SIMULATING THE SOLUTION OF THE DISTRIBUTED ORDER FRACTIONAL DIFFERENTIAL EQUATIONS BY BLOCK-PULSE WAVELETS

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1 U.P.. S. ull., Seres A, Vol. 79, ss., 7 SSN 3-77 SMULANG HE SOLUON OF HE SRUE ORER FRACONAL FFERENAL EQUAONS Y LOCK-PULSE WAVELES M. MASHOOF, A. H. REFAH SHEKHAN s pper, we roue eos se o operol rx o rol orer ero ro xe po (l vlue po) or srue orer rol erel equos (FE). We use lo-pulse wveles r uos rx o rol orer ero ro rrr l po, were rol ervve s ee e Cpuo or. e use o s eo we rsle (FE) o ler ler equos w e solve e. e propose eo s ee ese soe uerl exples. Kewors: srue orer rol erel equo, wvele, lo pulse, r uo, operol res.. rouo e sor o rol lulus s ore ree eures ol; owever, ol e ls wo ees e el s reeve prl eo eres; see [], [], [3] [4] or ore els o s rer. Frol lulus s e eerlzo o lulus, w e orer o ervves erls e rrr uers. e srue-orer operors e oe we we ere e rol-orer lulus operors w respe o e orer vrle. e rs e o srue orer erel equo ws se Cpuo 969 ler evelope [5] [6]. ese srue-orer erel equos were l ore osuve equos o eler e [7], uso equos [8] e ulesol ro wl oels [9]. e erese reers reer o [], [], [], [3], [4] [5] or ore els. Here s pper, we oser e srue orer ler equos o e or = (), uer l oos,, () epre o Apple Mes, Ful o Mel Sees, Lj r, sl Az Uvers, Lj, r, e-l: soo.o@oo.o epre o Apple Mes, Ful o Mel Sees, Lj r, sl Az Uvers, Lj, r e-l (Correspo uor): _re@lu..r

2 94 M. Msoo, A. H. Re Se were s e,,,,..., rol orer ervve o () Cpuo sese ro. A ree evelope o pproxo eor s pproxo o rrr uo wvele polols. ere re ere pes o wvele su s lo-pulse wvele, Hr wvele, Mex-H wvele, So wvele, uees wvele, Meer s wvele, so or. s pper, we l ous o pproxo lo-pulse wvele r uos o se o lo-pulse wvele Se Leere polols. A e uo e sesze opleel o olerle eree o ur us se o orool uos. For su ure represeo o e uo, e orool se soul e oplee [6]. s pper, we wll ppl lopulse Hr uos se o lo-pulse wvele Se Leere polols o pproxe e soluo o () uer oos (). seo we prese uer o eos ou rol lulus, srue orer ervve, lo-pulse wveles, r uos s properes. seo 4 we wll roue uerl eo se o lo-pulse r operol rx, seo 5 we wll suss e overee o e esre eo. A e e, we wll prese soe uerl exples.. Prelres s seo, we prese soe s eos properes o rol lulus, srue orer ervve wveles [6]. eo.. A rel uo x, x s s o e spe C, p R ere exss rel uer p, su x x x were x,, s s o e e spe C C, N. eo.. e Re-Louvlle rol erl o orer ro w respe o s,. eo.3. e rol ervve o Sese ro s ee s es o Cpuo,, N,, C.

3 Sul e soluo o e srue orer rol equos lo-pulse wveles 95 e relo ewee e Re-Louvlle erl Cpuo ervve operor s ve e ollow expressos s [7] [8]:, ( ) ( ), (3)! eo.4. e rol ervve o srue orer e Cpuo ro o w sese w respe o orer-es uo s s. Rer.5. We see we, s r el uo, e rol ervve o srue orer-es uo rol ervve o orer re e se. eo.6. e -se o lo-pulse uos or,,,...,, o,, s ee s,, oerwse. e sow e uos re sjo orool [6]. L, e pproxe e eore.7. A uo lo-pulse uo s F, were F,. Proo: [6]. Rer.8. Fro ove eore we ve, x, s sows we pproxe F ; e e uo F s les oe zero e,. Now we ee e r uos o lo-pulse se Leere polols. Frsl, we rell e se Leere polols. eo.9. e se Leere polols re ee o e ervl, e eere w e o e ollow reurree orul P P P,,..., were P P.

4 96 M. Msoo, A. H. Re Se eo.. Hr uos o lo-pulse se Leere polols j,,,,..., j,,,..., re ee o, s Pj ;, j were P j s ; oerwse, polols o,. e j se Leere Now, or pprox e uo we se H H H j j j,, were H,,,, j j, were u v j, u v j,. Now we roue e operol rx eos se o lo-pulse uos. Frol ero ro o e lo-pulse uo veor s ve s F, were o e rol orer ero [8] F... F s e lo-pulse operol rx 3, Now, le H, H, Q H,. (4) ; e we osru operol rx or Hr uos sq F. e ollow les, we prese operol rx o rol orer ero ro rrr or lo-pulse wveles se Leere r uos. Le.. e operol res o e rol orer ero ro or o re ve s F F j,, were q q j j,, oerwse.,,

5 Sul e soluo o e srue orer rol equos lo-pulse wveles 97 Proo: F q F Rer.. s ler, j e F F. Le.3. e operol res o e rol orer ero ro or se Leere r uos veor H, o re ve s Q. F Proo: Le H Q H, so H Q,, H, oreover; F Q. F j,.,,, s Q F e ollow seo, we wll oser lo-pulse wveles se Leere r uos or solv e srue orer ler equos s () uer l oos (). e ove esrpos, we pproxe uo L, s 3. Nuerl eos, or w W. (5) For e se o e sulo o soluo o srue orer rol equo () uer oo (), rs, we oser e ero oruls o Newo Coes or e erl er e srue orer equo [9], ex we use lo-pulse uos esre e prevous seo or pproxo o rol ervve us rol operol res. e we wll ve sse o ler equos w e solve exse eos; or exple, we use oe erve eos [] or s er. Le we oser e ollow orul or pproxo e erl er e srue orer equo

6 98 M. Msoo, A. H. Re Se R R, )6(. ppl (6) o () we wll ve, us, ) (, so we ve ) (, ro equo (3) we ve ). (! Now us equo (5) we ve W w W G w W G w su, (7) were W w, W w, W s lo-pulse or r uos o lo-pulse se Leere veor uos, w s s orrespo veor oees, respevel, so W w su!, w, equo (7) ples su w G w G w. (8) Rer 3.. Noe, s poe ou [], e vewe s e l se o, were ver lre uer o ers re osere. O e oer, ro ove relos we ve

7 Sul e soluo o e srue orer rol equos lo-pulse wveles 99!, ereore; e r se o e ls relo es o w s equl o, s. we reple e pproxo w equl e equo (8), we wll ve ler ler equo w s solvle. solv, we w W. w e sule s 4. Coveree lss s seo we w o vese e overee o e eo C,,. e e spe o ll esre e prevous seo. Le ouous uos w or x. eore 4.. Le e rrr rel oue uo, w s squre erle e ervl,, e F ; e e. Proo: [6]. Le L, e e spe o squre erle uos o,, X Spj :,..., j,...,. s ler j s os uo o eree. Now, le L,. Se X s e esol veor spe, s e uque es pproxo ou o X su s p X ; s p Xq X : p q, were,. ereore, ere exss e uque oees su p j j j, were j re ee eo., or ore els reer o []. L, e es ouousl erele, eore 4.. Le M o,. H H ou o X, e we ve j s e es pproxo o j j, 3 M H H,. 3!

8 M. Msoo, A. H. Re Se Proo: Le e e lor polol o orer - or o ) (,,ereore;!, oreover; or e ere exss ) (, su!. Se H H, s e es pproxo o ou o X, X, ro e ls equl we ve.! 3! 3, M H H Now, we w o sow e overee o e lo-pulse wveles eo or (FE). eore 4.3. we use, Newo Coes or e erl er e operol rx o e lo-pulse uos or rol er equo (), e we overee o e ex soluo o (FE). Proo: Le o,. Now, we sow E, es o zero we,. ero o orer ro o E we ve E, we, we ve

9 Sul e soluo o e srue orer rol equos lo-pulse wveles E us, E ; ereore, ro eore 4. e ove reerre equl we see E, so E, e ls equl sows we e E ; s es e eo esre seo 3 s overe. A slr eore e oe ro eore 4.3 we we use e operol rx o e r o lo-pulse se Leere uos. eore 4.4. we use Newo Coes or e erl er e operol rx o e r uos o lo-pulse se Leere uos or rol er equo (), e we ve overee o e ex soluo o (FE). Proo: e proo s slr o e proo o eore llusrve Exples orer o sow e ee o e eos esre seo 3 sule e ex soluo o srue orer equos, we oser soe exples er ex soluos re ow. 5.. Exple. Coser..,, l e ex soluo o s exple s. s er we use rpezol rule [9] or erl er w., 3 64 or pprox. Now ro equo (8) we e sule 3, 64. Fure sows e uerl resuls eere lo-pulse

10 M. Msoo, A. H. Re Se veor uos 3 64 or e exple 5., ro we wll see we reses ro 3 o 64; e uerl soluo es o ex soluo. le sows e solue error soe pos F.. Nuerl ex soluos o exple 5. lo-pulse uos. F.. Nuerl ex soluos o exple 5. r uos H 8, 3.

11 Sul e soluo o e srue orer rol equos lo-pulse wveles Exple. Now oser 6 ' 5. e ex soluo s 5 l 3,,. Slr o exple 5., s exple we use rpezol rule [9] or erl er w. 3, 64 H, 3. ure we prese e uerl ex soluos eere r uos. For ure we see e uerl soluo eere r uos re so lose o e ex soluo. le, we opre solue error o soluos eere 3, 64 8 or pprox H 8, 3 soe pos. Fro le we see e errors o r uo re less lo-pulse uos. Mel pos eleel lses le e e e e-5 Mel pos eleel lses H ,3 le e e e e-5.98 e-4.4 e e e-3.7 e e e e e e-4.9 e e e-3 8. e e e e e e e-4.3 e-3.6 e-3.6 e-3

12 4 M. Msoo, A. H. Re Se F. 3. Error o lo-pulse pproxos eere = 3, 64 or exples 5., 5.. Also, ure 3 we prese e error or exples 5., 5. eere = 3, 64. Fro ure 3 we see we we oule, e uer o zeros o re oule e ervl,. s e ws suppore rer.8. Noe e exple we rsle FE o ler ler equos su s Ax e solve ese equos. 6. Colusos e rol erel equos pl por role pss, el x olol sses. e srue-orer operors e oe we we ere e rol-orer lulus operors w respe o e orer vrle. e uel ol o s wor s ee o ppl lopulse se Leere r uos operol rx eo o sule e soluo o FE w l oos. s eo rsles FE o ler ler equo w ws presee seo 3 e overee o s eo ws eosre seo 4, lso ro seo 4 we sw we reses, we o oo sulo o soluo o FE w l oos. Moreover, wo uerl exples were ve o ver e

13 Sul e soluo o e srue orer rol equos lo-pulse wveles 5 eeveess o e propose sees o sulo o soluos. Alou e propose uerl lors re que eeve se o eers erel equos w soo soluos, oe s o urer vese ow pulse wveles uerll eve se o sos erel equos (w rell re soures o rl sls). R E F E R E N C E S []. K.. Ol J. Sper, e Frol Clulus, Ae Press, New Yor, NY, USA, 974. []. M. Cpuo, Els e sspzoe, Zell, olo, l, 969. [3]. A. Asr, A. Re Se H. Ser Nj, Soluo o sse o prl rol erel equos us e rol expoel operors, M. Me. Appl. S., vol. 35, o., J., pp [4]. H. A, A. H. Re Se H. Rezze, Su-equo eo or e rol reulrze lo-wve equos w oorle rol ervves, Se r, vol. 3, o. 3, J. 6, pp [5]. M. Cpuo, srue orer erel equos oell eler uo uso, Frol Clulus Apple Alss, vol. 4, No. 4, J., pp [6]. R. L. le P. J. orv, O e exsee o e orer o e soluo o srue orer equos, erol Jourl o Apple Mes, vol., No. 7, J., pp [7]. E. Rees-Melo, J. Mrez-Ve, C. Guerrero-Slzr, U. Orz-Meez, Applo o rol lulus o e oel o eler relxo peoe poler erls, Jourl o Apple Poler See, vol. 98, o., O. 5, pp [8]. N. J. FOR, M. L. MORGAO AN M. REELO, A pl e eree pproxo or e soluo o e uso equo w srue orer e, Elero rsos o Nuerl Alss, vol. 44, Ju. 5, pp [9]. H. A, A. Re Se H. Rezze, Sl Alss o srue Orer Frol Ce Sse,e Se Worl Jourl, vol. 3, O. 3, pp. -3. []. H. A, A. Re Se H. Rezze, Sl lss o ler srue orer sse w ulple e els, U.P.. S. ull., vol. 77, o., J. 5, pp []. A. Asr, A. Re Se S. Korros, O e eer uo ex+ϕ() s rol lulus, Ce. Eur. J. Ps., vol., o., O. 3, pp []. A. Asr A. R. Se, New ees or e wr e leler uos us e lple rsor As-Europe Jourl o Mes, vol. 7, o. 3, Sep. 4, pp. -8. [3]. H. Ser Nj, A. Re Se A. Asr, Sl lss o srue orer rol erel equos, Asr Apple Alss, vol., Jul., pp. -. [4]. A. Re Se, H. Ser Nj, Alrez Asr Frs Merous, Al su o ler sse o srue orer rol erel equo, Le Mee, vol. 67, o.,, pp [5]. H. A, A. H. Re Se H. Rezze, Approxe ll soluos o srue orer rol R erel equo, A Ss Eeer Jourl, Arle press.

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