A New Technique For Systems Of Abel-Volterra Integral .

International Journal of the Physical Sciences Vol. 7(1), pp. 89 - 99, 2 January, 2012Available online at http://www.academicjournals.org/IJPSDOI: 10.5897/IJPS11.864ISSN 1992-1950 2012 Academic JournalsFull Length Research PaperA new technique for systems of Abel-Volterra integralequationsJafar Biazar1* and H. Ebrahimi21Department of Mathematics, Rasht branch, Islamic Azad University, P. O. Box 41335-3516, Rasht, Iran.Faculty of Mathematical Science, University of Guilan, P. O. Box. 41635-19141, P. C. 4193833697, Rasht, Iran.2Accepted 8 September, 2011Extension of a computational method for solving a special kind of singular system is the novelty ofthis paper. These systems are called systems of Abel Volterra integral equations. This method isbased on the application of Legendre wavelets, as a basis functions for numerical solutions. Someexamples are presented to illustrate the efficiency and the simplicity of the method.Key words: Systems of Abel Volterra integral equations, Legendre wavelets method, operational matrices.INTRODUCTIONMathematical modeling of many physical systems leadsto functional equations in various fields of physics andengineering. In recent years some methods have beenused by many authors to obtain approximate solutions(He, 1999; Biazar et al., 2003, 2009; Faraz et al., 2010;Khan and Faraz, 2011). System of Volterra integralequations arise in mathematical modeling of manyphenomena (Delves and Mohamed, 1988; Jerri, 1999;Linz, 1985) and several methods have been proposed inthe literature to solve these systems. These systemshave been solved by Adomian decomposition method(Biazar et al., 2003), homotopy perturbation method(Biazar et al., 2009), variational iteration method (Biazarand Ebrahimi, 2010) and radial basis function networks(Golbabai et al., 2009).In the present paper, special kind of singular systemsof Volterra integral equations, called systems of Abelintegral equations are studied. Historically, Abel is thefirst person who had studied integral equations, duringthe 1820 decade (Jerri, 1999; Linz, 1985). He obtainedthe following equation, when he was generalizing thetautochrone problem. x0u (t )dt f ( x).x t(1)function which could be determined. This equation is aparticular case of a linear Volterra integral equation of thefirst kind. The kernel of Abel integral equations has weaksingularity and Abel integral equations are somewhat illposed and any small changes in the measurement datamay cause unpredictable huge errors in the numericalapproximate solutions.Some methods for solving the system of Abel integralequations are known. The idea of the fractional calculushas been used for a special kind of these systems(Mandal et al., 1996), an operational matrix methodbased on block-pulse functions for singular integralequations has been introduced (Maleknejad and Salimi,2008). In Pandey and Mandal (2010), Bernsteinpolynomials have been used for numerical solutions ofsystems of generalized Abel integral equations.The method introduced in this paper consists ofreducing a system of Abel integral equations into asystem of algebraic equations, by expanding theunknown functions, as a series in terms of Legendrewavelets with unknown coefficients (Maleknejad andSohrabi, 2007; Mahmoudi, 2005; Yousefi, 2006; Biazarand Ebrahimi, 2010). The general form of these systemsis considered as the following.mwhere f ( x) is a known function and u ( x) is an unknown Fi jm *Corresponding author. E-mail: biazar@guilan.ac.ir. Tel: 981313233509. Fax: 981313233509.( x , u 1 ( x ) , ., u n ( x ) ) j 1j 1xaG i j ( u 1 (t ) ,., u n (t ) )αi jdt f i ( x ),( x t )i 1, 2 ,., n , m 1, 2 ,.,(2)

90Int. J. Phys. Sci.0 α i j 1 ,0 x 1 ,whereandalsof i ( x ), i 1, 2 ,., n, are known functions.2 k 1 M 1f ( x) cnmψ n m ( x ) C T ψ ( x ).(6)n 1 m 0This paper is organized as follows: Legendre waveletsmethod is explained, applications of the method forintroduced systems are studied, numerical examples arepresented and conclusions are given, finally.where C and ψ ( x ) are 2k 1C c 10 ,c 11 ,K,c 1M 1 ,c 20 ,c 21 ,K,c 2M 1,K,c 2 k 1 0 ,K,c 2 k 1 ,M 1 LEGENDRE WAVELETS METHODWavelets constitute a family of functions constructed fromdilation and translation of a single function called themother wavelet (Daubeches, 1992; Christensen andChristensen, 2004). When the dilation parameter a andthe translation parameter b , vary continuously thefollowing family of continuous wavelets will appear:M 1 matrices given by:T(7)T c 1 ,c 2 ,K,c M ,c M 1 , K,c 2 k 1 M , andψ ( x ) [ ψ10 (x ),K,ψ1,M 1(x ),ψ 20 (x ) ,K,ψa ,b(t ) a1 2ψ(t b),aa ,b , a 0.(3)Tψ 2,M 1(x ),K,ψ 2 0 (x ),K,ψ 2k 1k 1(x ) ,M 1 (8)T ψ1(x ),K,ψ M (x ),ψ M 1(x ), K,ψ 2k 1 M (x ) . Legendre wavelets are defined on the interval [ 0 ,1 ] asfollows:Also, a functionapproximated by:ψ n m (t ) ψ (t ; k , n , m ) 1 k2n 1nm 2 Pm (2 k t 2n 1), k 1 t k 1 , 222 0,otherwise. where n 1, 2 , ., 2k 1(4)nm, k is any positive integer, m isψ n m ( x ),(5)n 1 m 0where c n m ( f ( x ) ,ψproduct of f ( x ) andnm(9)will be obtain by: cbeHere the entries of the matrix K [ k i j ] k 12 M 2 k 1 Mand t is the normalized time. Pm ( t ) is the famousLegendre polynomial of order m . These polynomials areorthogonal with respect to the weight function w (t ) 1 .The set of Legendre wavelets are an orthonormal set.2A function f ( x ) L [ 0 ,1 ] may be expanded asfollows:f ( x) canf (x , y ) ψ T (x ) K ψ ( y ).((k i , j ψ i ( x ) , f ( x , y ) ,ψ j ( y )the degree of Legendre polynomials, m 0,1,., M 1 f ( x , y ) L 2 [ 0 , 1 ] 2)( x ) , stands for the inneri , j 1 , 2 , . , 2k 1) ),(10)M .The integration of the vector ψ ( x ) , defined in Equation8, can be achieved as the following.x 0ψ(11)( t ) dt P ψ ( x ).k 1where P is the 2M 2 k 1 M operational matrix forintegration (Razzaghi and Yousefi, 2001).The following property of the product of two Legendrewavelet vector functions is well known as:ψ ( x )ψ T ( x ) YY% ψ ( x ).(12)ψ n m ( x ) . Let’s consider truncatedseries in Equation 5, as the following.whereYisagivenvectorandY%isa

Biazar and Ebrahimis2 k 1 M 2 k 1 M matrix. This matrix is called themmFiTψ (x ) X iTjψ (x ) operational matrix of product.x0j 1j 1mmj 1j 1 k 0mmj 1j 1 k 0 Ai jktkk 0( x t )sαi jdttkdtα( x t ) i jSOLUTION OF SYSTEMS OF ABEL VOLTERRAINTEGRAL EQUATIONS X iTjψ (x ) Ai j k Here, two cases of these systems will be studied. X iTjψ (x ) Ai j k Z k i j (x ),Case 1x0s(15)αi 1,2,., n, s 2k 1M ,2k 1M 1,. .Consider the system (Equation 2) with the limits 0 andx for integral signs. To solve this system by Legendrewaveletsmethod,unknownfunctions,u i ( x ), i 1 , 2 , ., n91areconsideredasalinearαwhere Ai j k andZ k i j (x )following formulas:combination of these wavelets as the following.Ai j k (c10 ,K, c n ,2 k 1 M ) u i (x ) C i T ψ (x ), i 1, 2,., n .(13)wherecan be determined by thed k G (C1Tψ (t ),.,C 2Tψ (t ))t 0k!,(16)i 1,2,., n , j 1,2,., m , k 0,1,., s .andTC i c i ,1 ,c i ,2 ,K,c i ,M ,c i ,M 1 , K,ci ,2 k 1 M , i 1,2,., n .αi jZkOther terms also will be considered as the followinggeneral expansions:tk(x ) α dt0( x t ) i jΓ (k 1)Γ (1 α i j ) k α i j 1 x,Γ (k α i j 2)x(17)k 0,1,., s .f i (x ) F iT ψ (x ),F i j ( x , u 1(x ),.,u n (x ) )Now let’s consider:X Ti j ψ (x ),s As) Ai j k t k ,(G i j u 1(t ), u 2 (t ),., u n (t ) (14)k 0i 1, 2,., n , j 1, 2,., m ,s 2 k 1M , 2 k 1 M 1,. .αi jkZ k i j (x ) Y iTj ψ ( x ),k 0i 1, 2 ,., n , j 1, 2,., m .Substitution of Equation 18 into the system (Equation 15),leads to the following system:mk 1where F i are the 2M 1 matrices and X i j arek 1the 2M 1 matrices with the entries which are interms of the components of the vectors C i fori 1 , 2 , ., n , j 1 , 2 , ., m .By substituting Equations 13 and 14 into the system,one gets:(18)Fi T ψ (x ) X iT j Y iTj ψ (x ), i 1, 2,., n , (19)()j 1Multiplyingψ T ( x) ,in both sides of the system(Equation 19) and applying1 0 ( . ) dx , a linear or non-linear system in terms of the elements of

92Int. J. Phys. Sci.whereC i , i 1, 2 , ., n , will be obtained.52914 x 2 16 x 2243 ( x 1 ) 3 xf 1(x ) 2 x 1515440Case 2If in Equation 2, limits of integral are different from 0 andx , then the formula (Equation 17) would not beapplicable, and another approach should be considered.One can write the kernels as:1αijψ T (x ) K i j ψ (t ),( x t )i 1, 2,., n , j 1, 2,., m .where K i , j are 2k 12Mci nψ n (x ) .3(21)n 1If one set x x j , where x j [ 0 , 1] , the error values canbe obtained. Therefore we can check the accuracy of themethod by using error functions.NUMERICAL EXAMPLESTo illustrate the method, some systems are consideredand solved by the proposed method.The19524exactareu (x ) x 2 1,andv (x ) x . This system is of the case 2. Let’s writethe system (Equation 22) as the following.x111 2 ( x 1) 0 x t u (t )dt x 0 3 x t v (t )dt x 1v (t )dt f 1(x ), x 3 0x t x 3 1 1 u (t )dt x 1 u (t )dt 0 4 x t 0 4 x t x 1 (1 x ) 5v (t )dt f 2 (x ), 0x t Consider k 1 , M 8 , andu (x ) C 1T ψ (x ) ,v (x )f 1 (x )TF1 ψ (x ),Tx 2 1 A 11ψ (x ),x3Consider the following system of Abel integral equationson [ 0,1] :3A T21ψ ( x ),1x t14(22)solutions3Example 1x111 2(x 1)u(t)dt xv (t )dt f1(x), 0x3x tx t 1x11 x 3u(t )dt (1 x ) 5v (t )dt f 2(x), 4x0 x tx t356 ( x 1 ) 4 x 3 128 ( x 1 ) 4 xf 2 (x ) 332313M 2 k 1 M complex matrices 232 ( x 1 ) 4 x 4 625x 5 625x 5. 7715961596considered as approximate solutions.Since the truncated Legendre wavelets series areapproximate solutions of system (Equation 2), one hasan error function e ( u i ( x ) ) as follows:e (u i (x ) ) u i (x ) 281 ( x 1 ) 3 x 3 27 (x 1) 3 x 2 3 ( x 1) 3 x, 2208811(20)with the entries according to the Equation 10. Inapplying Legendre wavelets method, obtained solutionswill be complex, because of complex entries of matricesK i j , and the real parts of the solutions can bek 124x tx 0C T2 ψ (x ),Tf 2 (x ) F 2 ψ (x ),xTA 12ψ (x ),1 xψ T (x ) K 1ψ (t ),ψ T (x ) K 2 ψ (t ),1u (t ) dt Y 1T ψ (x ),x tA T22 ψ ( x ),(23)

Biazar and Ebrahimi93Figure 1. The exact and approximate solutions of Example 1 (a1 and b1).1x 0345x tTY 2 ψ ( x ),c 2 ,6 0.0000632693604 0.0001176234874I ,c 2 ,7 0.0001577272089 0.00001908555302 I .Tu (t ) dtY 3 ψ ( x ),v (t ) dtY 4T ψ ( x ).x t1x 0x t1x 0v (t ) dtSince entries of the matrices K 1 and K 2 are complex,the real parts are considered as approximate solutions.u (x ) 0.1059683526 x 7 0.0279973931 x 6 0.4092470195 x 5 0.5281462460 x 4Substituting into the system (Equation 23) and solving theobtained system, the following results will be achieved. 0.2415872910 x 3 0.9532702993 x 2 0.00157711934 x 0.9998653975,c 1 , 0 1 .332139718 0 .0001712234033 I ,c 1 , 1 0 .2881733408 0 .0003360200620 I ,v (x ) 2.096522496 x 7 7.127044981x 6c 1 , 2 0 .07495349421 0 .0003778882289 I , 9.334519464 x 5 5.928096298 x 4c 1 , 3 0 .0004166935511 0 .00007348614476 I , 0.9036584798 x 3 0.2968255414 x 2c 1 , 4 0 .00009264304313 0.0002144572776 I , 0.01824363652 x 0.0002621496533.c 1 , 5 0 .000126694 0664 0 .00009326947553 I ,c 1, 6 0.0001029232530 0.00001070354157 I ,c 1, 7 0.0000079722934141 0.00001920837799 I ,Plots of the exact and approximate solutions are shownin Figure 1, and plots of error functions are shown inFigure 6.c 2 ,0 0.2501175456 0.001356238161 I ,c 2 ,1 0.2595052996 0.0009835260024 I ,c 2 , 2 0.1112783354 0.0002757174589 I ,c 2 ,3 0.01885257238 0.0006609553538 I ,c 2 , 4 0.0003527252537 0.0002674685299 I ,c 2 ,5 0.0002311584955 0.000081849724430 I ,Example 2Consider the following linear system of Abel integralequations, with the exact solutions u ( x) x , andv ( x) x on [ 0,1] (Maleknejad and Salimi, 2008).