A Collocation Method for Solving Abel s Integral Equations of First and Second Kinds

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1 A Collocato Method for Solvg Abel s Itegral Equatos of Frst ad Secod Kds Abbas Saadatmad a ad Mehd Dehgha b a Departmet of Mathematcs, Uversty of Kasha, Kasha, Ira b Departmet of Appled Mathematcs, Faculty of Mathematcs ad Computer Scece, Amrkabr Uversty of Techology, No. 424, Hafez Ave., Tehra, Ira Reprt requests to M. D.; E-mal: mdehgha@aut.ac.r Z. Naturforsch. 63a, (28); receved Jue 23, 28 A umercal techque s developed for solvg Abel s tegral equatos. The solutos of such equatos may exhbt a sgular behavour the eghbourhood of the tal pot of the terval of tegrato. The proposed method s based o the shfted Legedre collocato techque. Illustratve examples are cluded to demostrate the valdty ad applcablty of the preseted techque. Key words: Volterra Itegral Equato; Abel s Itegral Equatos; Shfted Legedre Polyomals; Collocato Method. 1. Itroducto I recet years, may dfferet methods have bee used to approxmate the soluto of Volterra tegral equatos wth weakly sgular kerels (see, for example, [1 3]). I the preset paper, we cosder the followg Volterra tegral equatos of the frst ad secod kds, respectvely: λ dt = f (x), t x 1, (1) y(x)+λ dt = f (x), t x 1, (2) where f (x) s L 2 (R) o the terval x 1ad < α < 1. Here λ,α ad the fucto f (x) are gve, ad y(x) s the soluto to be determed. For < α < 1 the tegral equatos (1) ad (2) are weakly sgular ad called Abel s tegral equatos of the frst ad secod kds, respectvely. The specal case α = 1/2ofte arses physcal problems. We assume that (1) ad (2) have a uque soluto. I 1823, Abel, whe geeralzg the tautochroe problem, derved (1). Ths equato s a partcular case of a lear Volterra tegral equato of the frst kd. Abel s tegral equatos frequetly appear may physcal ad egeerg problems, e. g., semcoductors, scatterg theory, sesmology, heat coducto, metallurgy, flud flow, chemcal reactos ad populato dyamcs [4]. May dfferet authors preseted umercal solutos for Abel s tegral equato of frst kd (see, for example, [5 7] ad the refereces there). I [6] the authors developed hghaccuracy mechacal quadrature methods ad, to avod the ll-posedess of the problem, the frst kd Abel tegral equato was trasformed to the secod kd Volterra tegral equato wth a cotuous kerel ad a smooth rght-had sde term expressed by weakly sgular tegrals. Also the author of [7] developed a umercal techque based o Legedre wavelet approxmatos for solvg (1) ad (2). The umercal treatmet s more dffcult for frst kd tha for secod kd Abel tegral equatos, whch have bee wdely studed [8 11]. I the preset paper, we apply the shfted Legedre collocato method for solvg Abel s tegral equatos. Our method cossts of reducg Abel s tegral equato to a set of lear algebrac equatos by expadg the approxmate soluto as shfted Legedre polyomals wth ukow coeffcets. The propertes of shfted Legedre polyomals are the utlzed to evaluate the ukow coeffcets. The paper s orgazed as follows: I Secto 2 we descrbe the basc formulato of Legedre ad shfted Legedre polyomals requred for our subsequet developmet. I Secto 3 the applcato of the shfted Legedre collocato method to the soluto of (1) ad (2) s summarzed. As a result a set of algebrac equatos s formed ad a soluto of the cosd / 8 / $ 6. c 28 Verlag der Zetschrft für Naturforschug, Tübge

2 A. Saadatmad ad M. Dehgha Abel s Itegral Equatos 753 ered problem s troduced. I Secto 4 the proposed method s appled to umercal examples ad the accuracy of our method usg several examples s checked. Secto 5 eds wth a cocluso. 2. Shfted Legedre Polyomals The well kow Legedre polyomals are defed o the terval z [ 1,1] ad ca be determed wth the help of the followg recurrece formulae [12]: L (z)=1, L 1 (z)=z, L +1 (z)= zl (z) + 1 L 1(z), = 1,2,... I order to use these polyomals o the terval x [,1] we defe the so-called shfted Legedre polyomals by troducg the chage of varable z = 2x 1. Let the shfted Legedre polyomals L (2x 1) be deoted P (x). TheP (x) ca be obtaed as follows: P (x)=1, P 1 (x)=2x 1, (2 + 1)(2x 1) P +1 (x)= P (x) ( + 1) + 1 P 1(x), = 1,2,... (3) The aalytcal form of the shfted Legedre polyomal P (x) of degree s gve by P (x)= k= x k +k ( + k)! ( 1) ( k)! (k!) 2. (4) Note that P ()=( 1) ad P (1)=1. A fucto u(x), o [,1], may be approxmated the form of a seres wth + 1 terms as u(x)= = c P (x), where the coeffcets c ( =,...,) are costats. 3. Soluto of the Sgular Volterra Itegral Equato I ths secto we solve the sgular Volterra tegral (1) ad (2) by usg the shfted Legedre collocato method. Frst of all we approxmate y(x) as y (x)=a x α + = c P (x), (5) where A ad the coeffcets c ( =,...,) are ukow. Substtutg (5) to (1) we have λ A x t α dt + λ = Now, we kow that ad c x P (t) dt = f (x). (6) t Γ ( + 1)Γ (1 α) dt = x +1 α (7) Γ ( + 2 α) t α πα dt = x. (8) s(πα) Employg (4) ad (7) we obta where P (t) dt = k= a (α) k xk+1 α, (9) a (α) +k ( + k)!γ (k + 1)Γ (1 α) k =( 1) (k!) 2 ( k)!γ (k + 2 α) =( 1) +k ( + k)!γ (1 α) k!( k)!γ (k + 2 α). By usg (8) ad (9), (6) ca be wrtte as (1) πα λ A s(πα) x+λ c a (α) k xk+1 α = f (x). (11) = k= Smlarly by substtutg (5) to (2) ad by usg (8) ad (9) we get ) A (x α πα + λ s(πα) x + c P (x) = (12) + λ c a (α) k xk+1 α = f (x). = k= To fd the soluto of the frst kd Abel tegral equato (1) or the secod kd Abel tegral equato (2) we collocate (11) or (12) at (+2) pots, respectvely. For sutable collocato pots we use the shfted Legedre roots z ( = 1,..., + 1) of P +1 (t) ad addtoal pot z = 1. The resultg equato geerates a set of (N + 2) lear algebrac equatos whch ca be solved for the ukow coeffcets c j ( j =,...,)

3 754 A. Saadatmad ad M. Dehgha Abel s Itegral Equatos ad A. Cosequetly y(x) gve (5) ca be calculated. 4. Numercal Expermets Ths secto s devoted to computatoal results. We apply the method preseted ths paper ad solve several examples. Those examples are chose whose exact solutos exst. Test 1. Cosder the frst kd Abel tegral equato [7, 13] dt = 2 x(15 56x x 3 ), (13) 15 whch has the exact soluto y(x) =x 3 x For ths problem we use (11) wth α = 1/2ad = 3. We obta A =, c = 11 12, c 1 = 1 2, c 2 = 1 12, c 3 = 1 2. Therefore usg (5) we have y 3 (x)= P (x) 1 2 P 1(x) P 2(x)+ 1 2 P 3(x) = (1) 1 1 (2x 1) (6x2 6x + 1) (2x3 3x x 1) = x 3 x 2 + 1, whch s the exact soluto. Test 2. I the secod example, we solve the secod kd Abel tegral equato [7, 13] y(x)=x x5/2 dt. (14) For ths problem we use (12) wth α = 1/2ad = 2. We obta A =, c = 1 3, c 1 = 1 2, c 2 = 1 6. Therefore usg (5) we have y 2 (x)= 1 3 P (x)+ 1 2 P 1(x)+ 1 6 P 2(x) = 1 3 (1) 1 2 (2x 1)+1 6 (6x2 6x + 1) = x 2, whch s the exact soluto. Tab. 1. Computatoal results of the absolute error y(x) y (x) of Test 1. x = 3 = 5 = 7 = 9 = Test 3. I ths example we apply the ew method to fd the soluto of the sgular Volterra tegral equato [7] y(x)=2 x dt. (15) The exact soluto of ths problem s y(x) =1 e πx erfc( πx), where erfc(x) s the complemetary error fucto defed by erfc(x)= 2 x x e u2 du. I Table 1 we preset the absolute error y(x) y (x) for some values of x usg the preset method wth = 3,5,7,9,11. From Table 1 we see that the approxmate soluto computed by the preset method coverges to the exact soluto. I Fg. 1, the absolute error fucto y(x) y (x) s plotted for = 7 ad = 1. Test 4. Cosder the lear Volterra tegral equato wth algebrac sgularty, preseted [13 15], y(x)= 1 2 πx + x dt, (16) wth the exact soluto y(x) = x. For ths example we use (12) wth α = 1/2ad = 1. We have A = 1, c =, c 1 =. Therefore usg (5) we get the exact soluto of ths example.

4 A. Saadatmad ad M. Dehgha Abel s Itegral Equatos 755 Fg. 1. Plot of the absolute error of Test 3 for = 7 (top) ad = 1 (bottom). Test 5. I ths example we cosder the secod kd Abel tegral equato wth α = 1/3, preseted [2], y(x) 1 dt 1/3 1 () (17) = x 2 (1 x) x14/ x11/ x8/3. For ths example usg (12) wth α = 1/3 ad = 4, we obta y(x)=x 4 2x 3 + x 2, whch s the exact soluto of the problem. Test 6. Cosder the Volterra tegral equato [16, 17] y(x)= 1 + π x x 4 s x 1 dt 4 (18)

5 756 A. Saadatmad ad M. Dehgha Abel s Itegral Equatos Table 2. The error y y s for s = 1,2 ad some values of of Test 6. y y 1 y y wth the exact soluto y(x)=1/ x + 1. I Table 2 the error y y s s llustrated for s = 1,2 ad x 1 ad some values of. I ths example we acheved a very good approxmato wth the exact soluto of the equato by usg oly a few terms of shfted Legedre polyomals. The umercal results obtaed ths secto demostrate that the preset method s capable of solvg Abel s tegral equatos (1) ad (2) ad ca be cosdered as a effcet method. Note that we have computed the umercal results by Maple programmg. Iterestg applcatos of some tegral equatos are gve [18 2]. 5. Cocluso We preseted a umercal scheme for solvg Abel s tegral equatos of the frst ad secod kds. Our method cossts of reducg Abel s tegral equatos to a set of lear algebrac equatos by expadg the approxmate soluto as shfted Legedre polyomals wth ukow coeffcets. The obtaed results showed that ths approach ca solve the problem effectvely, ad t eeds less CPU tme. The ew descrbed techque produces very accurate results eve whe employg a small umber of collocato pots. Ackowledgemet A. S. would lke to thak the Uversty of Kasha for support of ths research. [1] T. Dogo, N. J. Ford, P. Lma, ad S. Valtchev, J. Comput. Appl. Math. 189, 412 (26). [2] K. Malekejad ad N. Aghazadeh, Appl. Math. Comput. 161, 915 (25). [3] T. Dogo, S. McKee, ad T. Tag, Proc. R. Soc. Edburgh A 124, 199 (1994). [4] R. Goreflo ad S. Vessella, Abel Itegral Equatos, Aalyss ad Applcatos. Lecture Notes Mathematcs, Vol. 1461, Sprger, Hedelberg [5] R. F. Camero ad S. McKee, IMA J. Numer. Aal. 5, 339 (1985). [6] Y. P. Lu ad L. Tao, J. Comput. Appl. Math. 21, 3 (27). [7] S. A. Yousef, Appl. Math. Comput. 175, 57 (26). [8] S. Abelma ad D. Eyre, J. Comput. Appl. Math. 34, 281 (1991). [9] H. Bruer, M. R. Crsc, E. Russo, ad A. Veccho, J. Comput. Appl. Math. 34, 211 (1991). [1] P. Baratella ad A. P. Ors, J. Comput. Appl. Math. 163, 41 (24). [11] T. Lu ad Y. Huag, J. Math. Aal. Appl. 282, 56 (23). [12] C. Cauto, M. Y. Hussa, A. Quartero, ad T. A. Zhag, Spectral Methods Flud Dyamcs, Pretce- Hall, Eglewood Clffs, NJ [13] A. M. Wazwaz, A Frst Course Itegral Equatos, World Scetfc Publshg Compay, Sgapore [14] L. Tao ad H. Yog, J. Math. Aal. Appl. 324, 225 (26). [15] Q. Hu, SIAM J. Numer. Aal. 34, 1698 (1997). [16] P. Lz, SIAM J. Numer. Aal. 6, 365 (1969). [17] E. A. Galper, E. J. Kasa, A. Makroglou, ad S. A. Nelso, J. Comput. Appl. Math. 115, 193 (2). [18] M. Dehgha, It. J. Comput. Math. 83, 123 (26). [19] M. Dehgha ad A. Saadatmad, It. J. Comput. Math. 85, 123 (28). [2] M. Shakourfar ad M. Dehgha, Computg 82, 241 (28).