mathematics Article O the Geeralized Riesz Derivative Chekua Li * ad Joshua Beaudi Departmet of Mathematics ad Computer Sciece, Brado Uiversity, Brado, MB R7A 6A9, Caada; beaudijd3@bradou.ca * Correspodece: lic@bradou.ca Received: 3 Jue ; Accepted: July ; Published: 3 July Abstract: The goal of this paper is to costruct a itegral represetatio for the geeralized Riesz derivative RZ Dx s ux) for k < s < k + with k =,,, which is proved to be a oe-to-oe ad liearly cotiuous mappig from the ormed space W k+ R) to the Baach space CR). I additio, we show that RZ Dx s ux) is cotiuous at the ed poits ad well defied for s = + k. Furthermore, we exted the geeralized Riesz derivative RZ Dx s ux) to the space C k R ), where k is a -tuple of oegative itegers, based o the ormalizatio of distributio ad surface itegrals over the uit sphere. Fially, several examples are preseted to demostrate computatios for obtaiig the geeralized Riesz derivatives. Keywords: Riesz derivative; fractioal Laplacia; ormalizatio; distributio; Gamma fuctio. Itroductio Durig the past few decades, fractioal calculus 4 has bee a emerget tool which uses fractioal differetial ad itegral equatios to develop more sophisticated mathematical models that ca accurately describe complex systems. Fractioal powers of the Laplacia operator arise aturally i the stu of partial differetial equatios related to aomalous diffusio, where the fractioal operator plays a similar role to that of the iteger-order Laplacia for ordiary diffusio 5,6. By replacig Browia motio of particles with Lévy flights 7, oe obtais a fractioal diffusio equatio i terms of the fractioal Laplacia operator 8 of order s, ) via the Cauchy pricipal value P.V. for short) itegral 9, give as ) s ux) = C,s P.V. where = / x + + / x, ad the costat C,s is give by ) cos y C,s = R y +s = π / ux) uζ) dζ, ) R x ζ +s +s s Γ ) s. ) Γ s) Let x = x, x,, x ) R. For a give -tuple α = α, α,, α ) of oegative itegers or called a multi-idex), we defie α = α + α + + α, α! = α!α! α! x α = x α xα xα, α u = α α α u = x α xα α u. xα The Schwartz space SR ) space of rapidly decreasig fuctios o R ) is the fuctio space defied as SR ) = { ux) C R ) : ux) α,k C α,k cost) α, k N }, Mathematics, 8, 89; doi:.339/math8789 www.mdpi.com/joural/mathematics
Mathematics, 8, 89 of where N = {} N is the set of oegative itegers ad ux) α,k = sup x α k ux). x R Let r = x + x + + x. The fuctio space C k R ) is defied i Referece as follows. C k R ) = { ux) is bouded ad k ux) is cotiuous o R : M k cost) >, such that k ux) M } k as r where k = k, k,, k ) is a -tuple of oegative itegers. Applyig the ormalizatio i distributio theory, Pizzetti s formula, ad surface itegrals o R, Li very recetly exteded the fractioal Laplacia ) s ux) over the space C k R ) which cotais SR ) as a proper subspace) for all s > ad s =,,, ad obtaied Theorem below. Theorem. Let i =,, ad i < s < i +. The the geeralized fractioal Laplacia ) s is ormalized over the space C k R ) as ) s ux) = C,s r s Sr) r Ω ux) r i Ω i ux) i i! + ) + i ) dr, 3) where Ω = π / /Γ ) is the area of the uit sphere Ω R, k = k, k,, k ) with k + + k = i +, ad Sr) = ux + rσ) ux) + ux rσ)dσ. I particular for =, we have the followig. Ω Theorem. Let k < s < k + ad k =,,,. The the fractioal Laplacia operator ) s is ormalized over C k+ R) as ) s ux) = C,s y s Sy) u ) x)y yk k)! uk) x), 4) r where Sy) = ux + y) ux) + ux y). Defiitio. For a sufficietly ice fuctio ux) defied o R the left- ad right- sided Riema-Liouville derivatives of order α, with m < α < m N, give by RLD α,xux) = Γm α) d m x dx m ut) dt, t x) α m+ ad respectively. RLD α x,ux) = )m Γm α) d m dx m x ut) dt t x) α m+
Mathematics, 8, 89 3 of From itegratio by parts we have α m RL D,xux) α = u m) x), ad α m ) + RL D,xux) α = u m ) x). Defiitio. The α-order Riesz derivative of a fuctio ux) x R) is defied as RZD α xux) = Ψ α RLD α,x + RL D α x,) ux) where for α =, 3,. Ψ α = cos απ I geeral, the followig defiitio regardig the Riesz derivative o R ca be give. Defiitio 3. The Riesz fractioal derivative is defied for suitably smooth fuctio ux) x R ) i arbitrary dimesios by, RZDxux) α = l yu)x) d,l α) R y +α, < α < l where l is a arbitrary iteger bigger tha α, ad l yu)x) deotes either the cetred differece or o-cetred differeces l yu)x) = l yu)x) = l k= ) k ) l ux + l/ k)y), k l k= ) k ) l ux ky). k The d,l α) are ormalizig costats which are idepedet of the choice of l > α, ad are aalytic fuctios with respect to the parameter α by d,l α) = α π +/ Γ + α ) ) A lα) + α πα ), Γ si ad l k= ) k ) l k α, i the case of o-cetred differece, k A l α) = l k= ) k ) ) l l α k k, i the case of cetred differece, for a eve umber l > α. It is well kow that the Riesz derivative plays a importat role i aomalous diffusio 3 5 ad space of fractioal quatum mechaics. For example, the Riesz derivative satisfies the fractioal diffusio equatio, which has lots of physical applicatios 3: P L x, t; α) t σ α RZ D α x P L x, t; α) =,
Mathematics, 8, 89 4 of where P L x, t; α) is the α-stable Lévy distributio ad α, < α, is called the Lévy idex. There are also may studies, icludig umerical aalysis 6 9, scietific computig ad Fourier trasform methods,, o differetial equatios ivolvig the Riesz derivative with applicatios i several fields, icludig mathematical physics ad egieerig. It is widely cosidered that the Riesz derivative is equivalet to the fractioal Laplacia i arbitrary dimesios 4. Cai ad Li 5 showed that for s, ) ) s ux) = RZ Dx s ux), ux) SR) ad s = /, ) s ux) = RZ Dx s ux), ux) SR ) with >. Furthermore, o page 5 ad 6 i the same referece they stated i) for the case with α = 3, 5,, the Riesz derivative of the give fuctio ux) x R) ca be defied i the form ) RZDxux) α = α αγ +α ux + y) ux) + ux y) π Γ α ) y +α, which is suitable for positive values of α =, 4, 6,. ii) For k =,,. The, α 4k+ RZ Dxux) α = 4k 4k + )Γk + ) ) π / Γ 4k ad for k =,,,, α 4k+3 RZ Dxux) α = 4k+ 4k + 3)Γk + ) ) π / Γ +4k ux + y) ux) + ux y) y 4k+, ux + y) ux) + ux y) y 4k+4. We would like to recosider cases i) ad ii) i this paper as the itegrals o the right-had side do ot exist eve for a sufficietly good fuctio ux) SR). Ideed, by Taylor s expasio ux + y) ux) + ux y) = u x + θy) + u x θy) y u x)y as y +! where θ, ). This clearly makes all the itegrals o the right-had side diverget ear the origi. As outlied i the abstract, we establish a itegral represetatio for the geeralized Riesz derivative RZ Dx s ux) for k < s < k + with k =,,, as a liearly cotiuous mappig from the ormed space W k+ R) to the Baach space CR). The we stu the geeralized Riesz derivative i arbitrary dimesios ad further show that RZ Dx s ux) is cotiuous at the ed poits based o the ormalizatio of distributio ad the surface itegrals. I particular, the derivative RZ Dx s ux) is well defied for all s = k +, which exteds Defiitio.. The Geeralized Riesz Derivative o R Let CR) be the space of cotiuous fuctios o R give as CR) = {ux) : ux) is cotiuous o R ad u < } where u = sup ux). x R
Mathematics, 8, 89 5 of Clearly, CR) is a Baach space. The followig space will play a importat role i defiig the geeralized Riesz derivative o R. Let k =,,. We defie the ormed space W k R) as where Clearly, ux) = W k R) = u k = max { } ux) : u k) x) is cotiuous o R ad u k < { sup x R xux), sup x R x x + W kr) but ux) = } xu x), sup x + )u k) x). x x + x R / SR), ad SR) W k R) C k R) CR) for all k =,,. We are rea to prove the followig theorem which establishes a iitial equivalece betwee the Riesz derivative ad the fractioal Laplacia o the space W R). Theorem 3. Let u W R). The both RZ D s x ux) ad ) s ux) exist ad for < s <. RZD s x ux) = ) s ux) = C,s ux + y) ux) + ux y) y +s Proof of Theorem 3. Makig the variable chage z = x ζ, we derive from Equatio ) that for = ) ) s ux) ux z) ux) = C,s P.V. R z +s dz. Settig w = z o the right-had side of the above equality, we come to P.V. R ux) ux z) z +s dz = P.V. R ux) ux + w) w +s dw. Therefore, P.V. = P.V. R = P.V. ux) ux z) dz R z +s ux) ux z) z +s dz + P.V. R ux + y) ux) + ux y) y +s R ux) ux + w) dw w +s after relabelig y = z ad y = w. This implies that ) s ux) = C,s P.V. R ux + y) ux) + ux y) y +s. Note that the above itegral is well defied for ux) W R). Ideed, a secod order Taylor expasio ifers ux + y) ux) + ux y) y +s sup y R u y) y s.
Mathematics, 8, 89 6 of Hece, it is absolutely itegrable ear the origi. Furthermore, ux) W R) implies that there exists a costat C > such that y + )u y) C as y. as This idicates that the itegral is absolutely itegrable at ifiity. I summary, ) s ux) = C,s R = C,s ux + y) ux) + ux y) y +s ux + y) ux) + ux y) y +s, ux + y) ux) + ux y) y +s is a eve fuctio with respect to y. Assume < s < /. Itegratio by parts yields ux + y) ux) + ux y) y +s = ux + y) ux) + ux y) s y s + y= s = d ux + y) s dx y s d ux y) s dx y s by applyig the facts that all four itegrals ux + y) y s, ux y) y s, ad u x + y) y s are uiformly coverget with respect to x usig the coditios are bouded. Sice we come to From the formula 6 d dx d dx sup x R ) s ux) = C,sΓ s) s C,s = xux) ad sup xu x) x R u x + y) u x y) y s u x y) y s ux + y) y s = Γ s) RL Dx,ux), s ad ux y) y s = Γ s) RL D,xux) s ) RLDx,ux) s + RL D,xux) s. cos y y +s = Γ s) cosπs), s we have ) ) s ux) = Ψ s RLD,x s + RL Dx, s ux) = RZ Dx s ux).
Mathematics, 8, 89 7 of Fially we assume / < s <. Applyig we deduce that ) s ux) = = = d dx d dx ux + y) y s = Γ s) RL Dx,ux) s ad ux y) y s = Γ s) RL D,xux), s C,s ss ) d I particular for s = /, we have u x + y) + u x y) y s C,s ux + y) ss ) dx y s + d dx C,s ss ) Γ s) RLD,x s + RL Dx, s = C,s s Γ s) RLD s,x + RL D s x, RZD xux) = ) / ux) = C,/ = π which is well defied ad exteds Defiitio to the value α =. Remark. ux) ux y) y s ux) = RZ D s x ux). ux + y) ux) + ux y) y ux + y) ux) + ux y) y, a) Usig the formula Γz)Γ z + ) = π z Γz) for z =,,,, we have for u W R) that RZDx s ux) = ) s si πs ux) = Γ + s) π ux + y) ux) + ux y) y +s b) for < s <. This expressio has symbolically appeared i several existig literatures, such as Refereces 3,,4, for a suitable smooth fuctio ux). Cai ad Li preseted Theorem 3 i Referece 5 uder the coditios that ux) SR) which is a proper subspace of W R), ad s, ) with s = /. I order to stu the geeralized Riesz derivative, we briefly itroduce the followig basic cocepts i distributio ad the ormalizatio of x+ λ. Let DR) be the Schwartz space 7 of ifiitely differetiable fuctios or so-called the Schwartz space of testig fuctios) with compact support i R, ad D R) be the space of distributios liearly cotiuous fuctioals) defied o DR). Furthermore, we shall defie a sequece φ x), φ x),, φ m x), which coverges to zero i DR) if all these fuctios vaish outside a certai fixed ad bouded smooth set i R ad coverge uiformly to zero i the usual sese) together with their derivatives of ay order. We further assume that D R + ) is the subspace of D R) with support cotaied i R +. The fuctioal δ is defied as δ, φ) = φ), where φ DR). Clearly, δ is a liear ad cotiuous fuctioal o DR), ad hece δ D R).
Mathematics, 8, 89 8 of Let f D R). The the distributioal derivative f o D R) is defied as: f, φ) = f, φ ) for φ DR). I particular, δ m) x), φx)) = ) m φ m) ), where m is a oegative iteger. The distributio x+ λ o DR) is ormalized i Referece 7 as: x λ +, φx)) = x λ φx) φ) xφ ) xm m )! φm ) ) dx, 5) where m < λ < m m N) ad φ DR). Let τx) be a ifiitely differetiable fuctio o, +) R satisfyig the followig coditios: i) τx), ii) τx) = if x /, iii) τx) = if x. Let r = x + x + + x. We costruct the sequece I m r) for m =,, as: if r m, I m r) = m m ) τ + m +m r mm+ + m +m if r > m. Clearly, I m r) is ifiitely differetiable with respect to x, x,, x ad r, ad I m r) = if r m + m m, as m m + m +m m + m m ) mm+ =. + m+m Furthermore, I m r). Applyig Equatio 5) ad the idetity sequece I m r) for m =,,, Li established Theorems ad outlied i the itroductio. Based o Theorems ad 3, the geeralized Riesz derivative o R is well defied, for k < s < k + with k =,,,, as RZDx s ux) = C,s y s Sy) u ) x)y yk k)! uk) x), where ux) W k+ R), ad Sy) = ux + y) ux) + ux y). The followig theorem is to costruct a relatioship betwee the ormed space W k+ R) ad the Baach space CR) by the geeralized Riesz derivative. Theorem 4. Let k < s < k + with k =,,,. The the geeralized Riesz derivative RZ Dx s RZDx s ux) = C,s y s Sy) u ) x)y yk k)! uk) x) is a oe-to-oe ad liearly cotiuous mappig from W k+ R) to CR). give by
Mathematics, 8, 89 9 of Proof of Theorem 4. From the above itegral expressio, the geeralized Riesz derivative RZ Dx s is a liear mappig o the space W k+ R). Let u m x) W k+ R) ad u m x) i W k+ R). It follows from Taylor s expasio that S m y) = u m x + y) u m x) + u m x y) = u ) m x)y + + yk k)! uk) m x) + yk+ k + )! where θ, ). Clearly, u k+) m x + θy) + u m k+) ) x θy), RZD s x u m x) = = C,s k + )! C,s k + )! y s+k+ u k+) m y s+k y u k+) m x + θy) + u k+) ) m x θy) x + θy) + y u k+) ) m x θy). Therefore, RZ Dx s u m x) C,s k + )!k + s) sup u k+) m y) y R C +,s k + )!s k) sup y u k+) m y), y R u k+) which coverges to zero, as u m x) k+ implies that both sup y R m y) ad y sup y R u k+) m y) go to zero as m. It remais to show that RZ Dx s is oe-to-oe from W k+ R) to CR). Assume u x), u x) W k+ R) such that RZDx s u x) = RZ Dx s u x). This ifers that = Usig the formula 8 we arrive at = y s y s S y) u ) x)y yk k)! uk) x) S y) u ) x)y yk k)! uk) x) s k+) y s Γ s) = δk+) y), δ k+) y) S y) u ) x)y yk k)! uk) x) δ k+) y) S y) u ) x)y yk k)! uk) x).. Hece, S k+) ) = S k+) ),
Mathematics, 8, 89 of by otig that Evidetly, δ k+) y) which further claims that S y) u ) x)y yk k)! uk) x) S k+) ) = u k+) x) = u k+) x) = S k+) ), u x) = u x) + P k+ x), = S k+) ). where P k+ x) is a polyomial of degree k + i the space W k+ R), which must be zero due to the coditio sup xp k+ x) <. x R Remark. At this momet, we are uable to describe a subspace say C s R)) of CR) such that the geeralized Riesz derivative RZ Dx s is bijective ad liearly cotiuous mappig from W k+ R) to C s R). This further stu is of iterest sice we ca defie a iverse operatio of the Riesz derivative o C s R) if it exists. I additio, we have the followig theorem regardig the its at the ed poits for the geeralized Riesz derivative RZ D s x ux) over the space W k+ R). Theorem 5. Let ux) W k+ R) ad k < s < k + with k =,,,. The, s k+) RZ Dx s ux) = ) k u k+) x), ad s k + RZ Dx s ux) = ) k+ u k) x) i the space CR). I particular, for all k =,,. s k RZ Dx s ux) = ) k+ u k) x) Proof of Theorem 5. Let k < s < k + with k =,,,. The, RZ D s s k+) x ux) ) k u k+) x) = sup s k+) x R C,s y s Sy) u ) x)y yk k)! uk) x) ) k u k+) x). Usig s k+) Γ Γ s) Γ s) = k! ) k+, ad k + )! ) + k + k + )! = π, k+ k + )!
Mathematics, 8, 89 of we derive that ) C,sΓ s) = π / k+) k + )Γ s k+) + k + k! ) k+ k + )! = )k. Furthermore, the itegral coverges uiformly with respect to s. Hece, y s Sy) u ) x)y yk Γ s) k)! uk) x) s k+) = δ k+) y) y s Γ s) = S k+) ) = u k+) x). Sy) u ) x)y yk k)! uk) x) Sy) u ) x)y yk k)! uk) x) I summary, we get RZ D s s k+) x ux) ) k u k+) x) =, which implies that s k+) RZ Dx s ux) = ) k u k+) x) i the space CR). O the other had, RZ D s s k + x ux) ) k+ u k) x) = sup s k + x R C,s y s Sy) u ) x)y yk k)! uk) x) ) k+ u k) x). Usig we derive that Γ s) k )! ) = k, ad s k + Γ s) k)! ) Γ + k = k)! π k k! ) k )! C,sΓ s) = π / k kγ s k + + k ) k k)! = )k+.
Mathematics, 8, 89 of Thus, from y s + Γ s) = δs) y), s y s u ) x)y + + yk k)! uk) x) = Sy) u ) x)y + + yk k)! uk) x) δ s) y) s k + = s k + Ss) ) = S k) ) = u k) x) for s > k, ad it follows that Therefore, i the space CR). RZ D s s k + x ux) ) k+ u k) x) =. s k + RZ Dx s ux) = ) k+ u k) x) Remark 3. a) From Theorem 5, we have s k+ RZ Dx s ux) = s k+) + RZ Dx s ux) = s k+) RZ Dx s ux) = u 4k+) x) for all k =,,,, ad for all k =,,. b) Clearly for k =,,, s k RZ Dx s ux) = s k) + RZ Dx s ux) = RZDx k+ ux) = k+ k + /)k)! 4) k π y k s k) RZ Dx s ux) = u 4k) x) ux + y) ux) + ux y) u ) x)y yk k)! uk) x) usig the idetity Γ k + ) = 4)k k! π. k)! I particular, RZD xux) = π RZD 3 xux) = 6 π ux + y) ux) + ux y), ad y y 4 ux + y) ux) + ux y) u ) x)y. To ed off this sectio, we use the followig example to demostrate computatios of the geeralized Riesz derivative.
Mathematics, 8, 89 3 of Theorem 6. Let s > ad s =,,. The, Furthermore, ) Γ + s RZDx s e x = s π Γ s) se x j= x) j Γj s). j)! s + RZ Dx s e x = e x ad RZDx s e x k+ dk = ) s k dx k e x, where k =,,. Proof of Theorem 6. We first assume < s < 3. Lettig ux) = e x we come to By Theorem 4 as e x W 3 R)), RZD s x e x = C,s = C,s u ) x) = e x 4x e x, ad u 4) x) = e x 48x e x + 6x 4 e x. y s Sy) u ) x)y y4 4! u4) x) y s { e x+y) e x + e x y) u ) x)y y4 4! u4) x) }. Clearly, e x+y) e x + e x y) u ) x)y y4 4! u4) x) = e x+y) e x + e x y) e x 4x y + e x y y 4 e x + 4y 4 x e x 4 3 y4 x 4 e x = e x e y + y y4 ) + 4x y e x e y + y ) + 4 3 x4 y 4 e x e y ) + e x e y j=3 xy) j j)! usig e xy + e xy = xy) j j)! j= = + 4x y + 4 3 x4 y 4 + j=3 xy) j. j)! Makig the variable chage u = y, y s e y + y ) y4 = u s e u + u ) u du.
Mathematics, 8, 89 4 of Usig itegratio by parts, we get by otig that u s e u + u ) u du = e u + u u s u s + s u= u s e u + udu = u s e u + udu s e = u + u s s + ) u s + u s+ e u du s s + ) e = u s s + ) s + ) u s+ + u s+3 e u du ss ) s + ) = Γ s + 3) ss ) s + ) Γ s) = ss ), e u + u u u u s e u + u u u if < s < 3. Similarly, we obtai u s e u u s+ = u + e u + u u u s =, e = u + u u + u s =, ad = u + e u u s+ = y s e y + y ) Γ s) = s ), ) y 3 s e y = y j s e y = Γ s), Γj s), for j = 3, 4,.
Mathematics, 8, 89 5 of Hece, RZD s x e x = C,s e x y s e y + y y4 ) +4C,s x e x y s e y + y ) + 4 3 C,sx 4 e x y 3 s e y ) +C,s e x j=3 x) j j)! y j s e y = s π Γ + s ) e x + s+ s π Γ + s ) x e x ) + s+ π Γ s) s 3 Γ s) Γ + s x 4 e x ) Γ + s + s π Γ s) se x j=3 ) Γ + s = s π Γ s) se x j= x) j Γj s) j)! x) j Γj s). j)! Clearly, the series s π ) Γ + s Γ s) se x j= x) j Γj s) j)! ca be exteded to all values of s > ad s =,,. For example, a similar calculatio leads to RZDx s e x = s Γs + )e x + s+ sγs + ) x e x π π + s sγs + ) Γ s) π e x j= ) Γ + s = s π Γ s) se x x) j Γj s) j)! j= x) j Γj s) j)! if < s <. I additio, ) Γ + s s + RZ Dx s e x = s π s + Γ s) se x Γ s) + ) Γ + s = s π s + Γ s) se x Γ s) Γ + s + s π s + ) Γ s) se x j= x) j j)! Γj s) = e x, j= x) j Γj s) j)! by applyig the formula sγ s) = Γ s).
Mathematics, 8, 89 6 of Clearly for j =, 3,, k, Γj s) s k Γ s) = s k j s)j s) s)γ s) Γ s) = ) j k )k ) k j + ). Hece for k =,,, ) ) s k RZ Dx s e x = k π Γ + k e x + k+ k π Γ + k x e x ) k π k Γ + k x) e x j j)! )j kk ) k j + ) j= = ) k+ dk e x k dx by Theorem 5, which ca be verified directly by mathematical iductio. Remark 4. From the physicists Hermite polyomials give by d H x) = ) e x dx e x, we derive s k RZ Dx s e x = ) k+ e x H k x). 3. The Geeralized Riesz Derivative o R with I this sectio, we begi to stu the geeralized Riesz derivative RZ D s x ux) for s > o R, ad obtai its itegral represetatio usig Theorem metioed i the itroductio. I particular, we derive explicit itegral expressios for RZ D k+ x ux) whe k =,,,. Theorem 7. Let < s < ad k = k, k,, k ) be a -tuple of oegative itegers with k + + k =. The for ux) C k R ) defied i the itroductio), RZDx s ux) = ) s ux) = C,s r s Sr)dr 6) where Sr) is the surface itegral o the uit sphere Ω R, give by Sr) = Ω ux + rσ) ux) + ux rσ)dσ. Proof of Theorem 7. We let l = i the case of cetred differece from Defiitio 3 ad derive that yu)x) = k= ) k ) ux + k)y) = ux + y) ux) + ux y) k ad direct computatio implies that d,l s) = s sγ + s ) π Γ s) = C,s by makig use of the idetity Γ z)γz) = π si πz
Mathematics, 8, 89 7 of for ay o-iteger z. Hece, RZDx s ux) = C,s ux + y) ux) + ux y) R y +s, 7) which is well defied for ux) C k R ). Ideed, a secod order Taylor expasio derives ux + y) ux) + ux y) y +s D u L, < s <, y +s which is itegrable ear zero. Furthermore, ux) C k R ) implies that sup y D uy) y R is bouded as y. This deduces that the itegral coverges at ifiity. Usig the spherical coordiates below y = r cos θ y = r si θ cos θ y 3 = r si θ si θ cos θ 3 y = r si θ si θ cos θ y = r si θ si θ si θ, where the agles θ, θ,, θ rage over, π ad θ rages over, π. The Equatio 7) turs out to be where Clearly, the itegral RZDx s ux) = C,s Sr) = Ω r Sr) r +s dr = C,s ux + rσ) ux) + ux rσ)dσ. Sr) r +s dr = Sr) S) r +s dr Sr) dr, r+s coverges as S) = ad Sr) is a eve fuctio with respect to r. It follows from Theorem for < s < that RZDx s ux) = ) s ux) = C,s r s Sr)dr. Remark 5. There is a sig differece betwee Defiitio ad Defiitio 3 for =. Ideed for u W R) ad < s <, from Defiitio, ad RZD s x ux) = ) s ux) = C,s RZD s x ux) = ) s ux) = C,s ux + y) ux) + ux y) y +s ux + y) ux) + ux y) y +s
Mathematics, 8, 89 8 of by Equatio 7), which is directly from Defiitio 3. Let i =,, ad i < s < i +. Applyig Theorem 7 ad Theorem, we ca exted the geeralized Riesz derivative RZ Dx s over the space C k R ) as RZD s r s x ux) = ) s ux) = C,s Sr) r Ω ux) r i Ω i ux) i i! + ) + i ) where k = k, k,, k ) is a -tuple of oegative itegers with k + + k = i +. I particular, RZD xux) = Γ ) + RZDxux) 3 = 3Γ +3 RZD k+ x π + ) π + Sr) r dr, r 4 Sr) r Ω ux) ux) = k k + )k)!γ + + k π + 4) k k! Sr) r Ω ux) ) The followig theorem ca be foud i Referece. dr, r s r k Ω k ux) k k! + ) + k ) Theorem 8. Let ux) C k R ) with > ad i < s < i + for i =,,. The, dr, 8) dr. ux) = ) i+ i+ ux), s i+) )s ux) = ) i i ux) s i + )s ad where k = k, k,, k ) is a -tuple of oegative itegers ad k + k + + k = i +. From Theorem 8 we have s i+) RZ Dx s ux) = ) i+ i+ ux), s i + RZ Dx s ux) = ) i i ux). ad Hece, RZDx s ux) = ) i i ux). s i for i =,,. A a example, we are goig to compute RZ Dxux), where ux) = e x x. from Referece that Sr) = Ω ux + rσ) ux) + ux rσ)dσ = 4πe x x e r k= r k x + x )k k!). It follows
Mathematics, 8, 89 9 of usig The, RZDxux) = Sr) 4π r dr = e x x r e r + e r k= = e x x e x x k= e r r dr x + x )k k!) = πe x x e x x k= e r r k dr r k x + x )k k!) x + x )k k!) Γ k ) = π x πe x x e x x + x )k k )! k!) k= 4 k k )! Γ e r r dr = Γ /) = π, ) e r r k dr = Γ k, k ) k )! = π. 4 k k )! Let ux) C R ). The ux)i m r) has a compact support ad belogs to the space C k R ) for all -tuple of oegative itegers k where the idetity sequece I m r) is give i Sectio. Let i < s < i + with i =,,, ad set S m r) = Sr)I m r). Applyig Equatio 8) we ca defie the geeralized Riesz derivative RZ Dx s C R ) as over the space RZD s x ux) = ) s ux) = C,s m S m r) r Ω ux) m+m m r s if the it exists. To complete this sectio, we preset the followig theorem. Theorem 9. Let s > ad >. The RZ D s x x x ) = o R. r i Ω i ux) i i! + ) + i ) Proof of Theorem 9. We first ote that the fuctio x x C R ), but ot bouded. Clearly, x x ) = / x + + / x )x x ) = x, ) x x ) =. dr, 9)
Mathematics, 8, 89 of ad Assume < s < first. The from Equatio 9), RZDx s x x ) = m+m m C,s m To compute S m r) we come to Clearly, = C,s m m+m m r s S m r) r Ω ux) r s S m r) x r Ω dr. S m r) = I m r) ux + rσ) ux) + ux rσ)dσ Ω Ω ux + rσ) ux) + ux rσ) = x + rσ ) x + rσ ) x x + x rσ ) x rσ ) = 4x r σ σ + x r σ. x r σ dσ = x r where V is the volume of the uit ball i R. Furthermore, Ω Ω σ σ dσ = σ dσ = x r V = x r Ω, Ω σ σ dσ dσ = due to the itegral cacellatio over the uit sphere. Hece, dr Sr) = x r Ω, ad RZDx s x x ) = m C,s m C,s m r s Sr) x r Ω m+m m m = x Ω C,s m dr r s I m r)sr) x r Ω m+m m m r s I m r)dr =. dr It follows from ) x x ) = ) 3 x x ) = = that the result still holds for s >. 4. Coclusios A itegral represetatio is costructed for the geeralized Riesz derivative RZ Dx s ux) for k < s < k + with k =,, i arbitrary dimesios by applyig the ormalizatio of distributio ad the surface itegrals. We further show that RZ Dx s ux) is cotiuous at the ed poits ad well defied for s = + k. I additio, several examples are preseted to demostrate computatios for obtaiig the geeralized Riesz derivatives.
Mathematics, 8, 89 of Author Cotributios: Coceptualizatio, C.L. ad J.B.; methodology, C.L.; software, C.L.; validatio, C.L. ad J.B.; formal aalysis, C.L.; resources, C.L.; writig origial draft preparatio, C.L.; writig review ad editig, C.L.; visualizatio, C.L. All authors have read ad agreed to the published versio of the mauscript. Fudig: This work is supported by NSERC Caada 9-397) ad BURC. Ackowledgmets: The authors are grateful to the academic editor ad reviewers for the careful readig of the paper with productive suggestios ad correctios. Coflicts of Iterest: The authors declare o coflict of iterest. Refereces. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory ad Applicatios of Fractioal Differetial Equatios; Elsevier: North-Hollad, The Netherlads, 6.. Goreflo, R.; Maiardi, F. Fractioal Calculus: Itegral ad Differetial Equatios of Fractioal Order. I Fractals ad Fractioal Calculus i Cotiuum Mechaics; Spriger: New York, NY, USA, 997; pp. 3 76. 3. Podluby, I. Fractioal Differetial Equatios; Academic Press: New York, NY, USA, 999. 4. Srivastava, H.M.; Buschma, R.G. Theory ad Applicatios of Covolutio Itegral Equatios; Kluwer Academic Publishers: Dordrecht, The Netherlads; Bosto, MA, USA; Lodo, UK, 99. 5. Metzler, R.; Klafter, J. The radom walk s guide to aomalous diffusio: A fractioal amics approach. Phys. Rep., 339, 77. CrossRef 6. Metzler, R.; Klafter, J. The restaurat at the ed of the radom walk: Recet developmets i the descriptio of aomalous trasport by fractioal amics. J. Phys. A Math. Ge. 4, 37, R6. CrossRef 7. Madelbrot, B. The Fractal Geometry of Nature; Hery Holt ad Compay: New York, NY, USA, 98. 8. Kwaśicki, M. The equivalet defiitios for the fractioal Laplacia operator. Fract. Calc. Appl. Aal. 7,, 7 5. CrossRef 9. Saichev, A.I.; Zaslavsky, G.M. Fractioal kietic equatios: Solutios ad applicatios. Chaos Iterdiscip. J. Noliear Sci. 997, 7, 753 764, doi:.63/.667. CrossRef. Barros-Neto, J. A Itroductio to the Theory of Distributios; Marcel Dekker, Ic.: New York, NY, USA, 973.. Li, C. O the geeralized fractioal Laplacia. Submitted.. Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractioal Itegrals ad Derivatives: Theory ad Applicatios; Gordo ad Breach: Philadelphia, PA, USA, 993. 3. Bayı, S.Ş. Defiitio of the Riesz derivative ad its applicatio to space fractioal quatum mechaics. J. Math. Phys. 6, 57, 35. CrossRef 4. Cushma, J.H. Dyamics of Fluids i Hierarchical Porous Media; Academic Press: Lodo, UK, 99. 5. Beso, D.A. The Fractioal Advectio-Dispersio Equatio: Developmet ad Applicatio. Ph.D. Thesis, Uiversity of Nevada, Reo, NV, USA, 998. 6. Dig, H.F.; Li, C.P.; Che, Y.Q. High-order algorithms for Riesz derivaive ad their applicatios I). Abstr. Appl. Aal. 4, 4, 653797. CrossRef 7. Dig, H.F.; Li, C.P. High-order algorithms for Riesz derivative ad their applicatios V). Numer. Meth. Part. Differ. Equ. 7, 33, 754 794. CrossRef 8. Yag, Q.; Liu, F.; Turer, I. Numerical methods for fractioal partial differetial equatios with Riesz space fractioal derivatives. Appl. Math. Model., 34, 8. CrossRef 9. Huag, Y.; Oberma, A. Numerical methods for the fractioal Laplacia: A fiite differece-quadrature approach. SIAM J. Numer. Aal. 4, 5, 356 384. CrossRef. Muslih, S.I.; Agrawal, O.P. Riesz fractioal derivatives ad fractioal dimesioal space. It. J. Theor. Phys., 49, 7 75. CrossRef. Maiardi, F.; Luchko, Y.; Pagii, G. The fudametal solutio of the space-time fractioal diffusio equatio. Fract. Calc. Appl. Aal., 4, 53 9.. Miller, K.S.; Ross, B. A Itroductio to the Fractioal Calculus ad Fractioal Differetial Equatios; Wiley: Hoboke, NJ, USA, 993. 3. Pozrikidis, C. The Fractioal Laplacia; CRC Press: Boca Rato, FL, USA, 6. 4. Li, C.; Li, C.P.; Humphries, T.; Plowma, H. Remarks o the geeralized fractioal Laplacia operator. Mathematics 9, 7, 3. CrossRef 5. Cai, M.; Li, C.P. O Riesz derivative. Fract. Calc. Appl. Aal. 9,, 87 3. CrossRef
Mathematics, 8, 89 of 6. Gradshtey, I.S.; Ryzhik, I.M. Tables of Itegrals, Series, ad Products; Academic Press: New York, NY, USA, 98. 7. Gel fad, I.M.; Shilov, G.E. Geeralized Fuctios; Academic Press: New York, NY, USA, 964; Volume I. 8. Li, C. Several results of fractioal derivatives i D R + ). Fract. Calc. Appl. Aal. 5, 8, 9 7. CrossRef c by the authors. Licesee MDPI, Basel, Switzerlad. This article is a ope access article distributed uder the terms ad coditios of the Creative Commos Attributio CC BY) licese http://creativecommos.org/liceses/by/4./).