International Journal of Mathematical Archive-5(7), 2014, Available online through ISSN

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1 International Journal of atheatical Archive-5(7) Available online through wwwijainfo ISSN SOE NEW RELATIONSHIPS BETWEEN THE FRACTIONAL DERIVATIVES OF FIRST SECOND THIRD AND FOURTH CHEBYSHEV WAVELETS Saeer Qasi Hasan* and Dalia Raad Abd Departent of atheatics Collage of education Alustansiriyah University Iraq (Received On: 9-7-4; Revised & Accepted On: 5-7-4) ABSTRACT In this paper the shifted first second third and fourth kind Chevelets wavelets ΨΨ nn (tt) ΨΨ nnnn (tt) ΨΨ 3 nnnn (tt) and ΨΨ 4 nnnn (tt) properties are presented The ain ai is: Generalize the first second operational atrix to the fractional derivatives In this approach a truncated first second atrix of fractional derivatives are used Presented a new proposal forula expressing of fractional derivative α> operational atrix of shifted first kind Chybeshev wavelets D α ΨΨ nn (tt) inters of D α (x) (x) = [T (x) T (x) T n (x)] T and a forula expressing the fractional derivative α> of second kind Chebyshev wavelets D α ΨΨ nnnn (tt) inters of D φ(x) φ(x) = [U (x) U (x) U n (x)] T 3 Presented a new proposal forula expressing of fractional derivative α> operational atrix of shifted third and fourth kind Chybeshev wavelets D α ΨΨ 3 nnnn (tt) D α ΨΨ 4 nnnn (tt)inters of ΨΨ nnnn (tt) andψψ nnnn (tt) and a forula expressing of fractional derivative α> of first kind Chebyshev wavelets D α ΨΨ nn (tt)inters of ΨΨ nnnn (tt) and (tt) ΨΨ nnnn All the proposed results are of direct interest in any applications INTRODUCTION The Chebyshev polynoials are one of the ost useful polynoials which are suitable in nuerical analysis including polynoial approxiation integral and differential equations and spectral ethods for partial differential equations [4 9 7] One of the attractive concepts in the initial and boundary value probles is differentiation and integration of fractional order [ ] any researchers extend classical ethods in studies of differential and integral equations of integer order to fractional type of these probles [5 ] One of the wide classes of researches focuses to constructing the operational atrix of derivative in soe spectral ethods Recently a lot of attention has been devoted to construct operational atrix of fractional derivative[44]for exaple the fractional type first kind chebyshev polynoials are used to solving fractional diffusion equations[37] also are used to solve ulti-order fractional equation[] In this paper we use shifted chebyshev polynoials of first second third and fourth kind and recall soe iportant properties Next we used obtain the operational atrix of fractional derivative Wavelets theory is a relatively new eerging in atheatical research [5 6 3] It has been applied in a wide range of engineering disciplines particularly shifted first kind chebyshev wavelets play an iportant role in establishing algebraic ethods for the solution of ulti-order fractional differential equations [ ] and initial and boundary values probles of fractional order[3]new Spectral Second Kind Chebyshev Wavelets Algorith for Solving Linear and Nonlinear Second-Order Differential Equations[] Corresponding author: Saeer Qasi Hasan* Departent of atheatics Collage of education Alustansiriyah University Iraq International Journal of atheatical Archive- 5(7) July 4 8

2 and Fourth Chebyshev Wavelets / IJA- 5(7) July-4 PROPOSAL OPERATIONAL ATRIX OF CHEBYSHEV'S FRACTIONAL DERIVATIVE A It well know the first kind chebyshev polynoial T n (z) of degree n [] which defined on [-] by: T n (z) = Cos(n θ) where z= Cos(θ)θ [π] and can be deterined with the aid of the following recurrence forula: T n+ (z) = zt n (z) - T n- (z) n= 3 T (z) = T (z) =z The analytic for of chebyshev polynoial T n (x) of degree (n) is given by: n T n (z) = ( ) i z n-i n= () n-i- n(n i )! i= i!(n!)! and are orthogonal on [-] with respect to the weight function ω(t) = zz that is: π i = j = T i (z)t j (z) π dz = i = j z i j In order to use these polynoials on the interval [ ] we define the shifted chebyshev polynoials by introducing the change variable z=x- then T n (x) can be obtained as follows: T i+ (x) = (x-) T i (x) - T i (x) i= wheret (x)=t (x) =x- and the analytic for is T n (x) = with respect to the weight function ω(x)= xx xx that is: π i = j = T i (x)t j (x) x x π dddd = i = j i j n i= ( ) i n-i n(n i )! i!(n!)! x n-i n= 3 and are orthogonal The function u(x) square integrable in [ ] ay be expressed in the ter of shifted first kind chebyshev polynoial as: i= c i T i (xx) where the coefficients c i are given by c i = πσσ i uu(xx)t i (xx)dddd σσ i = ii = ii In practice only the first (+)-ters shifted first kind chebyshev polynoial are considered Then we have: u (xx) = c i T i (xx) = CC TT (xx) i= where the shifted first kind chebyshev coefficient vector C and the shifted first kind chebyshev vector (xx) are given by:cc TT = c (x) c (x) cc (x) (x) = [T (x) T (x) T (x)] T For the Caputo's derivative we have []: D α C = C is a constant Γ(n+) D α x n = Γ(n+ α) xn α n α for nϵn () n < α for nϵn for N = { } In the following theore we will define the fractional derivative of the vector (x) Theore : Let (x) be shifted first kind chebyshev vector defined as (x) = [T (x) T (x) T n (x)] T and also suppose α> then D α (x) = (α) (x) where (α) is (+) X (+) is an operational atrix of fractional derivative of order α> in the caputo sense and is defined as follows : 4 IJA All Rights Reserved 8

3 and Fourth Chebyshev Wavelets / IJA- 5(7) July-4 : α w () () () i w i w i = : n α () n α () n α () i= w n α i i= w n α i i= w n α i : () () w ii () i= i= w i i= w i and () is given by: w n α ji σσ j = j π k= ( )k+i j-k+n-i n(n i )!j(j k )! (n i α+j k+ ) i!(n!)! (n i α+)k!(j k)! (n i α+j k+) () w n α ji where n= α and σσ j = j = j Note that in α the first α rows are all zero Proof: Let T (x) be shifted first kind chebyshev polynoial then by using () and () we can find that: D α T n (x) = n < α and for n α D α T n (x) = n i= ( ) i n-i n(n i )! i!(n!)! Dα x n-i n α = i= ( ) i n-i n(n i )! i!(n!)! (n i α+) x n-i-α Now approxiate (x n-i-α ) by (+)-ters of shifted first kind chebyshev polynoial we have x n-i-α = j= d n ij T j (x) whered n ij = σσ j xn i α π x x T j (x) dx T j j (x) = k= ( ) k j-k j(j k )! k!(j k)! x j-k then d n ij = σσ j j π k= ( )k j-k j(j k )! xn i α +j k k!(j k)! x x = σσ j j π k= ( )k j-k j(j k )! (n i α+j k+ ) π k!(j k)! (n i α+j k+) where σσ j = j = j then n α i= j= n-i n(n i )!(n i)! n α () D α T n (x) = ( ) i = [ dx i!(n!)! (n i α+) d n ijt n (x) j= i= w nji ] T n (x) for n α n α () n α () n α = i= w n α i i= w n α i i= w n α i (x) for n α and D α T n (x) = [ ] (x) n < α B The second kind of degree (n) [6] which defined on the interval [ ] as: sin (n+)θ U n (z) = where z = cos θ θ nπ + kπ B sin θ These polynoials satisfy the following recurrence relation U (z) = U (z) = zz U n (z) = zu n (z) U n (z) n = 3 () For using this polynoials on interval () which called shifted chebyshev polynoials[6] by introducing the change variable (x ) and satisfy the following U n (x) = (4x )U n (x) U n (x) n = 3 where U (x) = U (x) = 4x 4 IJA All Rights Reserved 83

4 and Fourth Chebyshev Wavelets / IJA- 5(7) July-4 and the analytic for of shifted chebyshev polynoials u n (x) of degree (n) is given by U n (x) = n+ r= r ( ) n+ r (n+r)! r (n+ r)! r! x r (3) and are orthogonal with respect to the weight function ω(x)= xx xx that is: U (x) U n (x) π = n dx = 8 x x n The function f(x) square integrable in [ ] ay be expressed in the ter of shifted first kind chebyshev polynoial as: c i U i (xx) i= where the coefficients c i are given by c i = 8 ffff)t π i (xx)dddd In practice only the first (+)-ters shifted first kind chebyshev polynoial are considered Then we have: f (xx) = i= c i T i (xx) = CC TT φ(xx) where the shifted first kind chebyshev coefficient vector C and the shifted first kind chebyshev vector φ(xx) are given by:cc TT = c (x) c (x) cc (x) (x) = [U (x) U (x) U (x)] T In the following theore we will define the fractional derivative of the vector φ(x) Theore : Let φ(x) be shifted second kind chebyshev vector defined in φ(x) = [U (x) U (x) U n (x)] T also suppose > then D φ(x) = ( ) φ(x) where is the ( + ) ( + ) operational atrix of fractional derivative of order in the Caputo sense and defined as follows: = and () w n+ p r + () w + r n+ () w n+ r () w r p+ () w + r r= n+ () w n+ r () w r () w + r r= n+ () w n+ r () w r = 8 π r l( )n+p+ (l+r) (n + r)! (p + l)! (r )! (l+r) r +l + (n + r)! r! (r )(p + l)! l! (r +l + ) l= Note that in α the first α + rows are all zero Proof: Let UU (x) be shifted scondt kind chebyshev polynoial then by using () and (3) we can find that: D U n (x) = n and for (n = + ) we have D U n (x) = n+ r= r ( ) n+ r (n+r)! r (n+ r)! r! D x r = r n+ ( )n+ r (n+r)! r (r )! (n+ r)! r! (r ) xr 4 IJA All Rights Reserved 84

5 and Fourth Chebyshev Wavelets / IJA- 5(7) July-4 Now approxiate (x r ) by ( + )-ters of shifted second kind chebyshev series we have x r = p= d r p U p (x) xr d r p = 8 x x π U p (x) dx = 8 p+ l( ) p + l (p+l)! l π l= x x x r (p+ l)! l! = 8 p+ l( ) p + l (p+l)! l Γ r +l+ π π l= (p+ l)! l! Γ(r +l+) dx n+ (n D U + r)! r (r )! n (x) = r( ) n+ r d (n + r)! r! (r ) r p U n (x) p= n+ n+ () () = p= w n+ p r U n (x) n+ () n+ () = w n+ r w n+ r w n+ r φ(x) for n= + and D U n (x) = [ ]φ(x)n 3 PROPOSAL OPERATIONAL ATRIX OF CHEBYSHEVWAVELETS FRACTIONAL DERIVATIVE A This shifted first kind chebyshev wavelets Ψ n (t) = Ψ (kn t) have four arguents; k N n= k and n = n ; oreover is the order of the chebyshev polynoials of the first kind and t is the noralized tie and they are defined on the interval [) as []: kk ΨΨ nn (t) = TT ( kk tt nn ) nn kk oo ww TT = ππ where TT = TT ππ > t nn kk = - n = k- and nn = kk - the weight function ww = w(t ) and ww nn (t) = w( k t nn ) where w( k t nn ) = A function f(t) defined over [) y be expanded as follows f(t) = cc nn (t) where nn = = ΨΨ nn kk tt nn (kk tt nn ) cc nnnn = (f(t) ΨΨ nn (t) ) w = ww (tt) ff(tt) ΨΨ nn (t) dt and f(t) = nn = = cc nn where C = [ cc cc cc kk cc kk cc kk ]T ΨΨ nn (t) = c T ΨΨ nn (t) Thus ΨΨ nn (t) = [ΨΨ ΨΨ ΨΨ ΨΨ kk ΨΨ kk ΨΨ kk ]T Theore 3: Let ΨΨ nn (t) be shifted of first kind chebyshev vector and also suppose α > ) then D α ΨΨ nn (tt)(tt) = D α ( cc kk T ππ (( kk t - nn )) = ( ) ΨΨ nnnn (t) such that C = = > where is the (+)x(+) operational atrix derivative of order (α) in the caputo sense and is defined as follows: : ww = ii ww ii ww ii : αα αα αα ii= ww αα ii ii= ww αα ii ii= ww αα ii : ww ii ii= ii= ww ii ii= ww ii And ww αα jj ii is given by 4 IJA All Rights Reserved 85

6 and Fourth Chebyshev Wavelets / IJA- 5(7) July-4 ww αα jj ii = σσ j j π where σσ j = j = j k= ( )k+i (j+)-(k +i) Note that in α the first α rows are all zero Proof: DD TT ( kk t - nn ) = < αα and for ( αα ) DD TT ( kk t - nn ) = ii= ( ) ii -i ( ii )! ii!( ii)! αα = ii= ( ) ii -i ( ii )! ii!(nn ii)! ( i )!( i)!j(j k )! ( i α+j k+ ) i!( i)! ( i α+)k!(j k)! ( i α+k+) DD ( kk t - nn ) -i ( kk t - nn ) -i-α Now approxiate ( kk t - nn ) -i-α by (+) ters of shifted first kind chebyshev wavelet we have ( kk t - nn ) -i-α = jj = dd nn iijj TT jj ( kk t - nn ) dd iijj = σσ j ππ (kk tt nn ) nn ii αα TT jj ( kk tt nn ) dddd kk tt nn ( kk tt nn ) = σσ j jj ππ kk= ( )kk j-k jj (jj kk )! ( kk tt nn ) kk!(jj kk)! kk tt nn ( kk tt nn ) where σσ j = j = j then dd iijj = σσ j Therefore j π k= ( )k j-k j(j k )! ( i α+j k+ ) k!(j k)! ( i α+k+) dt D α T ( k t - n ) = ( ) i α i= j= -i ( i )!( i)! i!( i)! ( i α+) α j= i= ] T ( k t - n ) = [ w ji d ij T ( k t - n ) α α D α ΨΨ nn (t) = i= w α i i= w α i w α i and D α ΨΨ nn (t) = [ ] ΨΨ nn (t) for < α α i= ΨΨ nn (t) for α B Second kindchebyshev wavelets Ψ n (t) =Ψ (k n t) have four arguentsk n can assue any positive integer is the order of second kind chebyshev polynoials and t is the noralized tie They are defined on the interval [ ] by: [] k +3 Ψ n (t) = u π k t n t [ n n+ ] k k = n= kk - o w and w(t-) has to be dilated and translated as follows: ww nn ( kk t -nn) = ( kk tt nn) ( kk tt nn) and the function approxiation A function (tt) defined over [ ] ay be expanded in ters of second kind Chebyshevwavelets as f(t) = nn= = cc nnnn ΨΨ nnnn (t) cc nnnn = (f(t) ΨΨ nnnn (t)) = ww(tt) ff(tt) ΨΨ nnnn (tt) dt and (tt) = tt tt If the infinite series is truncated then (tt) can be approxiated as f(t) = kk nn= = cc nnnn ΨΨ nnnn (t) = cc TT ΨΨ nnnn (tt) where CC and Ψ(tt) are kk ( + ) atrices defined by 4 IJA All Rights Reserved 86

7 and Fourth Chebyshev Wavelets / IJA- 5(7) July-4 c = [ cc cc cc kk cc kk cc kk ]T ΨΨ nnnn (t) = [ΨΨ ΨΨ ΨΨ ΨΨ kk ΨΨ kk ΨΨ kk ] Theore 3: Let ΨΨ nnnn (tt) be second kind chebyshev wavelets and suppose α> then DD ΨΨ nnnn (tt) = ππ DD (uu ( kk t n) = ( ) ΨΨ nnnn (tt) where is (+) (+) operational atrix derivative of order in the caputo sense and defined as follow: = and + w + r + w + r w r w + p r = 8 w + r r= n+ w + r w r w + r r= n+ w + r w r rl( ) +p + (l+r) (rr )!(p+l)! (l+r) p+ Γ r +l+ π l= ( + rr)! rr! (rr αα(p+ l)! l! Γ(r +l+) Proof: Fro ( ) we have that UU ( kk + t n) = + rr ( +rr)! rr rr= rr ( ) ( + rr)! rr! (kk tt nn) rr Also we have that DD UU ( kk t n) = n and for (n = + ) we have DD UU ( kk t n) = + rr= rr + rr ( +rr)! rr ( ) ( + rr)! rr! DD ( kk tt nn) rr + = r ( ) + rr ( +rr)! rr (rr )! ( kk tt nn) rr αα ( + rr)! rr! (rr αα) Now approxiate ( kk tt nn) rr αα by (+) ter of shifted second kind chebyshev wavelets we have: ( kk tt nn) rr αα = pp= dd rr pp UU pp ( kk t n) dd rr pp = 8 ππ (kk tt nn) UU ( kk t n) ( kk tt nn) ( kk tt nn) dt = 8 pp+ l ( ) pp + l (pp+l)! l ππ l= ( kk tt nn) ( kk tt nn) ( kk tt nn)dddd (pp+ l)! l! (rr+l αα) = 8 pp+ l ( ) pp + l (pp+l)! l (rr αα+l+ ) ππ ππ l= (pp+ l)! l! (rr+l αα+) DD UU ( kk t n) = + r dd ( + rr)! rr! (rr αα) rr pp UU ( kk t n) + = p= w n+ p r UU ( kk t n) pp= ( ) + rr ( +rr)! rr (rr )! DD ΨΨ nnnn (t)= + w ππ p= n+ p r UU ( kk t n) thus + + D Ψ n (t)=[ w +r w +r w +r ]Ψ n (t) k +3 and D Ψ n (t) = D U ( k t n) = [ ] Ψ n (t) n π + 4 IJA All Rights Reserved 87

8 and Fourth Chebyshev Wavelets / IJA- 5(7) July-4 4 Operational atrices of Fractional Derivative Chebyshev wavelets The third and fourth kind chebyshev wavelets Ψ 3 n (t) = Ψ 4 n (t)=ψ(k n t) has four arguent k n N is the order of the polynoial V (t) or W (t) and t is the noralized tie they are defined explicitly on the interval [] as: [] kk+ kk+ 3 ΨΨ nnnn = vv ππ nn ( kk 4 t n ΨΨ nnnn = ww ππ nn ( kk t n) for t [ n n+ ] = n= k k k - and Ψ 3 4 n =Ψ n = otherwise and the weight function: w = (k t n) w ( k t n) = (k t n) ( k t n) A function nd the function f(t) defined over [] ay be expanded in ters of chebyshev wavelets as: n= = Ψ n f(t) = c n 34 (t) where c n = w f(t) Ψ 34 n (t) dt k n= = Ψ 34 n (t) f(t) = c n c = [c c c c k c k ]T Ψ 34 n (t) = [Ψ 34 Ψ Ψ k Ψ k Ψ k 34 Ψ k ] T 4 New Relation between Operational atrices of Fractional Derivative for ΨΨ nnnn kk+ ΨΨ nnnn (t) = ππ UU kk tt nn tt [ nn nn+ kk kk ww (t) and ΨΨ 33 nnnn (t) = n= kk - UU (t) = ππ UU (t) kk+ ΨΨ 3 nnnn (t) = VV kk tt nn tt [ nn nn+ ππ kk kk ] ww = n= kk - VV (t) = ππ VV (t) VV ( kk tt nn) = UU ( kk tt nn) - UU ( kk tt nn) ππ VV ( kk tt nn) = ππ UU ( kk tt nn) - ππ UU ( kk tt nn) VV kk tt nn = UU kk tt nn - UU kk tt nn VV ππ kk tt nn = UU ππ kk tt nn - ΨΨ 3 nnnn (t) = ΨΨ nnnn (t) - ΨΨ nnnn (t) Theore 4: Let ΨΨ nnnn 3 ππ UU kk tt nn (t) be third kind chebyshev vector and suppose (α >) then: + ( w + r w r ) DD ΨΨ 3 nnnn (t) = ππ DD (UU kk tt nn -UU kk tt nn ) = ΨΨ 3 nnnn (t) where is the (+)x(+) operational atrix derivative of order in the caputo sense and defined as follow 4 IJA All Rights Reserved 88

9 and Fourth Chebyshev Wavelets / IJA- 5(7) July-4 = where + ( w + r w + p r = 8 w p r = w + r ) ( w + r w r ) ( w r w r ) ππ l= + ( w + r + + w + r ) ( w + r w r ) ( w r w r ) rrl( )pp + + (l+rr) (rr )! ( +rr)!(pp+l)! (rr+l) pp+ rr +l+ ( + rr)! rr! (rr )(pp+ l)! l! (rr+l +) pp+ + ( w + r + + w + r ( w + r w r ) ( w r w r ) ππ rr l( )pp+ + (l+rr) (rr )! ( + rr)! (pp + l)! (rr+l) rr +l + ( rr)! rr! (rr )(pp + l)! l! (rr + l +) l= Note that in α the first α + rows are all zero Proof: Frotheore() we have that DD UU kk tt nn = DD UU ( kk tt nn) 8 rrl( )pp + + (l+rr) (rr )! ( +rr)!(pp+l)! (rr+l) + rr +l+ pp+ rr= αα + pp= ππ l= and ( + rr)! rr! (rr )(pp+ l)! l! (rr+l +) pp+ = 8 ππ rr l( )pp+ + (l+rr) (rr )! ( + rr)! (pp + l)! (rr+l) rr +l + ( rr)! rr! (rr )(pp + l)! l! (rr + l +) rr= αα + pp= l= Then DD UU kk tt nn DD UU kk tt nn = + rr= αα + pp= rr= αα + ( pp= w + p r ) UU ( kk tt nn) ( w p r )UU ( kk tt nn) ) Thus DD ΨΨ nnnn 3 (t)= + w ππ + p r p= w p r UU ( kk t n) then DD ΨΨ 3 nnnn (t)= [ ( + ww +rr + r w r ) ( r ww +rr + w r ) ( r ww +rr w r ]ΨΨ 3 nnnn (t) and DD ΨΨ 3 nnnn (tt) = ππ (DD UU ( kk tt nn) DD UU ( kk tt nn)) = [ ] Ψ n 3 (t) n 4 New Relation between Operational atrices of Fractional Derivative for ΨΨ nnnn Ψ nnnn (tt) = ππ UU ( kk tt nn) tt nn nn + kk oo ww = nn = kk kk (t) and ΨΨ 44 nnnn (t) 4 IJA All Rights Reserved 89

10 and Fourth Chebyshev Wavelets / IJA- 5(7) July-4 UU (tt) = ππ UU (tt) kk+ Ψ 4 nnnn (tt) = ππ WW ( kk tt nn) tt nn nn + kk oo ww = nn = kk WW (tt) = ππ WW (tt) kk WW ( kk tt nn) = UU ( kk tt nn) + UU ( kk tt nn) ππ WW ( kk tt nn) = ππ UU ( kk tt nn) + UU ( kk tt nn) WW ( kk tt nn) = UU ( kk tt nn) + UU ( kk tt nn) ππ WW ( kk tt nn) = ππ UU ( kk tt nn) + ππ UU ( kk tt nn) kk+ ππ WW ( kk tt nn) = Ψ nnnn (tt) Ψ nnnn (tt) Ψ 4 nnnn (tt) = Ψ nnnn (tt) Ψ nnnn (tt) Theore 4: Let φφ 4 nnnn (tt) be fourth kind chebyshev vector and suppose > Then: DD Ψ 4 nnnn (tt) = ππ DD UU ( kk tt nn) + UU ( kk tt nn) = ( ) Ψ 4 nnnn (tt) where ( ) is the ( + ) ( + ) operational atrix derivative of order in the Caputoense and defined as follow: ( ) = where + ( w + r + + w + r ) ( w + r w r ) w + p r = 8 w p r = 8 ( w r w r ) ππ l= pp+ + ( w + r + 4 IJA All Rights Reserved 9 + w + r ( w + r w r ) ( w r w r ) rrl( )pp + + (l+rr) (rr )! ( +rr)!(pp+l)! (rr+l) pp+ rr +l+ ( + rr)! rr! (rr )(pp+ l)! l! (rr+l +) ) + ( w + r + + w + r ( w + r w r ) ( w r w r ) ππ rr l( )pp+ + (l+rr) (rr )! ( + rr)! (pp + l)! (rr+l) rr +l + ( rr)! rr! (rr )(pp + l)! l! (rr + l +) l= )

11 and Fourth Chebyshev Wavelets / IJA- 5(7) July-4 Note that in α the first α + rows are all zero 43 New Relation between Operational atrices of Fractional Derivative forψψ nnnn Ψ nnnn (tt) = ππ UU ( kk tt nn) tt nn nn + kk oo ww kk = nn = kk UU (tt) = ππ UU (tt) (tt) ΨΨ 33 nnnn (tt) aaaaaa ΨΨ 44 nnnn (tt) kk+ Ψ 3 nnnn (tt) = ππ VV ( kk tt nn) tt nn nn + kk oo ww kk = nn = kk VV (tt) = ππ VV (tt) kk+ Ψ 4 nnnn (tt) = ππ WW ( kk tt nn) tt nn nn + kk OO WW kk = nn = kk WW (tt) = ππ WW (tt) UU ( kk tt nn) = VV ( kk tt nn) + WW ( kk tt nn) ππ UU ( kk tt nn) = ππ VV ( kk tt nn) + ππ WW ( kk tt nn) ππ UU ( kk tt nn) = ππ VV ( kk tt nn) + ππ WW ( kk tt nn) kk+ ππ UU ( kk tt nn) = ππ VV ( kk tt nn) + ππ WW ( kk tt nn) Ψ nnnn (tt) = Ψ 3 nnnn (tt) + Ψ 4 nnnn (tt) kk+ Theore 43: Let Ψ nnnn (tt)ψ 3 nnnn (tt)andψ 4 nnnn (tt) are shifted second third and fourth kind chebyshev vector respectively and suppose > Then: kk+ DD Ψ nnnn (tt) = ( ) Ψ nnnn (tt) = ππ DD (vv ( kk tt nn) + ww ( kk tt nn)) = ( ( ) φφ 3 nnnn (tt) + ( ) Ψ 4 nnnn (tt)) where ( ) is the ( + ) ( + ) operational atrix derivative of order in the Caputo derivative 44 New Relation between Operational atrices of Fractional Derivative for ΨΨ nn kk Ψ nn (tt) = ππ TT ( kk tt nn ) tt nn kk nn kk oo ww where (tt) aaaaaa ΨΨ nnnn (tt) ππ TT (tt) = TT (tt) = ππ TT (tt) > 4 IJA All Rights Reserved 9

12 and Fourth Chebyshev Wavelets / IJA- 5(7) July-4 Ψ nnnn (tt) = ππ UU ( kk tt nn) tt nn nn + kk oo ww kk UU (tt) = ππ UU (tt) TT ( kk tt nn) = UU ( kk tt nn) + UU ( kk tt nn) = 3 ππ TT ( kk tt nn) = ππ UU ( kk tt nn) ππ UU ( kk tt nn) TT ( kk tt nn) = UU ( kk tt nn) + UU ( kk tt nn) ππ TT ( kk tt nn) = ππ UU ( kk tt nn) + ππ UU ( kk tt nn) cc = = 4 Ψ CC nn (tt) = Ψ nnnn (tt) Ψ nnnn (tt) = 3 Theore 44: Let ΨΨ nn (tt) be fourth kind chebyshev vector and suppose > Then: DD ΨΨ nn (t)(tt) = CC 4 ππ DD UU ( kk tt nn) + UU ( kk tt nn) = ( ) ΨΨ nnnn (tt) where ( ) is the ( + ) ( + ) operational atrix derivative of order in the Caputo sense and defined as follow: ( ) CC = 4 + ( w + r + + w + r ) ( w + r w r ) ( w r w r ) + ( w + r + + w + r ( w + r w r ) ( w r w r ) ) + ( w + r + + w + r ( w + r w r ) ( w r w r ) ) where w + p r = 8 ππ l= rrl( )pp + + (l+rr) (rr )! ( +rr)!(pp+l)! (rr+l) pp+ rr +l+ ( + rr)! rr! (rr )(pp+ l)! l! (rr+l +) pp+ w p r = 8 ππ rr l( )pp+ (l+rr) (rr )! ( + rr)! (pp + l)! (rr+l) rr +l + ( rr)! rr! (rr )(pp + l)! l! (rr + l +) l= Note that in α the first α + rows are all zero 4 IJA All Rights Reserved 9

13 and Fourth Chebyshev Wavelets / IJA- 5(7) July-4 Proof: DD UU kk tt nn = DD UU kk tt nn = 8 rrl( )pp + + (l+rr) (rr )! ( +rr)!(pp+l)! (rr+l) + rr +l+ pp+ rr= αα + pp= ππ l= and ( + rr)! rr! (rr )(pp+ l)! l! (rr+l +) rr= αα + 8 rrl( )pp + (l+rr) (rr )! ( +rr)!(pp+l)! (rr+l) rr +l+ pp+ pp= ππ l= ( rr)! rr! (rr )(pp+ l)! l! (rr+l +) Then DD UU kk tt nn DD UU kk tt nn = + ( pp= w + p r ) UU ( kk tt nn) ( w p r )UU ( kk tt nn) Thus rr= αα + pp= rr= αα + DD ΨΨ nn (t)= CC + w 4 ππ + p r p= w p r UU ( kk t n) then + DD ΨΨ nn (t)= CC [ ( ww 4 +rr r w r ) ( r ww +rr + w r ) ( r ww +rr w r ]ΨΨ nnnn (t) and DD ΨΨ nn (tt) = CC 4 ππ (DD UU kk tt nn DD UU kk tt nn ) = CC [ ] Ψ 4 n (t) n REFERENCES [] Abd-Elhaeed W Doha E H and Youssri Y H "New Wavelets Collocation ethod for Solving Second- Order ultipoint Boundary Value probles Using Chebyshev Polynoials of Third and Fourth Kinds" Hindawi Publishing Corporation Abstract and Applied Analysis Vol 3 Article ID pages [] Abd-Elhaeed W Doha E H and Youssri Y H "New Spectral Second Kind Chebyshev Wavelets Algorithfor Solving Linear and Nonlinear Second- Order Differential Equations Involving Singular and Bratu Type Equations" Hindawi Publishing Corporation Abstract and Applied Analysis Vol 3 Article ID pages [3] Azizi H Loghani G B "Nuerical Approxiation For space Fractional Diffusion equations Via Chebyshev Finite Difference ethod" Journal of Fractional Calculus and Applications vol 4() July 3 pp 33-3 [4] BhrawyAHAlofi AS "The operational atrix of fractional integration for shifted Chebyshev polynoials" Applied atheatics Letters 6(3) 5-3 [5] Biazar J Ebrahii H "A Strong ethod for Solving Syste of Integro-Differential Equations "Applied atheatical Vol() 5-3 [6] ChiK& Liu "A Wavelet approach to fast approxiation of power Electronics circuits" International Jornal of Circuit theory and Applications 3(3) 59-6 [7] Darani A Nasiri A fractional type of the Chebyshev polynoials for approxiation of solution of linear fractional differential equations" Vol No 3 pp 96-7 [8] Diethel K "The analysis of fractional differential equations" Berlin: Springer-Verlag [9] Doha E H Bhrawy AH Ezz-Eldien SS "A Chebyshev spectral ethod based on operational atrix for initial and boundary value probles of fractional order" Coput ath Appl 6() [] Fox Lslie and Ian Bax Parker "Chebyshev Polynoials in Nuerical Analysis" Oxford university press London vol [] Jie WLiu "Application of Wavelet Transfor to steady-state Approxiation of power electronics Wavefors" Departent of Electronic Hong Kong polytechnic University (3) [] Heydari HHooshandasl R aalekghaini F and ohaadi F "Wavelet Collocation ethod for Solving ultiorder Fractional Differential Equations" Hindawipuplishing Corporation journal of applied atheatics vol /Article ID IJA All Rights Reserved 93

14 and Fourth Chebyshev Wavelets / IJA- 5(7) July-4 [3] KazeiNasab Kilican A pashazadehatabakan Z and Abbasbandy S "Chebyshev Wavelet Finit Difference ethod: A New Approach for Solving Initial and Boundary Value Proble of Fractional Order" Hindawi Publishing Corporation Abstract and Applied Analysis vol 3 Article ID pages [4] Lakestani Dehghan Irandoust-pakchin S "The construction of operational atrix of fractional derivatives using B-spline functions" Coun Nonlinear Sci Nuer Siulat 7 () 49-6 [5] Li X "Nuerical solution of fractional differential equations using cubic spline wavelet collocation ethod Coun Nonlinear Sci Nuer Siulat 7() [6] iller KS Ross B "An Introduction to the Fractional calculus and Fractional Differential Equations" Wiley New York 993 [7] NkwantaA and Barnes ER "Two Catalan-type Riordan arrays and their connections to the Chebyshev polynoials of the first kind" Jornal of Integer Sequences 5()-9 [8] Oldha KBSpanier J The Fractional Calculus" Acadeic press New York 974 [9] Podlubny I Fractional Differential Equations" Acadeic Press San Diego 999 [] Saadatandi A Dehghan "A new operational atrix for solving fractional- order differential equations" Coput ath Appl 59 () [] Saadatandi A Dehghan zizi R The Sine-Legendre collocatio n ethod for a class of fractional convection-diffusion equations with variable coefficients Coun Nonliner Sci Nuer Siulat 7() [] Seifollahi Shaloo A S "Nuerical Solution of Nonlinear ulti-order Fractional Differential EQUATIONS BY Operational atrix of Chebyshev Polynoials" World Applied Prograing Vol(3) Issue(3) arch [3] Sohrabi S "Coparison Chebyshev Wavelets ethod with BPFs ethod For Solving Abel's Integral equation" in Shas Engineering Journal Vol () Source of support: Nil Conflict of interest: None Declared [Copy right 4 This is an Open Access article distributed under the ters of the International Journal of atheatical Archive (IJA) which perits unrestricted use distribution and reproduction in any ediu provided the original work is properly cited] 4 IJA All Rights Reserved 94