VARIOUS phenomena occurring in the applied sciences

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1 roceedigs of he Ieraioal MuliCoferece of Egieers ad Compuer Scieiss 8 Vol I IMECS 8 March -6 8 Hog Kog Exac Soluios ad Numerical Compariso of Mehods for Solvig Fracioal-Order Differeial Sysems Nachapo Lekdee Sekso Sirisubawee ad Saoe Koopraser Absrac I his paper we apply he Laplace-Adomia- ade mehod ( which is based o he Laplace-Adomia decomposiio mehod ( ad he Adams-Bashforh- Moulo ype predicor-correcor scheme o solve a fracioalorder model of he glucose-isuli homeosasis i ras for aalyical ad umerical soluios respecively. Moreover he exac soluios of his fracioal-order model which are solved usig he Laplace rasform are employed o umerically ad graphically compare wih he resuls obaied usig he wo mehods. The ad he predicor-correcor scheme ca also be applied simply ad efficiely o oher fracioal-order differeial sysems arisig i egieerig problems. Idex Terms Laplace-Adomia decomposiio mehod adé approximaio Adams-Bashforh-Moulo predicorcorrecor scheme Capuo fracioal-order derivaive Exac soluio I. INTRODUCTION VARIOUS pheomea occurrig i he applied scieces [] [] ad egieerig [] [] have recely bee modeled by fracioal-order differeial equaios (FDEs. Several mehods are ow beig used o aalyically solve FDEs for example he Adomia decomposiio mehod (ADM [6] he Laplace-Adomia decomposiio mehod ( [] he Dua-Rach modified ADM [8] he homoopy aalysis mehod (HAM [9] ad he mulisep geeralized differeial rasform mehod (MSGDTM []. Numerical soluios of FDEs ca also be obaied via may approaches such as he Adams-Bashforh-Moulo ype predicor-correcor scheme or (redic Evaluae Correc ad Evaluae mehod [] he Galerki fiie eleme mehod [] he Legedre waveles mehod [] ad he specral collocaio mehod []. I he prese paper we sudy he sysem of fracioal order differeial equaios i (. This sysem is a geeralizaio of a ieger-order sysem proposed by Lombare e al. [] as a model for glucose-isuli homeosasis i healhy Mauscrip received December ; revised Jauary 8. Nachapo Lekdee is a hd sude i he Deparme Mahemaics Kig Mogku s Uiversiy of Techology Norh Bagkok Bagkok 8 Thailad.( karaho@gmail.com. Sekso Sirisubawee is a lecurer i he Deparme Mahemaics Kig Mogku s Uiversiy of Techology Norh Bagkok Bagkok 8 Thailad.( sekso.s@sci.kmub.ac.h Saoe Koopraser is a Associae rofessor i he Deparme Mahemaics Kig Mogku s Uiversiy of Techology Norh Bagkok Bagkok 8 Thailad ad a researcher wih he Cere of Excellece i Mahemaics CHE Si Ayuhaya Road Bagkok Thailad ( saoe.k@sci.kmub.ac.h ras. CD α a i( = c g( c α 6 i( CDa α g( = c (i( c c i( + c d( c ( CDa α d( = c α d( wih iiial codiios i( = i g( = g d( = d. ( I ( C Da α is he Capuo fracioal derivaive operaor of order α ( ] sarig from = a. The sae variables are he blood isuli coceraio i( he blood glucose coceraio g( ad he amou of glucose i he iesie d(. The cosas c i i =... are o-egaive model parameers. I his paper we solve Eq. ( by hree mehods ad compare he soluios. The firs mehod is a aalyical mehod based o he Laplace-Adomia-adé mehod ( [] []. The secod mehod is a umerical mehod based o he Adams-Bashforh Moulo predicor-correcor mehod ( []. The hird mehod uses Laplace rasforms o fid a exac soluio of he sysem. The paper is orgaized as follows. I secio prelimiary defiiios ad properies are give. I secio a descripio of he mehods used i our work are briefly give. I secio he hree mehods for solvig he sysem ( ad he soluios obaied are give. Fially secio icludes a discussio of he resuls ad he coclusios. II. RELIMINARY DEFINITIONS AND ROERTIES I his secio we provide defiiios of fracioalorder operaors such as he Riema-Liouville fracioal iegral ad he Capuo fracioal derivaive. The defiiio of he Miag-Leffler fucios ad heir impora properies are briefly give. A fucio f( ( > is said o be i he space C α (α R if i ca be expressed as f( = p g( for some p > α where g( is coiuous i [. The fucio is also said o be i he space Cα m if f (m C α m N (for furher deails see [6]. Defiiio.: [6]. The Riema-Liouville fracioal iegral operaor of order α > of a fucio f C α wih a is defied as RLJ α a f( = Γ(α a ( τ α f(τdτ > a ( where Γ( is he gamma fucio. Defiiio.: [6]. For a posiive real umber α he Capuo fracioal derivaive of order α wih a is defied ISBN: ISSN: 8-98 (ri; ISSN: (Olie IMECS 8

2 roceedigs of he Ieraioal MuliCoferece of Egieers ad Compuer Scieiss 8 Vol I IMECS 8 March -6 8 Hog Kog i erms of he Riema-Liouville fracioal iegral i.e. C Dα a f( = RL Ja m α f (m ( or i ca be expressed as CD α a f( = Γ (m α a f (m (τ α m+ dτ ( ( τ where m < α < m a ad f C m m N. A impora propery of he Riema-Liouville fracioal iegral ad he Capuo fracioal derivaive of he same order γ ca be wrie as [6] RLJ α a m CDa α f( = f( f (k (a ( ak ( k! where m < α < m f Cα m for m N ad α. The Laplace rasforms of a fracioal derivaive i he Capuo sese ad of some ypes of he Miag-Leffler fucios [6] are as follows. Lemma.: [6] The Laplace rasform of he Capuo fracioal derivaive of order m < α < m is m L { C Da α f(} = s α F (s s α k f (k ( (6 where F (s = L {f(}. Defiiio.: [6] Give α β > ad z C. The oe parameer Miag-Leffler fucio E α is defied as E α (z = j= z j Γ(jα + ( ad he Miag-Leffler fucio wih wo parameers is defied as z j E αβ (z = Γ(jα + β. (8 j= Lemma.: [] The Laplace rasforms for several Miag-Leffler fucios are give by L {E α ( λ α } = sα s α + λ (9 L { β E αβ ( λ α } = sα β s α + λ ( provided ha s > λ /α where λ is a cosa parameer. III. ALGORITHMS OF THE METHODS I his secio we will prese he mehods ad he algorihms for he ad he mehods ha we use o aalyically ad umerically solve he fracioal-order sysem (. A. The Laplace Adomia Decomposiio Mehod The Laplace Adomia Decomposiio Mehod ( [8] [9] for solvig FDEs or a sysem of FDEs is as follows. Cosider he followig fracioal-order iiial value problem: CD α a u( + R(u + N(u = g( ( where m < α < m m N ad he soluio u( saisfies some give iiial codiios. I Eq. ( C D α a deoes he Capuo fracioal derivaive of order α wih respec o R ad N are liear ad oliear operaors of u respecively ad g is a source erm. Takig he Laplace rasform of boh sides of Eq. ( ad he applyig he formula (. o he resulig equaio we obai L { C D α a u(} + L {R(u} + L {N(u} = L {g(} L {u(} = m s α s α k u (k ( + {g(} {R(u} L {N(u}. ( sα I he we defie he soluio u( as a ifiie series u( = u i ( ( i= ad represe he oliear erm N by a ifiie series of Adomia polyomials N(u = A i ( i= where he A i polyomials ca be deermied by he followig formula ( A i = d i i! dλ i N λ k u k i. ( λ= Subsiuig ( ad ( io ( we ge { } L u i ( = m s α s α k u (k ( + {g(} i= { ( } { R u i ( } A i. i= i= (6 The we have he Adomia recursio scheme as follows L {u } = m s α s α k u (k ( + {g(} ( L {u + } = {R(u (} {A }. Applyig he iverse Laplace rasform o Eq. ( we ca evaluae he soluio compoes u (. The he -erm approximaio of he soluio is ϕ ( = u i ( (8 i= which i he lim yields he exac soluio of Eq. ( as u( = lim ϕ (. (9 Someimes he exac soluio u( i Eq. (9 may be wrie i a closed form. If he exac soluio u( i Eq. (9 ca be wrie as a power series i which a idepede variable is raised o fracioal powers ad he radius of covergece of he series is quie small he he soluio migh o be valid for he eire domai of ieres. Therefore a echique of aalyical coiuaio o obai a soluio valid i he domai of ieres is required. The adé approxima mehod cosrucs a raioal fucio i as a approximaio for ISBN: ISSN: 8-98 (ri; ISSN: (Olie IMECS 8

3 roceedigs of he Ieraioal MuliCoferece of Egieers ad Compuer Scieiss 8 Vol I IMECS 8 March -6 8 Hog Kog a slowly covergig or divergig power series i. I is oe of he well-kow covergece acceleraio echiques which ca be applied o a -erm polyomial approximaio φ (. We deoe he [m/m] diagoal adé approxima of φ ( i by [m/m] {φ (} i.e. adé [m/m] {φ (} = [m/m] {φ (} where m = ( / if =... ad m = / if = However if each variable i he -erm approximaio φ ( has a fracioal power he we mus chage such fracios o ew ieger powers usig a rasformaio before applyig he adé approximas. The improved by he adé approximas is called he Laplace-Adomia-adé mehod (. B. Adams-Bashforh-Moulo predicor-correcor scheme Currely he Adams-Bashforh-Moulo ype predicorcorrecor scheme or he [] mehod is exesively employed o umerically solve FDEs. I our work we will use his mehod o obai approximae umerical soluios for sysem (. The releva formulas of he mehod are as follows. Cosider he fracioal-order iiial value problem (FIV CDa α u( = f( u( T u (k ( = u (k k =...m α (m m ( where f is a oliear fucio ad m is a posiive ieger. The FIV ( ca be rasformed o he followig Volerra iegral equaio u( = m u (k k k! + Γ(α ( τ α f(τ u(τdτ. ( I order o approximae he iegral i ( we discreize he eire ime T as he uiform grid { = h : =...N} for some ieger N ad he sep size h := T/N. Le u h ( deoes he approximaio o u(. Suppose ha we have already calculaed approximaios u h ( j j =... he he approximaio u h ( + of he FIV ( ca be compued usig he mehod as follows: u h ( + = m h α Γ(α + k + u (k h α + k! Γ(α + f( + u h ( + + a j+ f( j u h ( j ( j= where α+ ( α( + α if j = ( j + a j+ = α+ + ( j α+ ( j + α+ ( if j if j = +. The iiial approximaio u h ( + i Eq. ( is called a predicor ad is give by u h ( + = where m k + u (k + k! Γ(α b j+ f( j u h ( j j= ( b j+ = α (( + jα ( j α. ( ISBN: ISSN: 8-98 (ri; ISSN: (Olie IV. MAIN RESULTS I his secio we demosrae he use of he ad he as described above o solve he FIV i Eqs. (-(. However we will firs obai he exac soluio of he fracioal-order model of glucose-isuli homeosasis i healhy ras usig he Laplace rasform mehod. Takig he Laplace rasform of sysem ( we have L { C D α a i(} = L {c g( c α 6 i(} L { C D α a g(} = L { c (i( c c i( + c d( c } L { C D α a d(} = L { c α d(}. (6 Subsiuig he iiial codiios ( io (6 we obai s α I(s s α i = c α 6 I(s + c G(s s α G(s s α g = (c + c I(s + c D(s + c c c s s α D(s s α d = c α D(s ( where I(s = L {i(} G(s = L {g(} ad D(s = L {d(}. Algebraically maipulaig he resulig sysem we obai he followig liear sysem for he variables I(s G(s ad D(s: sα + c α 6 c c + c s α c s α + c α I(s G(s = D(s s α i s α g +c c c s s α d. (8 From he las equaio i sysem (8 we ca easily obai D(s as follows D(s = sα d s α + c α. (9 Takig he iverse Laplace rasform of (9 ad he usig he formula (9 we obai d( = d E α ( c α α. ( Applyig Cramer s rule o sysem (8 o obai he remaiig wo variables we fid I(s = sα i + c (s α + c α (g s α + µ + c c d s α s(s α + c α (sα β (s α β ( [ G(s = s(s α + c α c d s α (s α + c α 6 sα 6 + c (c + c + (s α + c α 6 (s α g + c c c (s α + c α ] s(s α + c α (c + c s α i ( Separaig he soluios i Eqs. (-( io parial fracios we have I(s = i s + φ ( s β β s α φ ( s β β β s α β ( s φ s α + c α ( G(s = g s + ω ( s β β s α ω ( s β β β s α β ( s ω s α + c α ( IMECS 8

4 roceedigs of he Ieraioal MuliCoferece of Egieers ad Compuer Scieiss 8 Vol I IMECS 8 March -6 8 Hog Kog where φ = βi + c (g β + µ + c c d β β + c α ( φ = βi + c (g β + µ + c c d β β + c α (6 c c d c α φ = (β + c α (β + c α ( β = cα 6 + (c α 6 µ c (8 β = cα 6 (c α 6 µ c (9 ω = g β + c α 6 µ + β µ + c α 6 g + (β + c α 6 c β d β + c α µ β i ( ω = g β + c α 6 µ + β µ + c α 6 g + (β + c α 6 c β d β + c α µ β i ( ω = (cα 6 c α c c α d (β + c α (β + c α ( µ = c + c ad µ = c c c. ( Takig he iverse Laplace rasforms for I(s ad G(s i Eqs. ( ad ( respecively ad he usig he formula ( we fially obai he soluios i erms of he Miag-Leffler fucios as follows: i( = i + φ β β E αα+ (β α φ β β E αα+ (β α φ E αα+ ( c α α ( g( = g + ω β β E αα+ (β α ω β β E αα+ (β α ω E αα+ ( c α α. ( Therefore Eqs. ( ( ad ( are he exac soluios of he FIV (-(. I paricular for he special case of ieger order α = he Miag-Leffler fucios ca be reduced o expoeial fucios []. Thus for α = he exac soluios ( ( ad ( reduce o he followig: ( ( e β e β i( = i + φ β β ( e c φ c g( = g + ω β β ( e c ω c d( = d e c. β ( e β β φ β β A. The Laplace-Adomia-adé mehod ω β β β ( e β β (6 The prese secio is devoed o he use of he Laplace- Adomia-adé mehod ( o obai a aalyical soluio for he FIV i Eqs. (-(. We begi by usig he Laplace Adomia Decomposiio mehod ( o obai a series soluio of he FIV. We he show ha he radius of covergece of he series is very small ad ha he series diverges over a large regio of he domai of ieres. Thus he is required o obai a soluio which ca be used over he whole domai by replacig he series soluio by a adé approxima i.e. by a raioal fucio. The for solvig he FIV i Eqs. (-( is as follows. We begi he from Eq. (. Afer some sraighforward algebraic maipulaio ad akig he iverse Laplace rasforms we obai he followig implici formulas for he soluios { } i( = i + L s L {cg( α cα 6 i(} { g( = g + L L { c(i( c ci( + cd( c} sα { } d( = d + L s L α { cα d(}. ( We ca he expad he implici formulas i( g( d( io ifiie series by ieraio. The ifiie series for he soluios are he give by i( = i k ( g( = g k ( d( = d k (. (8 Foruaely he model does o have ay oliear erms so we do o eed o replace hem by he Adomia polyomials. Subsiuig Eq. (8 io Eq. ( for i( g( d( we obai } { { }} i k ( = i + L c g k ( c α 6 i k ( { { ( g k ( = g + L c i k ( c c i k ( + c { { d k ( = d + L d k ( c }} c α }} d k (. (9 Machig he wo sides of Eq. (9 we ca deermie he soluio compoes from he followig recursio scheme. L {i (} = i s L {g (} = g s + c c c s α+ L {d (} = d s. L {i + (} = c {g (} cα 6 {i (} L {g + (} = c + c {i (} + c {d (} L {d + (} = cα {d (} =... ( Ieraively akig he iverse Laplace rasform of he recur- ISBN: ISSN: 8-98 (ri; ISSN: (Olie IMECS 8

5 roceedigs of he Ieraioal MuliCoferece of Egieers ad Compuer Scieiss 8 Vol I IMECS 8 March -6 8 Hog Kog sio scheme i Eq. ( we obai i ( = i g ( = g + ( α c Γ(α + + cc Γ(α + d ( = d ( i ( = α g c Γ(α + i c α 6 Γ(α + ( + α c c Γ(α + + cc c Γ(α + g ( = α ( d c Γ(α + i c Γ(α + i c Γ(α + d ( = d α c α Γ(α + ( i ( = α d c c Γ(α + g c c α 6 Γ(α + i c c Γ(α + i c c + α i c α 6 Γ(α + + α ( c c c α 6 Γ(α + cc ccα 6 Γ(α + Γ(α + g ( = α ( d c c α Γ(α + g c c Γ(α + g c c Γ(α + + i c c α 6 Γ(α + + α i c c α 6 Γ(α + + α ( c c c Γ(α + + d ( = d α c α Γ(α +. c c c Γ(α + cc c cc c c Γ(α + Γ(α + Sice he expressios of he soluio compoes are quie log for oly he firs wo soluio compoes are expressed as above. The -erm approximaios of he soluio i( g( ad d( are defied as I ( = i k ( G ( = g k ( D ( = d k ( ( respecively. The iiial codiios ad parameer values described i [] which are employed i our simulaios are as follows. i = g = d = c =. c =. c =. c =. c 6 =. c =. c =. ( Usig he symbolic algebra package MATHEMATICA he approximaig soluios of he FIV (-( for he special case α = wih he give iiial codiios ad parameer values i Eq. ( are I ( = G ( = D ( = The correspodig soluios usig he are adé [/] {I (} = adé [/] {G (} = ( adé [/] {D (} = ( The correspodig exac soluios i (6 obaied usig Laplace rasforms for α = he approximae umerical soluios usig he RK mehod he ad he are compared i Fig.. I is obvious from Fig. ha he soluios for i( g( obaied usig he are differe from hose obaied usig he oher mehods whe is approximaely close o = ad ha he soluio diverges for >. This divergece is due o he fac ha he ifiie series soluio for he diverges for >. Due o his divergece of he compared wih he i is clear ha he will be a beer mehod ha he for oher values of α. i( g( d( Exac Rk EXACT RK Exac RK Fig.. Simulaio comparisos of he soluios i( g( d( for he FIV (-( wih α = usig he exac soluios he RK mehod he ad he. The simulaio resuls i( g( obaied from he are divergig for. B. Adams-Bashforh-Moulo predicor-correcor scheme Applyig he Adams-Bashforh-Moulo predicorcorrecor scheme i Eqs. ( ( [] o he FIV (-( we discreize he ime ierval wih pois { } ad obai he formulas for i h = i h ( g h = g h ( ISBN: ISSN: 8-98 (ri; ISSN: (Olie IMECS 8

6 roceedigs of he Ieraioal MuliCoferece of Egieers ad Compuer Scieiss 8 Vol I IMECS 8 March -6 8 Hog Kog dh = dh ( as follows: ih+ dh+ c gh+ cα 6 ih+ Γ(α + X aj+ (c ghj cα 6 ihj ( Γ(α + j= = i + + gh+ For simpliciy le = z he we have c (i = g + h+ c Γ(α + c i h+ c + c dh+ X aj+ ( c (ihj c + Γ(α + j= c ihj c + c dhj = d + ( cα dh+ Γ(α + X + aj+ ( cα dhj Γ(α + j= G (z = +.8z z 6 D (z =.z....8 (6 z. Calculaig he ade [/] of he resulig soluios i Eq. (6 usig he commad adeapproxima i Mah emaica ad he recallig ha z = he leads us o he followig approximaig soluios ade [/] I ( = (6 ade [/] G ( = ade [/] D ( = (6 ( i which i h+ = i + X bj+ [c ghj cα 6 ihj ](8 Γ(α j= gh+ = g + X bj+ [ c (ihj c Γ(α j= d h+ I (z = + 9.z z 6 c ihj c + c dhj ] X = d + bj+ [ cα dhj ] Γ(α j= (9 (6 usig The simulaio resuls of he problem for α = all of he mehods i.e. he exac formulas ( ( ad ( he mehod i Eqs. (-(6 wih he sep size h = he i Eq. (6 ad he i Eq. (6 are show i Fig.. The soluio curves i( g( cosruced by he are divergig whe ad he umerical simulaios obaied by he mehod ad he are i very good agreeme wih he exac soluios. The absolue errors bewee umerical soluios which are compued usig he ad he mehod ad he exac soluios are show i Table I. I ca be umerically cocluded from Table I ha he mehod achieves a higher degree of accuracy ha he whe is larger. where for l = (6 Exac (( + jα ( jα j. (6 α We will use he discreized formulas (-(6 o obai he umerical soluios for he FIV (-( i he ex secio. blj+ = i( alj+ α+ ( α( + α if j = ( j + α+ + ( jα+ = ( j + α+ if j if j = + C. Simulaio resuls 8 Exac g( I his secio we will show he simulaio resuls of he FIV (-( wih α =. obaied usig he formulas for he exac soluios i Eqs. ( ( ad ( ad he fucio mlf for he Miag Leffler fucios which is implemeed by []. The approximaig soluios of he problem geeraed by he mehod he ad he are also described. I addiio he absolue errors of he umerical resuls obaied by he ad he mehod compared o hose obaied usig he exac soluio formulas are show. The followig simulaio resuls are for α =. Applyig he o he problem via he recursio scheme ( he -erm approximaios of he soluios are demosraed as 6 6 I ( = + 9./ G ( = +.8/ D ( =. / /....8 ISBN: ISSN: 8-98 (ri; ISSN: (Olie (6. IMECS 8

7 roceedigs of he Ieraioal MuliCoferece of Egieers ad Compuer Scieiss 8 Vol I IMECS 8 March -6 8 Hog Kog he followig approximaig raioal soluios d( Exac ade [/] I ( = ade [/] G ( = ade [/] D ( = (68 Fig.. Simulaio comparisos of he soluios i( g( d( for he FIV (-( usig he exac soluios he mehod he ad he for α =. TABLE I T HE ABSOLUTE ERRORS OF DIFFERENT METHODS CAOMARED WITH THE EXACT SOLUTIONS OF THE FIV (-( WITH α = α= i.6e-.e-.8e-.89e- 9.8E-.6E-.E-.E-.E- 6.E- Exac- g 9.9E-.89E-.869E-.6E- 9.6E-.E-.E-.E-.96E-.E- i 9.6E-6.69E-6 8.E-.E-.E-8.6E-.E-.6E-.E-.E- Exac- g.e-6.8e-.6e-.66e-8.e-8.e-8.86e-8.e-8.e-8.e-8 d.9e-6 8.8E-.6E-.8E-.E- 6.69E-.6E-.8E-.669E-.E- Exac i( α= The umerical simulaios of he problem for α = usig all of he mehods i.e. he exac formulas ( ( ad ( he mehod i Eqs. (-(6 wih he sep size h = he i Eq. (68 ad he i Eq. (66 are described i Fig.. The soluio curves i( g( obaied usig he are divergig whe ad he umerical simulaios obaied by he mehod ad he are i very good agreeme wih he exac soluios. The absolue errors bewee umerical soluios which are compued usig he ad he mehod ad he exac soluios are show i Table II. I is o difficul o observe from Table II ha he mehod aais a beer accuracy ha he whe is far away from he iiial poi. d.6e-6.8e-6.88e-.9e-.8e-.e-.e-8.e E-9.6E- 8 Exac g( 6 6 Nex we will simulae umerical resuls of he problem for α = as follows. Applyig he o he problem via he recursio scheme ( he -erm approximaios of he soluios are expressed as Exac (66 d( D ( =.8/ /. I ( = +./ G ( = +.86/ For simpliciy le = z he we ge I (z = +.z z G (z = +.86z D (z =.8z z (6 9 z. Similarly as above we compue he ade [/] of he obaied soluios i Eq. (6 ad he subsiue z = io he resulig equaios. The eveually brigs ISBN: ISSN: 8-98 (ri; ISSN: (Olie Fig.. Graphical comparisos of he soluios i( g( d( for he FIV (-( usig he exac soluios he mehod he ad he for α =. IMECS 8

8 roceedigs of he Ieraioal MuliCoferece of Egieers ad Compuer Scieiss 8 Vol I IMECS 8 March -6 8 Hog Kog TABLE II THE ABSOLUTE ERRORS OF DIFFERENT METHODS CAOMARED WITH THE EXACT SOLUTIONS OF THE FIV (-( WITH α = α = Exac- i g d..e-.8e- 6.8E-6.8E- 9.E- 8.9E-6..9E-.E- 8.E-6 6.E-.99E-6 8.E E-.9E- 8.9E E-.E-.9E-6..8E- 6.6E-.6E-6.8E- 8.6E-.E-6..6E- 9.6E-.E-6.E-.9E- 6.88E-6 α = Exac- i g d..e-.66e-6.e-.e-.9e-6.6e-6..6e-6.6e-6.6e-6 6.9e E-.698E-6..e-6 6.6E-.89E e-6.66E-.9E e-.9E- 9.E- 6.6e-.66E-.6E-..8e-.9E-.8E-.6e-.8E-.E- V. CONCLUSION I his aricle we have successfully obaied he exac soluios of he fracioal-order iiial value problem (- ( by usig he Laplace rasform. We have also obaied approximaig aalyical soluios of he problem via he ad he. The umerical soluios of he problem simulaed via he mehod have also bee compued. The simulaios of he soluios calculaed by he above approaches have bee compared for he ieger order α = ad he fracioal orders α =. The comparisos have show ha he approximae soluios geeraed via he ad he are i very good agreeme wih he exac soluios of he problem whereas he ifiie series soluios geeraed by he become diverge as he ime icreases. [] K. A. Gepreel T. A. Nofal ad N. S. Al-Sayali Exac soluios o he geeralized hiroa sasuma kdv equaios usig he exeded rial equaio mehod Egieerig Leers vol. o. pp [6] K. Kaaria ad. Vellaisamy Saigo space ime fracioal poisso process via adomia decomposiio mehod Saisics & robabiliy Leers. [] F. Haq K. Shah G. ur Rahma ad M. Shahzad Numerical soluio of fracioal order smokig model via laplace adomia decomposiio mehod Alexadria Egieerig Joural. [8] S. Sirisubawee ad S. Kaewa New modified adomia decomposiio recursio schemes for solvig cerai ypes of oliear fracioal wo-poi boudary value problems Ieraioal Joural of Mahemaics ad Mahemaical Scieces vol.. [9] Q. Liu J. Liu ad Y. Che Asympoic limi cycle of fracioal va der pol oscillaor by homoopy aalysis mehod ad memory-free priciple Applied Mahemaical Modellig vol. o. pp. 6. [] A. Freiha ad S. Momai Applicaio of mulisep geeralized differeial rasform mehod for he soluios of he fracioal-order chua s sysem Discree Dyamics i Naure ad Sociey vol.. [] K. Diehelm N. J. Ford ad A. D. Freed A predicor-correcor approach for he umerical soluio of fracioal differeial equaios Noliear Dyamics vol. 9 o. pp.. [] H. Wag D. Yag ad S. Zhu A perov galerki fiie eleme mehod for variable-coefficie fracioal diffusio equaios Compuer Mehods i Applied Mechaics ad Egieerig vol. 9 pp. 6. [] H. Lu. W. Baes W. Che ad M. Zhag The specral collocaio mehod for efficiely solvig pdes wih fracioal laplacia Advaces i Compuaioal Mahemaics pp. 8. [] M. Lombare M. Lupo G. Campeelli M. Basualdo ad A. Rigalli Mahemaical model of glucose isuli homeosasis i healhy ras Mahemaical bioscieces vol. o. pp. 69. [] N. A. Kha A. Ara ad N. A. Kha Fracioal-order riccai differeial equaio: aalyical approximaio ad umerical resuls Advaces i Differece Equaios vol. o. p. 8. [6] I. odluby Fracioal differeial equaios: a iroducio o fracioal derivaives fracioal differeial equaios o mehods of heir soluio ad some of heir applicaios. Academic press 998 vol. 98. [] R. Magi X. Feg ad D. Baleau Solvig he fracioal order bloch equaio Coceps i Mageic Resoace ar A vol. o. pp [8] S. Momai ad Z. Odiba Numerical compariso of mehods for solvig liear differeial equaios of fracioal order Chaos Solios & Fracals vol. o. pp. 8. [9] H. Ahmed M. S. Bahga ad M. Zaki Numerical approaches o sysem of fracioal parial differeial equaios Joural of he Egypia Mahemaical Sociey vol. o. pp.. [] H. J. Haubold A. M. Mahai ad R. K. Saxea Miag-leffler fucios ad heir applicaios Joural of Applied Mahemaics vol.. [] I. odluby Calculaes he miag-leffler fucio wih desire accuracy (. ACKNOWLEDGMENT This research is suppored by he Deparme of Mahemaics Kig Mogkus Uiversiy of Techology Norh Bagkok Thailad ad he Cere of Excellece i Mahemaics he Commissio o Higher Educao Thailad. REFERENCES [] Y. Cho I. Kim ad D. Shee A fracioal-order model for mimod milleium Mahemaical Bioscieces vol. 6 o. Suppleme C pp. 6. [] M. Asgari Numerical soluio for solvig a sysem of fracioal iegro-differeial equaios IAENG Ieraioal Joural of Applied Mahemaics vol. o. pp [] S. Zhag ad Y.-Y. Zhou Muliwave soluios for he oda laice equaio by geeralizig exp-fucio mehod IAENG Ieraioal Joural of Applied Mahemaics vol. o. pp. 8. [] H. Sog M. Yi J. Huag ad Y. a Numerical soluio of fracioal parial differeial equaios by usig legedre waveles Egieerig Leer vol. o. pp ISBN: ISSN: 8-98 (ri; ISSN: (Olie IMECS 8