FINITE DIFFERENCE APPROXIMATIONS FOR TWO-SIDED IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 Numerical Solutions of Fractional Integral and Fractional Integrodifferential Equations I. I. Gorial Department of Mathematics, College of Education Ibn Al–Haitham, University of Baghdad Abstract In this paper, we introduce and discuss an algorithm for the numerical solution of some kinds of fractional integral and fractional integrodifferential equations. The algorithm for the numerical solution of these equations is based on iterative approach. The stability and convergence of the fractional order numerical method are described. Finally, some numerical examples are provided to show that the numerical method for solving the fractional integral and fractional integrodifferential equations is an effective solution method. Introduction Various fields of science and engineering deal with the dynamical systems, which can be described by fractional-order equations. This topic has received a great deal of attention in the last decade (1, 4, 5). Numerical methods associated with integral order ordinary differential equations were treated extensively in the literature. On the other hand, theoretical studies of the numerical methods and the error estimate of fractional order differential equation are quite limited, because theoretical analysis of fractional–order numerical methods is very difficult(2). In this paper, we find the general solution of fractional integral equations of the form:  0 )( 0     bwwd iv n i i , where v > 0 and n+v > 0 .  ,0 )( 0   iv n i i wd where v > 0. we offer fractional differintegrations calculated by Nishimoto, since this definition enables us to calculate some fractional differintegrations that are easier to calculate than the other definitions. Fractional Integral Equations of Order n+v It's general form is: 0 )( 0     bwwd iv n i i , where v > 0 and n+v > 0 and , k m v   ,, zkm  k≠0 [1] Now, to find ai's that satisfies e az solution az ew  then: 0 0    bad n i k m i i IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 baad k m i n i i  0 km k i n i i baad        0 [2] Equation [2] is an an algebraic equation of order max(m,nk) in the unknown a and by finding its roots ai's we find the general solution of [1]:    ),max( 0 )( nkm i za i ieczw where sc' are arbitrary constants. Example 1: Consider fractional integral equations of the form [1] where n =1, 23v , d1=1, d0 =1 and ,2b eps=0.000006 Program 020 2 1 )23( 0 )(             wwwbwwd n i iv i Now, to find ai's that satisfies az ew  , then: 01 4 1 402 232 1 2 3                aaaaa Use Fixed point at 0 < a < 1 Choose 3 2 4 1 4 1 16 1  aaa 3 1 2 321321 )(),,,( acacccccaf  ),,,( )( ),,,( 321321 cccaf ad d cccaD  5.0a 25.0 1 c 25.0 2 c 063.0 3 c ),,,( 321 cccaD .19493451588085773769 .19493451588085773769 <1 then ),,,( 321 cccaf 0.73100443455321651638 f (0.73100443455321651638, ),, 321 ccc 0.77536871880939399242 f (0.77536871880939399242, ),, 321 ccc 0.78374322291556063677 f (0.78374322291556063677, ),, 321 ccc 0.78531902901931962366 f (0.78531902901931962366, ),, 321 ccc 0.78561536712613996803 f (0.78561536712613996803, ),, 321 ccc 0.78567108872455685163 f (0.78567108872455685163, ),, 321 ccc 0.78568156605114873639 f (0.78568156605114873639, ),, 321 ccc 0.78568353609401988289 IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 f (0.78568353609401988289, ),, 321 ccc 0.78568390651924141194 f (0.78568390651924141194, ),, 321 ccc 0.78568397616992137448 f (0.78568397616992137448, ),, 321 ccc 0.78568398926626797715 f (0.78568398926626797715, ),, 321 ccc 0.78568399172876071348 f (0.78568399172876071348, ),, 321 ccc 0.78568399219178071858 f (0.78568399219178071858, ),, 321 ccc 0.78568399227884189928 f (0.78568399227884189928, ),, 321 ccc 0.78568399229521192455 f (0.78568399229521192455, ),, 321 ccc 0.78568399229828996376 f (0.78568399229828996376, ),, 321 ccc 0.78568399229886872434 f (0.78568399229886872434, ),, 321 ccc 0.78568399229897754811 f (0.78568399229897754811, ),, 321 ccc 0.78568399229899801013 f (0.78568399229899801013, ),, 321 ccc 0.78568399229900185758 Stop condition: |0.78568399229900185758 - 0.78568399229899801013|=3.886E-15 < eps. Then 0.78568399229900185758 is approximate 1 st root So, and in the same way, we find other roots for the equation and put it in: zazaza ecececzw 321 321 )(  Where sc' are arbitrary costants and , i a 3,...,1i is the i-th approximate root of the equation. Fractional Integral Equations of Order nv It's general form is: ,0 )( 0   iv n i i wd dn ≠ 0 and k m v  > 0, ,, zkm  k≠0 [3] Now, to find ai’s satisfies the solution az ew  then: ,0 0   n i azvi i ead since e az ≠ 0, then 0 0   n i vi i ad Rewriting this equation in the form:   01 1 1 ... ddadada n n n n v      01 1 1 ... ddadada n n n n k m        kmkn n n n dadadad 01 1 1 ...    [4] Which is an algebraic equation of order max(m,nk) in the unknown a. Finding its roots ai's w solution of [3] in the form:    ),max( 0 )( nkm i za i ieczw IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 Where sc' e find, are arbitrary constants. Notice. We use numerical method to solve equations [2] and [4] since we can not obtain theoretical solution always because general roots don't exist to solve equation of order more than the one which equal 5. As a special case, consider the fractional integral equations of order 2v. ,0 )()2(  wbww vv  v > 0, , k m v   ,, zkm  k≠0 Example 2: Consider fractional integral equations of the form [3] where n =4, 21v , d0= -1, d1 = d2 = d3 = d4= 1 and eps=0.00005. Program 00 2 1 .4 2 1 .3 2 1 .2 2 1 .1 2 1 .0 0 )(                                       wwwwwwd n i vi i Now, to find ai's that satisfies az ew  , then:     013301 3691222 1 .3 12 1                  aaaaaaaa Use a fixed point at 1 < a < 2 Choose  12 1 369 133  aaaa 12 1 9 10 8 9 7 8 6 7 5 6 4 5 3 4 2 32110987654321 )(),,,,,,,,,,( acacacacacacacacaccccccccccccaf  ),,,,,,,,,,( )( ),,,,,,,,,( 1098765432110987654321 ccccccccccaf ad d ccccccccccaD  5.1a 1 1 c 0 2 c 0 3 c 1 4 c 0 5 c 0 6 c 3 7 c 0 8 c 0 9 c 3 10 c ),,,,,,,,,( 10987654321 ccccccccccaD 0.68025578889276714799 0.68025578889276714799 <1 then ),,,,,,,,,,( 10987654321 ccccccccccaf 1.5253332893390171162 f (1.5253332893390171162, ),,,,,,,,, 10987654321 cccccccccc 1.5425681077739530993 f (1.5425681077739530993, ),,,,,,,,, 10987654321 cccccccccc 1.5542943084644395413 f (1.5542943084644395413, ),,,,,,,,, 10987654321 cccccccccc 1.5622724402627148754 f (1.5622724402627148754, ),,,,,,,,, 10987654321 cccccccccc 1.5677002760803632163 f (1.5677002760803632163, ),,,,,,,,, 10987654321 cccccccccc 1.5713928861397996957 f (1.5713928861397996957, ),,,,,,,,, 10987654321 cccccccccc 1.5739049148507729346 f (1.5739049148507729346, ),,,,,,,,, 10987654321 cccccccccc 1.5756137644497782316 f (1.5756137644497782316, ),,,,,,,,, 10987654321 cccccccccc 1.5767762147279333541 f (1.5767762147279333541, ),,,,,,,,, 10987654321 cccccccccc 1.5775669638893329789 f (1.5775669638893329789, ),,,,,,,,, 10987654321 cccccccccc 1.5781048604366752736 f (1.5781048604366752736, ),,,,,,,,, 10987654321 cccccccccc 1.5784707548380016783 f (1.5784707548380016783, ),,,,,,,,, 10987654321 cccccccccc 1.5787196467007653374 IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 f (1.5787196467007653374, ),,,,,,,,, 10987654321 cccccccccc 1.5788889495273038146 f (1.5788889495273038146, ),,,,,,,,, 10987654321 cccccccccc 1.5790041135285882146 f (1.5790041135285882146, ),,,,,,,,, 10987654321 cccccccccc 1.5790824508319440980 f (1.5790824508319440980, ),,,,,,,,, 10987654321 cccccccccc 1.5791357376804271389 f (1.5791357376804271389, ),,,,,,,,, 10987654321 cccccccccc 1.5791719846033924980 f (1.5791719846033924980, ),,,,,,,,, 10987654321 cccccccccc 1.5791966405705758888 f (1.5791966405705758888, ),,,,,,,,, 10987654321 cccccccccc 1.5792134121050226560 f (1.5792134121050226560, ),,,,,,,,, 10987654321 cccccccccc 1.5792248204712984177 f (1.5792248204712984177, ),,,,,,,,, 10987654321 cccccccccc 1.5792325806915453767 f (1.5792325806915453767, ),,,,,,,,, 10987654321 cccccccccc 1.5792378593625278867 f (1.5792378593625278867, ),,,,,,,,, 10987654321 cccccccccc 1.5792414500293495853 f (1.5792414500293495853, ),,,,,,,,, 10987654321 cccccccccc 1.5792438924789067742 f (1.5792438924789067742, ),,,,,,,,, 10987654321 cccccccccc 1.5792455538860883029 f (1.5792455538860883029, ),,,,,,,,, 10987654321 cccccccccc 1.5792466840112748059 Stop condition: |1.5792466840112748059- 1.5792455538860883029| = 1.13E-6 < eps. Then 1.5792466840112748059 is approximate 1 st root As in the same way, we find the remaining roots for the equation and put them in: zazazazazazazazazazazaza ececececececececececececzw 121110987654321 121110987654321 )(  Where sc' are arbitrary costants and , i a 12,...,1i is the i-th approximate root of the equation. Example 3: Consider fractional integral equations of the form ,0 )()2(  wbww vv  where 23v , b= -1, 1 and eps = 0.00001 Program 00 2 3 2 3 .2 )()2(                wwwwbww vv  Now, to find ai's that satisfies az ew  , then: 01301 362 3 3          aaaa Use a fixed point at 0 < a < 1 Choose 3 6 3 1  a a 3 1 6 7 5 6 4 5 3 4 2 3217654321 )(),,,,,,,( acacacacacacccccccccaf  ),,,,,,,( )( ),,,,,,( 765432176,54321 cccccccaf ad d cccccccaD  5.0a 333.0 1 c 0 2 c 0 3 c 0 4 c 0 5 c 0 6 c 333.0 7 c IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 ),,,,,,( 76,54321 cccccccaD 4.2889469781380487238·10 -2 4.2889469781380487238·10 -2 <1 then ),,,,,,,( 7654321 cccccccaf 0.59674159596366847153 f (0.59674159596366847153, ),,,,,, 7654321 ccccccc 0.70364454056999046660 f (0.70364454056999046660, ),,,,,, 7654321 ccccccc 0.72034885326896709841 f (0.72034885326896709841, ),,,,,, 7654321 ccccccc 0.72425624597214515841 f (0.72425624597214515841, ),,,,,, 7654321 ccccccc 0.72523130986763758800 f (0.72523130986763758800, ),,,,,, 7654321 ccccccc 0.72547834796395703979 f (0.72547834796395703979, ),,,,,, 7654321 ccccccc 0.72554117386534124342 f (0.72554117386534124342, ),,,,,, 7654321 ccccccc 0.72555716687026665442 f (0.72555716687026665442, ),,,,,, 7654321 ccccccc 0.72556123905387340202 f (0.72556123905387340202, ),,,,,, 7654321 ccccccc 0.72556227598903008752 f (0.72556227598903008752, ),,,,,, 7654321 ccccccc 0.72556254003692763325 f (0.72556254003692763325, ),,,,,, 7654321 ccccccc 0.72556260727504995250 f (0.72556260727504995250, ),,,,,, 7654321 ccccccc 0.72556262439682887969 f (0.72556262439682887969, ),,,,,, 7654321 ccccccc 0.72556262875678689238 f (0.72556262875678689238, ),,,,,, 7654321 ccccccc 0.72556262986702404804 f (0.72556262986702404804, ),,,,,, 7654321 ccccccc 0.72556263014973928500 f (0.72556263014973928500, ),,,,,, 7654321 ccccccc 0.72556263022173102582 Stop condition: |0.72556263022173102582-0.72556263014973928500|= 7.199E-11 < eps Then 0.72556263022173102582 is approximate 1 st root So, and in the same way, we find the remaining roots for the equation and put them in: zazazazazaza ececececececzw 654321 654321 )(  Where sc' are arbitrary costants and , i a 6,...,1i is the i-th approximate root of the equation. 4. Fractional Integro-Differential Equations We study these of the form: 0 )( 0     bwwd iv n i i , , k m v   ,, zkm  k≠0 and n-1 < |v| < n [5] To solve such kind, suppose az ew  , then: bad n i iv i    0 and as in section 2,we get the solution. IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 Notice. Special kinds of fractional differintegral equations, of variable coefficient are solved theoretically by Nishimoto (3). 5. Existence and Uniqueness To show the existence of a fixed point of fractional integral and fractional integrodifferential equations we give the following theorem. Theorem. Let g(x) be a function in interval I= [a,b] for all Ix  such that continuous, differentiable and there exists a constant  , 0 ≤ λ<1 such that  )(xg Ix  .Then g has exactly a fixed point I and if Ix  0 , then the sequence defined from ),( 1 nn xgx   ,...1,0n .Converges to  . Proof. To show the existence of a fixed point . suppose ,)( aag  ,)( bbg  then a 0 Then, there exist ),( ba such that ,0)( h then )(xg by mean value theorem. To show ),( 1 nn xgx   let Ix  0 converges to  Let nn xe  (error) )()( 1  nn xgge  by mean value theorem between , 1n x ))(()()( 11   nn xcgxgg  Then 1 )(   nn ecge 1 )(   nn ecge 21   nnn eee  Then the sequence   ,0 n e then    n x . To show uniqueness of  let θ be another fixed point of g and θ I . let θ = x0, x1= g(θ)= θ , 0  e   11 xe Then 10 ee  but , 01 ee  then 1 e < 0 e which is a contradiction, then ,0 10  ee   . 6. Rate of convergence of iterative method Give any iterative method its order which is said to be equals p if p nn ece  1 for some number c depending on f where , nn xe  that is p nn ee  1 if, 1p is linear as p increases the method converges faster. Since ),(xgx  ,)(  xg 0 <1 nn ee  1 Then general iterative method is linear. Finally, the method converge if 0 <1 and c must be less then 1. 7. Stability The issue of stability is very important when implementing the method on a computer in finite- precision arithmetic because we must take into account the effects introduced by rounding errors. It is known (6) that the classical iterative method is a reasonable and practically useful compromise in the sense that its stability properties allow for a save application to mildly stiff IBN AL- HAITHAM J. FOR PURE & APPL. SCI VOL.22 (2) 2009 equations without undue propagation of rounding errors, whereas the implementation does not require extremely time consuming elements. From the results of (7), we can see that these properties remain unchanged when we look at the fractional version of the algorithm instead of the classical one, and therefore it is also clear that the behaviour does not depend on the order of the differential operators involved. References 1.Podlubuy, I., (1999) “Fractional Differential Equation”, Academic Press, San Diego , CA, USA. 2. Liu, F.;Anh, V. and Turner, I., (2004), J. comp. and Appl. Math. 166, 209-219 in press. 3. Yuan, L. and Agrawal, O.P., (1998), Journal of Vibration and Acoustics 124, 321-324. 4. Taha, B., (2004), M.Sc. Thesis, University of Technology. 5. Naji , S., (2004), Ph.D. Thesis, Ibn Al-Haitham, Baghdad University. 6. Hairer , E. and Wanner , G., (1991) “Solving Ordinary Differential EquationsII:Stiff and Differential –Algebraic Problems”, Springer, Berlin. 7. Lubich , C., (1986), SIAM.J. Math. Anal.17 ,704-719. 2002( 2) 22مجلة ابن الهيثم للعلوم الصرفة والتطبيقية المجلد الحلول العددية للمعادالت التكاملية الكسرية والتفاضلية التكاملية الكسرية كولاير ايمان ايشو ادجامعة بغد-ابن الهيثم-كلية التربية- قسم الرياضيات ةصالخال فييه اييلب ب قدييا وييقشنا خناوزيينا لخب اشعييي بدييا ب عييققع ييقعل بنييخب ب شعيياقال ب ضلاشبعييي ب لةيي عي خب ض ا ييبعي ب ضلاشبعييي ب لةيي عيو خبخ لخب اشعيييي ب ديييا ب عييققع ضبيييئ ب شعييياقال واقشيييي تبيير بةيييان ب ضايييا ق ب ضلييي ب عو لشييا ناوزييينا باةيييضا ب عي خب ضايييا ق وب ضقيب لةخ عي ع بط عاي ب عققعي ل ق ب عققعيي ديا ب شعياقال ب ضلاشبعيي ب لةي عي خب ض ا يبعي ب ضلاشبعيي بقيبلع ب وقشنا قعيل باشلبيي ب عققعيي ب ضيه ضلقيل بخ ب ط دا شؤل فعاا شاقخا و عق لبقب لة عي اه ط