knd The system of p Volterra-Fredholm ntegral equatons of the second knd (SVFI-2) s gven as follows: s b y (s) = f (s) + λ k k k (s, t)y k (t)dt + kk(s, t)y k (t)dt, (1) a for = 1, 2, 3,..., p.f (s) s a contnuous functon on [a, b], k k (s, t) and k (s, t) denote the k gven contnuous kernel functons on R = {(s, t), a t < s b} whle y (s) s the unknown functon to be determned. 4 FPM soluton for SVFI-2 wth symmetrc kernels The system of lnear Volterra- Fredholm ntegral equaton of the second knd gven n (1), ths system can be wrtten as follows: Y = F + λ k K k Y k + λ kkky k, (2) a λ k a

3 Fxed pont method for the system of Volterra-Fredholm ntegral equatons 193 Where Y = y (s), F = f (s), K k = k k (s, t) and Kk = k k (s, t), frst, we suppose that the ntal soluton Y = F, then the frst approxmaton s Y 1 = F + For second approxmaton, substtute Y 1 Y 2 = F + λ k K k Y k + m nto (3) we get λ k K k Y 1 k + m In general, the fxed-pont method can be wrtten as Y r+1 = F + λ k K k Y r k + m where max s the maxmum number of teratons. If the relatonshp defned n Equaton (5) converges then wth M(Y r ) = F + λ k K k Y k, (3) λ kk ky 1 k, (4) λ kkky r, for r = 1, 2, 3,..., max (5) k lm Y Y r =, (6) n Y r+1 = M(Y r ), (7) m λ k K k Yk r + k= k= where Y k = y k (s), Y r = y r k (s). The condton for the convergence property of the FPM s where M k = b a b Algorthm for FPM a λ k < 1 M k, [k k (s, t)] 2 dsdt and λ k < 1 M k where M k = λ k K k Y r k, (8) b Step 1 Let Y = F, for = 1, 2, 3,...p from equaton (1). Step 2 Fnd Y r+1 from Equaton (7), r = 1, 2, 3,..., max. Step 3 Compute the value of absolute error gven by e r = y (s) y r(s). Step 4 The approxmate solutons converges f where ε s a value close to. lm y (s) y r (s) = ε, r a b a [k k (s, t)]2 dsdt.

4 194 Talaat I. Hasan, Shaharuddn Salleh and Nejmaddn A. Sulaman Example 1 Consder the followng system y 1 (s) = 2 sn(s) s y 2 (t)dt, y 2 (s) = cos(s) wth the exact solutons y 1 (s) = sn(s) and y 2 (s) = cos(s). y 1 (t)dt, Soluton. By applyng FPM and ts program, we obtan the approxmate solutons of y 1 (s) and y 2 (s) as follows and shown n Table 1. s.2 r y 1(s) = 2 sn(s) y (s) = cos(s) y1(s) 1 = sn(s) + (.4597)s y 1 2(s) = cos(s) y1 2 (s) = sn(s) (.4597)s y 2 2(s) = cos(s) y 3 1(s) = sn(s) (.2298)s y 3 2(s) = cos(s).2298 Table 1: FPM results for example 1 and compared to the exact soluton Exact soluton of y 1 (s) = sn(s) Approxmate values of y 1 (s) Absolute error e r 1 y = 1 (s) y1 r(s) Exact soluton of y 2 (s) = cos(s) Approxmate values of y 2 (s) Absolute error e r 2 y = 2 (s) y2 r(s) Fgures 1(a) and 1(b) show a comparson between the exact and the approxmate solutons, gven n Example 1. Example 2 Consder the followng system y 1 (s) = 1 + s y 2 (s) = 2 + s + s s y 2 (t)dt y 1 (t)dt wth the exact solutons y 1 (s) = e s and y 2 (s) = 2e s. 1 1 (st)y 1 (t)dt, (st)y 2 (t)dt,

5 Fxed pont method for the system of Volterra-Fredholm ntegral equatons 195 Fgure 1(a): Graph y 1 (s) = sn(s) Fgure 1(b): Graph y 2 (s) = cos(s)

6 196 Talaat I. Hasan, Shaharuddn Salleh and Nejmaddn A. Sulaman Soluton By applyng FPM and ts program, we obtan the approxmate solutons of y 1 (s) and y 2 (s) as follows and shown n Table 2. y 1(s) = 1 + s y 2(s) = 2 + s y 1 1(s) = s +.25s 2 y 1 2(s) = 2 + (2.1867)s + (.5)s 2 y 2 1 (s) = 1 + (1.2153)s + (.5417)s2 + (.833)s 3 y 2 2(s) = 2 + (2.856)s + (2.1667)s 2 + (.5)s 3 y 3 1(s) = 1 + (.9984)s + (.12)s 2 + (.3611)s 3 + (.625)s 4 y 3 2(s) = 2 + (1.9981)s + (1.243)s 2 + (.6667)s 3 + (.125)s 4 s.2 r Table 2: FPM results for example 2 and compared to the exact soluton Exact soluton of y 1 (s) = e s Approxmate value of y 1 (s) Absolute error e r 1 y = 1 (s) y1 r(s) Exact soluton of y 1 (s) = 2e s Approxmate value of y 2 (s) Absolute error e r 2 y = 2 (s) y2 r(s) Fgures 2(a) and 2(b) show a comparson between the exact and the approxmate solutons, gven n Example 2. 5 IFPM to solve SVFI-2 wth symmetrc kernels Improved fxed-pont method s obtaned by addng α Y n both sdes of Equaton (7) where α 1: (1 + α )Y = α Y + M(Y ) (9) we get Y = α M(Y ) + α 1 + α Y = M α ( Y ). (1) If Y verfes Equaton (7) then t also verfes Equaton (1). Ths means Y s a soluton of Equaton (5) expressed n the teratve form, as follows: Y r+1 = M α ( Y r ), (11)

7 Fxed pont method for the system of Volterra-Fredholm ntegral equatons 197 Fgure 2(a): Graph y 1 (s) = e s Fgure 2(b): Graph y 2 (s) = 2e s

8 198 Talaat I. Hasan, Shaharuddn Salleh and Nejmaddn A. Sulaman The approxmate solutons n Equaton (11) should converge faster than the teratve form defned n Equaton (7) to the exact soluton of Equaton (1). The selecton for the optmal values of the scalar α s based on the followng role where α = (ν + ρ ) 2, (12) ν = sup {M(Y ) M(Y 1 )} and ρ = nf {M(Y ) M(Y 1 )}. (13) [a,b] [a,b] Algorthm for IFPM Step 1 Fnd the optmal values of the scalar α n Equatons (12) and (13). Step 2 Let Y = F, for = 1, 2, 3,..., p, from Equaton (1) whch s the ntal soluton. Step 3 Addng α Y to both sdes of Equaton (7). Step 4 Compute Y r+1 From Equaton (11). Step 5 Calculate e r = y (s) y r(s). Step 6 The approxmate solutons converges faster f lm y (s) y r r (s) = ε. 6 Numercal examples about SVFI-2 and results by usng IFPM The method of secton 5 s very useful for fndng the numercal solutons of SVFI-2. The computatons assocated wth the examples were performed usng MATLAB verson 12. Example 3 Fnd approxmate soluton ofa SVFI-2, n example 1 by usng IFPM. Soluton Applyng the numercal technque whch s IFPM, we obtaned the results for approxmate solutons of y 1 (s) and y 2 (s), where the optmal values are α 1 =.19 and α 2 = , as shown n Table 3. y1 (s) = 2 sn(s) y2(s) = cos(s).4597 y1 1 (s) = (.7654)sn(s) + (.5675)s y 1 2(s) = cos(s).859 y 2 1(s) = (1.55)sn(s) (.271)s y 2 2(s) = cos(s) +.25 y 3 1(s) = (.9871)sn(s) (.19)s y2 3 (s) = cos(s) +.17 Fgures 3(a) and 3(b) show a comparson between the exact and the approxmate solutons, gven n Example 3. Example 4 Fnd approxmate soluton ofa SVFI-2 n example 2 by usng IFPM. Soluton Applyng the numercal technque whch s IFPM we obtaned the results for approxmate solutons of y 1 (s) and y 2 (s), where the optmal values are α 1 =.128and

9 Fxed pont method for the system of Volterra-Fredholm ntegral equatons 199 s.2 Table 3: IFPM results for example 3 and compared to the exact soluton r Exact soluton of y 1 (s) = sn(s) Approxmate Values of y 1 (s) Absolute error e r 1 = y1 (s) y1 r(s) Exact soluton of y 2 (s) = cos(s) Approxmate Values of y 2 (s) Absolute error e r 2 = y2 (s) y2 r(s) Fggure 3(a): Graph y 1 (s) = sn(s)

10 2 Talaat I. Hasan, Shaharuddn Salleh and Nejmaddn A. Sulaman Fgure 3(b): Graph y 2 (s) = cos(s) α 2 =.315, as shown n Table 4. y 1 (s) = 1 + s y 2(s) = 2 + s y 1 1(s) = 1 + (1.1478)s + (.2216)s 2 y 1 2(s) = 12 + (1.8872)s + (.382)s 2 y 2 1(s) = 1 + (1.717)s + (.4434)s 2 + (.562)s 3 y 2 2 (s) = 2 + (2.21)s + (.886)s2 + (.964)s 3 y 3 1(s) = 1 + (1.265)s + (.498)s 2 + (.1259)s 3 + (.17)s 4 y 3 2(s) = 2 + (2.24)s + (.9618)s 2 + (.2281)s 3 + (.183)s 4 Fgures 4(a) and 4(b) show a comparson between the exact and the approxmate solutons, gven n Example 4. 7 Fxed pont and contractve mappng The orgnal teratve method s proposed as follows: Y r+1 (s) = F (s) + k= λk k Y r + k= λ K ky r, for r =, 1, 2,...,max (14)

11 Fxed pont method for the system of Volterra-Fredholm ntegral equatons 21 s.2 Table 4: IFPM results for example 4 and results compared to the exact soluton r Exact soluton of y 1 (s) = e s Approxmate Value of y 1 (s) Absolute error e r 1 y = 1 (s) y1 r(s) Exact soluton of y 1 (s) = 2e s Approxmate Value of y 2 (s) Absolute error e r 2 y = 2 (s) y2 r(s) Fgure 4(a): Graph y 1 (s) = e s

12 22 Talaat I. Hasan, Shaharuddn Salleh and Nejmaddn A. Sulaman Fgure 4(b): Graph y 2 (s) = 2e s The approach of the sequences Y r+1 (s) to Y (s) has been studed for solvng Equaton (1). Let T(Y r ) = F (s) + k= λk k Y r + k= λ K ky r. (15) Our soluton n ths work has been drected toward solvng Equaton (1). The ntegral equaton (15) s searched n the followng manner, the rght hand sde s supposed as a transformaton T on Y r r expressed by T(Y ), when the left hand sde denotes that the transformaton had left ths element Y r+1 unchanged. That s Y r+1 = T(Y r ), (16) whch mean that the soluton Y r whch we search for the ntegral equaton (15) expresses an especal values n the doman of the transformaton T, called that whch remans unaltered of stable under the transformaton T. Each element Y r as defned n equaton (16) s known a fxed pont of the transformaton T. Lemma 1 [12] If a map T satsfes the Lpchtz condton on the nterval [a, b] then there exsts a postve constant L, such that T(y) T(w) y w L, for all values y, w R. The constant L s called Lpchtz constant. Theorem 2 [3] Every contracton mappng s a Lpschtz functon on [a, b]. Theorem 3 [3] Every Lpschtz functon s a contnuous functon on [a, b]. Theorem 4 Let T be a contracton mappng on R then: 1. The relaton Y r+1 = T(Y r ) has a unque exact soluton Y R.

13 Fxed pont method for the system of Volterra-Fredholm ntegral equatons For any ntal soluton Y R the sequence defned n Equaton (16)converges to Y. Proof 1. Suppose that T s a contracton mappng on R. Then T s Lpchtz functon by Theorem 2 and T s contnuous functon by Theorem 3. Snce Y s exact soluton, then Y = T(Y ) (17) s contnuous on the regon R. Now we want to show that, the equaton (17) has a unque soluton. By contradcton, suppose that there exsts another soluton Z,such that Z Y whch satsfy Y = T(Y ) and Z = T(Z ) Now Y Z = T(Y ) T(Z ) Y Z L, because T s Lpchtz functon. Then we get Y Z Y Z L, Hence L 1 whch s a contradcton to the defnton of the Lpchtz constant, because L < 1. Snce T s contracton mappng on the regon R, then Z = Y. Therefore, Equaton (16) has a unque soluton. 2. By part one of ths theorem we gve the closure condton. That s mean If Y R then Y d R. Now Y d Y T(Y d ) T(Y ) Y d 1 Y L, also Y d 1 Y T(Y d 1 ) T(Y ) Y d 2 Y L 2, and so on, after d-tmes we get Y d Y Y d (d 1) Y L d 1 Take the lmt of both sdes we get Y d (d 1) 1 Y L d Y Y L d, lm Y d Y lm Y Y L d. d d Snce < L < 1, f d then L d. That means Y d Y for d. Hence, the sequence defned n equaton (16) converges to Y. 8 Concluson In ths work, we propose two methods called fxed pont method and mprove fxed pont method to solve SVFI-2. Several numercal examples were tested usng algorthms for FPM and IFPM for solvng a system SVFI-2. The results gven n Tables 1, 2, 3 and 4, ndcate clearly that both methods acheve good convergence as r ncreases when the error decreases. The mentoned examples demonstrated the valdty and applcablty of the technques. Fnally, we concluded that IFPM converges faster than FPM as shown n Tables 5 and 6.

14 24 Talaat I. Hasan, Shaharuddn Salleh and Nejmaddn A. Sulaman Table 5: Comparson between the absolute error of the present methods, FPM and IFPM methods n Example 1 s r Absolute errors of y 1 (s) by FPM Absolute errors of y 1 (s) by IFPM Absolute errors of y 2 (s) by FPM Absolute errors of y 2 (s) by IFPM Runnng Tme second Table 6: Comparson between the absolute error of the present methods, FPM and IFPM methods n Example 2 s r Absolute errors of y 1 (s) by FPM Absolute errors of y 1 (s) by IFPM Absolute errors of y 2 (s) by FPM Absolute errors of y 2 (s) by IFPM Runnng Tme by second

15 Fxed pont method for the system of Volterra-Fredholm ntegral equatons 25 Acknowledgments The authors would lke to extend ther sncere thanks to the referees for ther helpful remarks and suggestons. Many thanks are due to the Salahaddn Unversty and Unverst Teknolog Malaysa for provdng the fnancal support for ths research. The authors would also lke to thank the Malaysan Mnstry of Hgher Educaton for provdng fnancal support n FRGS grand number 4F625 for ths research. References [1] Javad, S. A. Modfcaton n successve approxmaton method for solvng nonlnear Volterra Hammersten ntegral equatons of the second. J. Mathematcal Extenson (1): [2] Mathwalu, M. S. and Sulaman, B. Numercal soluton of second knd lnear Fredholm ntegral equatons usng QSGS teraton method wth hgh-order Newton-Cotes quadrature schemes. Malaysa J. Mathematcal Scence (5) : [3] Maleknejad, K. Nour K. and Torkzandeh, L. Comparson projecton method for solvng system ntegral equatons. J. Bulletn of the Malaysan Mathematcal Scence Socety (34): [4] Yang, C. On the numercal soluton of Volterra-Fredholm ntegral equatons wth Abel kernel usng Legendre Polynomals. Internatonal Journal Advance Scence and Techncal Research (1): [5] Mrzaee, F. and Hosen, A. Numercal soluton of nonlnear Volterra-Fredholm ntegral equatons usng hybrd of block-pulse functons and Taylor seres. Journal of Alexandra Engneerng (52): [6] Hasan, T. I. Salleh, S. B. and Sulaman, N.A. Solvng a System of Volterra-Fredholm Integral Equaton of the second knd va seres soluton method. Journal of Appled Mathematc Scences (44): [7] Chnt, C. and Hou, J. Numercal method for solvng non Volterra ntegral equatons wth convoluton kernel. IAENG Internatonal Journal Appled Mathematcs (3): [8] Abdou, M. A. Volterra- Fredholm ntegral equaton of frst knd and contact problem. J. Appled mathematcs and computaton (8): [9] Jerr, A. J. Introducton to Integral Equaton wth Applcatons. New York and Basel: Marcel Dekker, Inc [1] Sulaman, N. A. Fxed pont method and ts mprovement to solvng Second knd Fredholm ntegral equaton. J. Dohuk Unversty (1): [11] Sulaman, N. A. and Hassan, T.I. Soluton of a certan system of Fredholm ntegral equatons of the frst knd wth symmetrc kernels. J. Koya Unversty : 3 16

16 26 Talaat I. Hasan, Shaharuddn Salleh and Nejmaddn A. Sulaman [12] Waz Waz, A. M. Lnear and Non-lnear Integral Equatons Methods and Applcatons. Berln: Sprnger-Verlag. 24. [13] Codngton, E. A. An Introducton to Ordnary Dfferental Equatons. Thrd edton New Jersey: Prentce Hall [14] Parpour, M. and Kamyar, M. Numercal soluton of non-lnear Volterra- Fredholm ntegral equatons by usng new bass functons. J. Communcatons n Numercal Analyss (17): [15] Zarebna, M. A numercal method of non-lnear Volterra- Fredholm ntegral equatons. Journal of Appled Analyss and Computaton :