An Introductory Overview of Fractional-Calculus Operators Based Upon the Fox-Wright and Related Higher Transcendental Functions

. This survey-cum-expository review article is motivated essentially by the widespread usages of the operators of fractional calculus (that is, fractional-order integrals and fractional-order derivatives) in the modeling and analysis of a remarkably large variety of applied scienti(cid:28)c and real-world problems in mathematical, physical, biological, engineering and statistical sciences, and in other scienti(cid:28)c disciplines. Here, in this article, we present a brief introductory overview of the theory and applications of the fractional-calculus operators which are based upon the general Fox-Wright function and its such specialized forms as (for example) the widely- and extensively-investigated and potentially useful Mittag-Le(cid:31)er type functions.


@TQA
where n is given y @TIAF iquivlentlyD sine the reltionship @TQA n e written s followsX where n is givenD s in @TQAD y @TIAF e give elow three exmples of how frtionlEorder derivtives re potentilly useE ful in the modeling nd nlysis of pplied proE lemsF Example 1. he following (rstE nd seondE order liner ordinry di'erentil equtionsX re usully referred to s the relaxation equation nd the oscillation equationD respetivelyF elsoD in the theory of prtil di'erentil equtionsD the following prtil di'erentil equtionsX re known s the diusion (or heat) equation nd the wave equationD respetivelyF vet us rell tht the si proesses of reE lxtionD di'usionD osilltions nd wve propE gtion hve een revisited y severl uthors y introduing frtionlEorder derivtives in the governing @ordinry or prtilA di'erentil equE tionsF his leds to superslow or intermediate processes thtD in mthemtil physisD we my refer to s fractional phenomenaF he nlyE sis of eh of these phenomenD when rried out y mens of frtionl lulus nd vple trnsformsD involves suh speil funtions in one vrile s those of the wittgEve1er nd poxEright typesF hese useful speil funtions re investigted systemtilly s relevnt ses of the generl lss of funtions whih re popE ulrly known s pox9s HEfuntion fter ghrles pox @IVWU!IWUUA who initited detiled study of these funtions s symmetril pourier kerE nels @seeD for detilsD IPS nd IPVAF es mtE ter of ftD s we shll see in etion S of this rtileD mthemtil modeling nd nlysis of relEworld nd other pplied prolems re eing omplished widely nd extensively y mking use of frtionlEorder derivtives insted of posE itive integerEorder derivtivesF e now summrize elow some reent invesE tigtions y qoren)o et al. QT who did indeed mke referenes to numerous erlier closelyrelated works on this sujetF I. The Fractional (Relaxation-Oscillation) Ordinary Dierential Equation with v 0 ≡ 0 for ontinuous dependene of the solution on the prmeter α lso in the trnE sition from α = 1− to α = 1+Du eing the timeEderivtive of uF Explicit Solution @in both sesAX where E α (z) denotes the fmilir wittgEve1er funtion de(nedD s in @WAD y @seeD for exmpleD IPWD pF RPD iqution ssFS @PQAA II. The Fractional (Diusion-Wave) Partial Dierential Equation with g (x) ≡ 0 for ontinuous dependene of the solution on the prmeter β lso in the trnsiE tion from β = 1 2 − to β = 1 2 +F Explicit Solution (in both cases): whih n redily e expressed in terms of right9s generlized fessel funtion or the fesselEright funtion J µ ν (z) de(ned y @seeD for exmpleD IPWD pF RPD iqution ssFS @PPAA Example 2. prtionlEorder kineti equtions of di'erent forms hve een widely usedD in reent yersD in the modeling nd nlysis of severl importnt prolems of physis nd sE trophysisF sn ftD during the pst dede or soD frtionlEorder kineti equtions seem to hve gined populrity due minly to the disovery of their reltion with the theory of g @gontinuousEime ndom lksA in RTF hese equtions re investigted with the ojetive to determine nd interpret erE tin physil phenomen whih govern suh proE esses s di'usion in porous mediD retion nd relxtion in omplex systemsD nomlous di'uE sionD nd so on @see lso RQD SW nd IHUAF por n ritrry retionD whih is hrterE ized y timeEdependent quntity N = N (t)D it is possile to lulte the rte of hnge dN dt to e lne etween the destrution rte d nd the prodution rte p of N D tht isD fy mens of feedk or other intertion mehE nismD the destrution nd the prodution deE pend on the quntity N itselfD tht isD ine the destrution or the prodution t time t depends not only on N (t)D ut lso on the pst history N (η) (η < t) of the vrile N D suh deE pendene isD in generlD omplitedF his my e formlly represented y the following equE tion @see QWAX ruold nd wthi QW studied speil se of the eqution @UIA in the following formX is the numer density of speies j t time t = 0 nd the onstnt c j > 0F his is known s stndrd kineti equtionF he solution of the eqution @UPA is esily seen to e given y N j (t) = N 0 e −cj t , @URA whihD upon integrtionD yields the following lE terntive form of the eqution @UPAX is the stndrd integrl opertor nd c is onstnt of integrtionF he frtionlEorder generliztion of the eqution @USA is given s in the following form @see QWAX in terms of the fmilir rightEsided iemnnE viouville frtionl integrl opertor of order ν de(nedD s in @SRAD y @seeD for exmpleD SR nd UHY see lso PTA de(ned y por onsiderly lrge numer of extensions nd further generliztions of the frtionlE order kineti eqution @UTAD the interested reder should refer to SW nd IHU s well s the other relevnt referenes whih re ited in eh of these pulitionsF rereD in this exmpleD we propose to investiE gte solution of fmily of frtionlEorder kiE neti equtions whih re ssoited with the generl funtion E α,β (ϕ; z, s, κ) de(ned y @QPAD whih we hve introdued in this rtileF he results presented here re generl enough nd ple of eing speilized ppropritely to inE lude solutions of the orresponding @known or newA frtionlEorder kineti equtions ssoiE ted with simpler funtionsF Theorem 1. Let c, µ, ν, ρ, σ ∈ R + . Suppose also that the general function E α,β (ϕ; z, s, κ), dened by @QPA, exists. Then the solution of the following generalized fractional kinetic equation: is given by provided that the right-hand side of the solution asserted by @UWA exists.
Proof. pirst of llD y the vple gonvolution heoremD it is oserved from the de(nition @UUA tht husD in view of the vple trnsform formul @RVAD we (nd upon tking the vple trnsform of eh memer of the generlized frtionl kiE neti eqution @UVA tht e now invert the vple trnsform ourE ring in @VPA y using the following wellEknown identityX L t λ : s = Γ(λ + 1) s λ+1 e re thus led to the solution @UWA sserted y heorem IF his evidently ompletes the proof of heorem IF Remark 8. he distint dvntge of using the generl funtion E α,β (ϕ; z, s, κ), de(ned y @QPA, in the nonEhomogeneous term of the frtionlE order kineti eqution @UVA lies in its generlE ity so tht solutions of other kineti equtions involving reltively simpler nonEhomogeneous terms n e derived y ppropritely speilE izing the solution @UWA sserted y heorem IF Theorem 2. Let c, µ, ν, ρ, σ ∈ R + . Suppose also that the general function E α,β (φ; z), dened by @IRA, exists. Then the solution of the following generalized fractional kinetic equation: is given by provided that the right-hand side of the solution asserted by @VSA exists.
Proof. heorem Q n e provenD long the lines nlogous to those of our demonstrtion of heE orem I nd heorem QD y pplying the de(E nition @PVA nd the vple trnsform formul @SQAF e hoose to skip the detils invovedF Example 3. sn this third exmpleD we hoose to rell n erlier investigtion of n initilEvlue prolem in whih rilfer @see RQA onsidered the following eigenvlue eqution for the generl @rilfer9sA two-parameter frtionl derivtive opE ertor H D α,β 0+ of order α (0 < α < 1) nd type β (0 β 1) de(ned y the eqution @SAX under the initil ondition givenD in terms of the orresponding two-parameter frtionl integrl opertor H I α,β 0+ D y where it is titly ssumed tht c 0 eing given onstnt nd with the pE rmeter λ eing the eigenvlueF he ondition x > 0 in iqF @VVA ws not mentioned expliE itly y rilfer RQD pF IISD iqF @IIVAF rowE everD sine the iemnnEviouvilleD the viouvilleE gputo nd the rilfer opertors of frtionl lE ulus re ll de(ned y de(nite integrls over the oviously nonEempty intervl (0, x)D suh onE dition s x > 0 is titly ssumed to e stis(ed in ll developments involving eh of these opE ertors of frtionl lulusF sn terms of the twoEprmeter wittgEve1er funtion de(ned y @WAD rilfer9s over two dedes old solution of @VVA under the initil ondition @VWA is given y @see RQD pF IISD iqF @IPRAAX . @WHA xowD upon setting β = 0 nd c 0 = 1 in rilfer9s solution @VWA @withD of ourseD x > 0AD we re led to orreted version of the limed solution @see IRVD pF VHPD iqF @QFIAA of the following initilE vlue prolemX whereD just s in iqF @VWAD in the form given y in terms of the twoEprmeter wittgEve1er funtion de(ned y @WAF Remark 9. sn eh of the ove exmplesD we hve mde use of the lssil vple trnsform in solving the onsidered frtionlEorder ordiE nry nd prtil di'erentil equtionsF yther known or lssil integrl trnsforms n posE sily lso e suitly pplied in some of these sesF xeverthelessD it my e immensely nd potentilly helpful to investigte the possiility of developing some kind of n integrl or other trnsformtion whih would enle us to (nd solutions of frtionlEorder di'erentil equE tions y (rst reduing them to the orresponding integerEorder di'erentil equtionsF

Developments in Recent Years
sn reent yersD remrkly wide vriety of relEworld prolems nd issues in mny res hve een modeled nd nlysed y mking use of some very powerful toolsD one of whih involves pplitions of opertors of frtionl lulusF sn ftD suh importnt de(nitions hve een introdued for frtionlE order derivtivesD inludingD for exmpleD the iemnnEviouvilleD the qrünwldEvetnikovD the viouvilleEgputoD the gputoEprizio nd the etngnEflenu frtionlEorder derivtives @seeD for exmpleD IQD PRD PTD SRD VP nd ISSAF fy using the fundmentl reltions of the iemnnEviouville frtionl integrlD the iemnnEviouville frtionl derivtive ws onstrutedD whih involves the onvolution of given funtion nd powerElw kernel @seeD for detilsD SR nd VPAF he viouvilleE gputo @vgA frtionl derivtive involves the onvolution of the lol derivtive of given funtion with powerElw funtion PSF gputo nd prizio PR nd etngn nd flenu IQ proposed some interesting frtionlEorder derivtives sed upon the exponentil dey lw whih is generlized powerElw funtion @see SD VD IHD IID IP nd ISAF he gputoEprizio @gpgA frtionlEorder derivE tive s well s the etngnEflenu @efgA frtionlEorder derivtive llow us to desrie omplex physil prolems tht followD t the sme timeD the power lw nd the exponentil dey lw @seeD for detilsD SD VD IHD IID IP nd ISAF sn noteworthy erlier investigtionD riE vstv nd d IQU investigted the model of the gs dynmis eqution @qhiA y extending it to some new models involving the timeEfrtionl gs dynmis eqution @pqhiA with the viouvilleEgputo @vgAD gputoEprizio @gpgA nd etngnEflenu @efgA timeEfrtionl derivtivesF hey emE ployed the romotopy enlysis rnsform wethod @rewA in order to lulte the pproximte solutions of pqhi y using vgD gpg nd efg in the viouvilleEgputo sense nd studied the onvergene nlysis of rew y (nding the intervl of onvergene through the hEurvesF rivstv nd d IQU lso showed the e'etiveness nd ury of this method @rewA y ompring the pproximte solutions sed upon the vgD gpg nd efg timeEfrtionl derivtivesF gonsider the following homogeneous timeE frtionl gs dynmis eqution @pqhiAX rivstv nd d IQU used the rew @seeD for exmpleD SV nd WHA in order to solve the vgD gpg nd efg nlogues of the pqhi @WRAF hey otined these nlogous equtions y repling the timeEfrtionl derivtive ∂ α ψ ∂τ α in the pqhi @WRA y suessivelyD where the order α of the timeE frtionl derivtives is onstrined y n − 1 < α n (n ∈ N).
he orresponding vgD gpg nd efg timeE frtionl nlogues of the pqhi @WRA re given y denote the timeEfrtionl derivtives of order α for suitly de(ned funtion f(τ )D whih re de(nedD respetivelyD y is known s the efg timeEfrtionl derivtive of order α in the viouvilleEgputo sense givenD for is the wittgEve1er funtion nd M (α) is normliztion funtion with the sme properE ties s in the viouvilleEgputo @vgA nd the gputoEprizio @gpgA sesF por the detils of this nd other loselyErelted investigtionsD the interested reder should see the work y rivstv nd d IQUAF sn the urrent onslught of the goron virusD whih is referred to s gyshEIW @seeD for detilsD PD UTD VQ nd WWAF es in the se of the goron virusD the iol virus n e trnsmitted to others y ontt with infeted ody )uidsD through roken skinD or through the muous memrnes of the eyesD nose nd mouthD ut the iol virus n lso e trnsE mitted through sexul ontt with person who hs the virus or hs reovered from it @seeD for detilsD PHY see lso the reentlyEpulished works USD VID IHVD IPID IPQD IPRD IPUD nd IRS for the frtionlEorder modeling of 152 c 2021 Journal of Advanced Engineering and Computation (JAEC) other diseses nd other iologil situtionsAF prtionl lulus is generliztion of the lssil @or ordinryA lulus nd mny reserhers hve pid ttention to this siene s nd when they enounter numer of issues in the rel worldF wost of these issues do not hve ext nlytil solutionF his sitution nturlly interests mny reserhers to look for nd pply numeril nd pproximte methods to otin solutions y using suh methodsF here re mny useful methodsD suh s the homotopy nlysis @see PWD QH VW nd TQAD re9s vritionl itertion method @see RI nd RQAD edomin9s deomposition method @see RSD RR nd RTAD the pourier spetrl methods RUD (nite di'erene shemes @see RVAD ollotion methods @see SPD SQ nd VVAD nd so onF sn order to (nd more out the frtl lulusD we refer the reders to the investigtions in SR nd VPF wore reentlyD new onept ws introdued for the frtionlE order opertor euse this opertor hs two ordersD the (rst representing the frtionl order nd the seond representing the frtl dimensionF ome reent developments in the re of numeril tehniques n e found in @for exmpleA WID IRH nd IRIF yur ttention in this setion is now drwn towrd the ide of the frtlEfrtionl derivE tive of the omposite order (ρ, k) on the frtlEfrtionl iol virus @ppiAF ith this ojet in view @see WAD we reple the derivtive of integer order with respet to ζ y the frtlEfrtionl derivtives sed on the power lw @ppAD the exponentilElw @ppiA nd the wittgEve1er lw @ppwA kernels whih orrespond to the viouvilleEgputo @vgA @see VRAD gputoEprizio @gpA @see WQA nd the etngnEflenu @efA @see WRA frtionl derivtivesD respetivelyF rereD just s we hve lredy mentioned in etion R oveD we use the term Liouvile-Caputo fractional derivative in order to give due redit lso to viouville whoD in ftD onsidered suh frtionl derivtives mny dedes erlier in IVQPF his topi hs ttrted mny reserhers nd hs een pplied to reserhes stemming from vrious relEworld situtions @seeD for exmpleD IRF QRD TP nd IRWAF huring the pst severl yersD mny reE serhers9 fous hs een direted towrds modeling nd nlysis of vrious prolems in iomthemtil sienesF his rnh of siene represents mny distinguished dt on iologil phenomen suh s the iol nd other relted virusesD the nervous system nd its impulse trnsmissionD the teril ell nd its spredD et cetera @see RI nd IQVAF his hs led to the modeling of mny relEworld prolemsF es result of prolems tht rise from the rel world on the sis of sttistil nlysis nd iologil experimentsD mthemtE il models of these prolems re proposed nd most of them were studiedF hese proposed models enle sientists nd reserhers to study nd verify the ehvior of these models seprtely in iologil lortory experiment @see UD PID SU nd VSAF efter modeling the iologil phenomenon mthemtillyD tht isD s funtion of time nd the prmeters involvedD the numeril solutions n e found nd these solutions n then e represented in tles nd (guresF elsoD if the lortory results re villeD omprison etween theoretil nd lortory results n e mdeF he prmeters 'eting this system n lso e ontrolled ppropritelyF elsoD one of the dvntges of mthemtil modeling is the posE siility of reEstudying the prolems mny times nd t ny time vlue without reEexperimentingF e egin y introduing the epidemiologil model of the iol virus s followsX sn the following tleD we de(ne the indepenE dent vriles nd the prmeters of the iol virusF Symbol denition β1(ζ) The susceptible population β2(ζ) The infected population β3(ζ) The recovery population The rate of natural death The rate of death from the disease δ The rate of recovery from the disease he new model is otined upon repling the ordinry derivtive D ζ in the ove epidemiologE il model in @WVA to @IHIA y the orresponding frtlEfrtionl derivtive involving the power lw kernel s in the erlier work @see WAF where the funtions β i (ζ) (i = 1, 2, 3, 4) re ontinuous in the intervl (a, b) nd frtl difE ferentile on (a, b) with order kF he frtlE frtionl derivtive of β i (ζ) of order ρ in the viouvilleEgputo @vgA sense with the power lw re given y @see WA tust s we pointed out ove @nd lso in etion RAD we hve used the term viouvileE gputo sense4 in order to give due redit lso to viouville who onsidered suh frtionl derivtives mny dedes erlier in IVQPF por the relevnt detils of the numeril solutions for the ove model of the frtlE frtionl iol virusD the reder is referred to the work of rivstv nd d IQWF he detiled nlysis of this model nd the numeril solutionsD whih re presented in IQW nd in other works ited in IQW re potentilly ene(il to iologil reserhers with view to linking these (ndings to the iologil lortory resultsF pinllyD in this setionD we turn to the ft tht mny experiments nd theories hve shown tht lrge numer of norml phenomen tht ours in the engineering nd pplied siE enes n e wellEdesried y using disrete frtionl lulusF sn prtiulrD frtionl difE ferene equtions hve een found to provide powerful tools in the modeling nd nlysis of vrious phenomen in mny di'erent (elds of siene nd engineering suh s those inD for exE mpleD physisD )uid mehnis nd het onE dutionF gonsiderle ttention hs een given in the existing literture to the sujet of frE tionl di'erene equtions on the (nite time sles @seeD for exmpleD QSAF sn the urrent literture on this sujetD there re few ppers whih investigte the existene nd uniqueness of frtionl di'erene equtions in the sense of the iemnnEviouville @vA frtionl lulusF e hoose to rell here the work of vu et al. TS investigted the existene nd uniqueness of the following unertin frtionl forwrd di'erE ene eqution @pphiA given y where RL ψ−1 ∆ ψ denotes frtionl iemnnE viouville type forwrd di'erene with 0 < ψ 1D nd H 1 nd H 2 re two relEvlued funtions de(ned on [1, ∞]×R, z ∈ N 0 ∩[0, T ]D a 0 ∈ R is risp numerD nd ε ψ , ε ψ+1 , · · · , ε T +ψ re (T + 1) ssh unertin vriles with symmetril unE ertinty distriution L( 1 , 2 )F he ove work ws generlized y wohmmed UP ndD suseE quentlyD y rivstv nd wohmmed IQQF sn dditionD wohmmed et al. UQ otined the exE istene nd uniqueness of the nl se @kE wrdA of the equtions @IHVA nd @IHWAF wotiE vted y these developmentsD rivstv et al. IQR onsidered generl fmily of unertin frtionl di'erene equtions of the viouvilleE gputo type @pvghiAF hey derived n unE ertin frtionl sum equtionD whih is equivE lent to the pvghi y using the si properE ties of the viouvilleEgputo type unertin frE tionl di'erene equtionsF efter introduing suessive ird itertion method for (nding solution to the pvghiD rivstv et al. IQR pplied the theory of fnh ontrtion under the vipshitz onstnt ondition nd suessfully investigted the struture of the lgers of exE istene nd uniqueness of the pvghiF hey lso presented three exmples to show the e'eE tiveness of the proposed investigtion @seeD for detilsD rivstv et al. IQRAF