The Sumudu transform and its application to fractional differential equations
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1 ISSN : (Olie) Iteratioal e-joural for Educatio ad Mathematics vol. 0, No. 05, (Oct. 03), 9-40 The Sumudu trasform ad its alicatio to fractioal differetial equatios I.A. Salehbhai, M.G. Timol Deartmet of Mathematics Veer Narmad South Gujarat Uiversity, Surat ibrahimmaths@gmail.com Article Ifo. A B S T R A C T Received o 0 Jue. 03 Revised o 0 Set. 03 Acceted o Set. 03 Keywords: Itegral Fractioal Sumudu Fractioal equatios. trasform, Derivatives, trasforms, differetial The Sumudu trasform rovides owerful oeratio methods for solvig differetial ad itegral equatios arisig i alied mathematics, mathematical hysics, ad egieerig sciece. A attemt is made to solve some fractioal differetial equatios usig the method of Sumudu trasform. AMS Subject classificatio: 6A33, 44A0. Itroductio: The methods of Itegral trasforms (Seddo [8]) have their geesis i ieteeth cetury work of Joseh Fourier ad Oliver Heaviside. The fudametal idea is to rereset a fuctio f x i terms of a trasform F( ) is b F ( ) K, x f xdx a (.) where the fuctios K, x is called kerel. 9
2 The Sumudu trasform ad its alicatio to fractioal differetial equatios. There are a umber of imortat itegral trasforms icludig Fourier, Lalace, Hakel, Laguerre, Hermit ad Melli trasforms. They are defied by choosig differet kerels K, x ad differet values for a ad b ivolved i (.). I 993, G. K. Watugala [9] itroduced the sumudu trasform to solve differetial equatios ad cotrol egieerig roblems. Weerakoo [0] has discussed this trasform by derivig the Sumudu trasform of artial derivatives. The Sumudu Trasform is defied as follows (Watugala [9]): If f t is a fuctio defied o the Real lie, the Sumudu Trasform of f t is defied by F t e ;Re 0 (.) Sf t f tdt 0 There has bee a great deal of iterest i fractioal differetial equatios (Miller ad Ross [3], Oldham ad Saier [4]). These equatios arise i mathematical hysics ad egieerig scieces. There are may defiitios of fractioal calculus are give by may differet mathematicias ad scietists (see Podluby [5]). Here, we formulate the roblem i terms of the Cauto fractioal derivative (see Cauto [],[]), which is defied as: If is a ositive umber ad is the smallest iteger greater tha such that, the the fractioal derivative of a fuctio f t is defied by (see Podluby [5]): x d f f C f t dx dt t x t 0 (.3) Further we used the result due to (Katatbeh ad Belgacem [6]): d f S F f dt r0 r r 0 (.4) 30
3 I.A. Salehbhai ad M.G. Timol/It. e- J. for Edu. & Math. vol. 0 (Oct. 03) 9-40 where is the smallest iteger greater tha.. Defiitios, Notatios & Some Results of Secial fuctios Some secial fuctios used i this thesis are as give below (See Raiville [7]) : The gamma fuctio is defied as x e x dx where, Re 0 (.) 0 Here ote that (.) ca be exteded to the rest of the comlex lae, excetig zero ad egative iteger. Alterative defiitios for Gamma Fuctio (Raiville [7]) are zz! lim z z... (.) e r e r r (.3) where is kow as Euler s costat ad is defied as followig: lim... log 3. (.4) The Pochhammer symbol (Raiville [7]) is defied by the equatios..., (.5) which is Geeralizatio of factorial fuctio. 3
4 The Sumudu trasform ad its alicatio to fractioal differetial equatios. If is either zero or a egative iteger, the we ca defie. (.6) The ordiary biomial exressio (Raiville [7]), defied as a ( a) z ( z). (.7)! 0 The Hyergeometric Fuctio (Raiville [7]) is defied as F a, b; c; x 0 c a b x (.8)! where the series o R.H.S. of (.8), whe c is either zero or a egative iteger, is absolutely coverget withi the circle of covergece x, ad diverget outside it; o the circle of covergece the series is absolutely coverget if c a b Re 0. The Geeralized Hyergeometric Fuctio (Raiville [7]) is defied as F q a, a,..., a; b, b,..., b ; x q a i i x. (.9) q 0! bj j Here ote that:. If q, the series (.9) coverges absolutely for every fiite x.. If q, the series (.9) coverges absolutely whe x ad diverges whe x. 3. If q, ad x, the series (.9) coverges absolutely whe 3
5 q Re bj ai0 j i. I.A. Salehbhai ad M.G. Timol/It. e- J. for Edu. & Math. vol. 0 (Oct. 03) If q, the series (.9) diverges for z 0, ad for z 0 its value is oe. 3. The solutio of fractioal Differetial Equatios: I this sectio, we obtai the solutio of some fractioal differetial Equatios usig Sumudu Trasform. (A) Cosider the fractioal Differetial Equatios is of the form 5 3 d x d x d x cosh t with iitial coditio x ; x0 0; x 0 0 (3.) dt dt dt Solutio: Alyig the Sumudu Trasform (.) o both the sides of equatio (3.) ad usig (.4), we have X x x x 3 x0 x0 x (3.) Further simlificatio yields, X 5 3 (3.3) Takig iverse Sumudu Trasform of (3.3) gives 33
6 The Sumudu trasform ad its alicatio to fractioal differetial equatios. xt e 6 3 t t 0 t t 0 u e cos 3 t u 3 si 3 t u du u u si 3 e t u t t t e du e erf 0 3 u 6 e 6 i e u e 3cos 3 t u 3 si 3 t u du u 3 si 3 e t u t t e 0 3 u t erf i t u du t (3.4) Equatio (3.4) is the solutio of (3.) Where erf t is the well-kow error fuctio (see Raiville [7]). (B) Cosider the fractioal Differetial Equatios is of the form 3 d y d y log 3 0 t with iitial coditio y0 ; y0 (3.5) dt dt Solutio: Alyig the Sumudu Trasform (.) o both the sides of equatio (3.5) ad usig (.4), we have 3 3 Y y0 y0 y0 loge (3.6) Further simlificatio yields, Y 3 3 log e 3 (3.7) 34
7 I.A. Salehbhai ad M.G. Timol/It. e- J. for Edu. & Math. vol. 0 (Oct. 03) 9-40 Takig iverse Sumudu Trasform of (3.7) gives t t e t yt u l 4u e du t t i erf i t t u e...(3.8) Where t is the well-kow Dirac Delta fuctio (see Raiville [7]). Equatio (3.8) is the solutio of (3.5) (C) Cosider the fractioal Differetial Equatios is of the form t e with iitial coditio z z z z d z d z d z 3 dt dt dt 0 ; 0 ; 0 ; 0 (3.9) Solutio: Alyig the Sumudu Trasform (.) o both the sides of equatio (3.9) ad usig (.4), we have Z z z0 z 0 z z 0 z0 z 0 z 0 z0 (3.0) Further simlificatio yields, Z (3.) Takig iverse Sumudu Trasform of (3.) gives 35
8 The Sumudu trasform ad its alicatio to fractioal differetial equatios. t t z t erf t e 4 ierf i t 8t e t 7 F, ; t t t 5 t (3.) Hece, (3.) is the required solutio of (3.9). (D) Cosider the fractioal Differetial Equatios is of the form d z d z d z 3 si t 3 dt dt dt with iitial coditio z z z z (3.3) Solutio: Alyig the Sumudu Trasform (.) o both the sides of equatio (3.3) ad usig (.4), we have A z z z z z0 3 z 0 3 z 0 z z (3.4) Further simlificatio yields, A (3.5) Takig iverse Sumudu Trasform of (3.5) gives 3 5 / 7 9 t 5 7 t 50 at 330 t 8t F ;, ; 60t F ;, ; t t i t i75 e erf i t i40 e erf (3.6) Hece, (3.6) is the required solutio of (3.3). 36
9 I.A. Salehbhai ad M.G. Timol/It. e- J. for Edu. & Math. vol. 0 (Oct. 03) Some Grahs: Figure 4.: Plot of zt Figure 4.: Plot of Re zt 37
10 The Sumudu trasform ad its alicatio to fractioal differetial equatios. Figure 4.3: Plot of Im zt Figure 4.4: Plot of Re at 38
11 I.A. Salehbhai ad M.G. Timol/It. e- J. for Edu. & Math. vol. 0 (Oct. 03) 9-40 Ackowledgemet First author is thakful to Uiversity Grats Commissio, New Delhi for awardig Dr. D. S. Kothari Postdoctoral Fellowshi (Award No.: File o. F.4-/006 (BSR)/3-803/0 (BSR) dated 09//0). Refereces [] Cauto M., Liear models of dissiatio whose Q is almost frequecy ideedetart II, Geohysical Joural of the Royal Astroomical Society, vol. 3, , 967. [] Cauto M., Elasticity ad Dissiatio, Zaichelli, Bologa, Italy, 969. [3] Miller K.S., Ross B., A Itroductio to the Fractioal Calculus ad Fractioal Differetial Equatios, Joh Wiley ad Sos, Ic., 993. [4] Oldham K.B., Saier J., Fractioal Calculus: Theory ad Alicatios, Differetiatio ad Itegratio to Arbitrary Order, Academic Press, Ic., New York- Lodo, 974. [5] Podluby I., Fractioal Derivatives: History, Theory, Alicatios. Utah State Uiversity, Loga, Setember 0, 005. [6] Katatbeh Q. D. ad Belgacem F.B.M., Alicatios of the Sumudu trasform to fractioal differetial equatios, Noliear Studies 8 (0) 99-. [7] Raiville E.D., Secial Fuctios, The Macmilla Comay, New York, 960. [8] Seddo I.N., The Use of Itegral Trasforms, McGraw-Hill Book Comay, New York,
12 The Sumudu trasform ad its alicatio to fractioal differetial equatios. [9] Watugala G. K., Sumudu trasform: a ew itegral trasform to solve differetial equatios ad cotrol egieerig roblems, Iteratioal Joural of Mathematical Educatio i Sciece ad Techology 4 (993) [0] Weerakoo S., Alicatio of Sumudu trasform to artial differetial equatios, Iteratioal Joural of Mathematical Educatio i Sciece ad Techology 5 (994)
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