fractional calculus wikipedia fractional calculus contents historical note edit nature fractional derivative edit heuristics edit fractional derivative basic power function edit laplace transform edit fractional integral edit riemann liouville fractional integral edit hadamard fractional integral edit atangana baleanu fractional integral edit fractional derivative edit riemann liouville fractional derivative edit caputo fractional derivative edit caputo fabrizio fractional derivative edit atangana baleanu derivative edit riesz derivative edit type edit generalization edit erdélyi kober operator edit functional calculus edit application edit fractional conservation mass edit groundwater flow problem edit fractional advection dispersion equation edit time space fractional diffusion equation model edit structural damping model edit pid controller edit acoustical wave equation complex medium edit fractional schrödinger equation quantum theory edit variable order fractional schrödinger equation edit edit fractional theory edit note edit reference edit source edit reading edit article history fractional calculus edit book edit external link edit navigation menu personal tool namespaces variant view search navigation contribute tool print export language wikipedia free encyclopedia jump navigation jump search branch mathematical analysis fractional application derivative integral series article calculus fundamental theorem leibniz integral rule limit function continuity mean value theorem rolle theorem differential definition derivative generalization differential infinitesimal function total concepts differentiation notation second derivative implicit differentiation logarithmic differentiation related rate taylor theorem rules identity sum product chain power quotient rule inverse general leibniz faà di bruno formula integral lists integral integral transform definition antiderivative integral improper riemann integral lebesgue integration contour integration integral inverse function integration parts discs cylindrical shell substitution trigonometric weierstrass euler euler formula partial fraction change order reduction formula differentiate integral sign series geometric arithmetico geometric harmonic alternating power binomial taylor convergence test summand limit term test ratio root integral direct comparison limit comparison alternate series cauchy condensation dirichlet abel vector gradient divergence curl laplacian directional derivative identity theorem gradient green stokes divergence generalize stokes multivariable formalisms matrix tensor exterior geometric definitions partial derivative multiple integral line integral surface integral volume integral jacobian hessian specialized fractional malliavin stochastic variations glossary calculus glossary calculus list calculus topic v t e fractional calculus branch mathematical analysis study different possibility define real number power complex number power differentiation operator d d f x d d x f x integration operator j note j f x x f s d s develop calculus operator generalize classical context term power refer iterative application linear operator d function f repeatedly compose d d n f d d d d n f d d d n f d example ask meaningful interpretation d d analogue functional square root differentiation operator expression linear operator apply twice function effect differentiation generally look question define linear functional d real number way take integer value n ℤ coincide usual n differentiation d n power j n motivation introduction study sort extension differentiation operator d set operator power d ℝ define way continuous semigroup parameter original discrete semigroup d n n ℤ integer n denumerable subgroup continuous semigroup develop mathematical theory apply branch mathematic fractional differential equation know extraordinary differential equation generalization differential equation application fractional calculus historical note nature fractional derivative heuristics fractional derivative basic power function laplace transform fractional integral riemann liouville fractional integral hadamard fractional integral atangana baleanu fractional integral fractional derivative riemann liouville fractional derivative caputo fractional derivative caputo fabrizio fractional derivative atangana baleanu derivative riesz derivative type generalization erdélyi kober operator functional calculus applications fractional conservation mass groundwater flow problem fractional advection dispersion equation time space fractional diffusion equation model structural damping model pid controller acoustical wave equation complex medium fractional schrödinger equation quantum theory variable order fractional schrödinger equation fractional theory note reference source read article history fractional calculus books external link historical note edit applied mathematic mathematical analysis fractional derivative derivative arbitrary order real complex appearance letter write guillaume de gottfried wilhelm leibniz time leibniz write bernoulli brother describe similarity binomial theorem leibniz rule fractional derivative product function citation need fractional calculus introduce niels henrik abel early paper element find idea fractional order integration differentiation mutually inverse relationship understanding fractional order differentiation integration consider generalize operation unified notation differentiation integration arbitrary real order independently foundation subject lay liouville paper autodidact oliver heaviside introduce practical use fractional differential operator electrical transmission line analysis circa theory application fractional calculus expand greatly th th century numerous contributor give definition fractional derivative integral nature fractional derivative edit confuse fractal derivative th derivative function f x point x local property integer case non integer power derivative word non integer fractional derivative function f x x depend value f far away expect fractional derivative operation involve sort boundary condition involve information function fractional derivative function order define mean fourier mellin integral transform heuristics edit fairly natural question ask exist linear operator h half derivative h f x d f x d d x f x f x turn operator exist operator p p f x f x way definition d n y dx n extend real value n let f x function define x form definite integral x j f x x f t d t repeat process give j f x x j f t d t x t f s d s d t extend arbitrarily cauchy formula repeat integration j n f x n x x t n f t d t lead straightforward way generalization real n gamma function remove discrete nature factorial function give natural candidate fractional application integral operator j α f x γ α x x t α f t d t fact define operator straightforward j operator satisfie j α j β f x j β j α f x j α β f x γ α β x x t α β f t d t proof j α j β f x γ α x x t α j β f t d t γ α γ β x t x t α t s β f s d s d t γ α γ β x f s s x x t α t s β d t d s step exchange order integration pull f s factor t integration change variable r define t s x s r j α j β f x γ α γ β x x s α β f s r α r β d r d s inner integral beta function satisfy follow property r α r β d r b α β γ α γ β γ α β substitute equation j α j β f x γ α β x x s α β f s d s j α β f x interchange α β show order j operator apply irrelevant complete proof relationship call semigroup property fractional differintegral operator unfortunately comparable process derivative operator d significantly complex show d commutative additive general fractional derivative basic power function edit half derivative purple curve function f x x blue curve derivative red curve animation show derivative operator oscillate antiderivative α y x derivative α y simple power function y x continuously let assume f x monomial form f x x k derivative usual f x d d x f x k x k repeat give general result d d x x k k k x k replace factorial gamma function lead d d x x k γ k γ k x k k k k obtain half derivative function x d d x x γ γ x γ γ x π x demonstrate fact half derivative h f x df x repeat process d d x x π π γ γ x π γ γ x π x π γ π expect result d d x d d x x d d x x negative integer power k gamma function undefined use follow relation d d x x k γ k γ k x k k extension differential operator nee constrain real power example th derivative th derivative yield second derivative set negative value yield integral general function f x α complete fractional derivative d α f x γ α d d x x f t x t α d t arbitrary α gamma function undefined argument real negative integer imaginary zero necessary apply fractional derivative integer derivative perform example d f x d d f x d d d x f x laplace transform edit section cite source help improve section add citation reliable source unsourced material challenge remove learn remove template message come question laplace transform know l j f s l t f τ d τ s s l f s s l j f s l j f s s l f s s assert j α f l s α l f s example j α t k l γ k s α k γ k γ α k t α k k expect give convolution rule l f g l f l g shorthande p x x α clarity find j α f t γ α l l p l f γ α p f γ α t p t τ f τ d τ γ α t t τ α f τ d τ cauchy give laplace transform work relatively function useful solve fractional differential equation fractional integral edit riemann liouville fractional integral edit classical form fractional calculus give riemann liouville integral essentially describe theory periodic function include boundary condition repeat period weyl integral define fourier series require constant fourier coefficient vanish apply function unit circle integral evaluate zero riemann liouville integral exist form upper low consider interval b integral define d t α f t t α f t γ α t t τ α f τ d τ t d b α f t t b α f t γ α t b τ t α f τ d τ valid t valid t b contrast grünwald letnikov derivative start derivative instead integral hadamard fractional integral edit hadamard fractional integral introduce jacques hadamard give follow formula d t α f t γ α t log t τ α f τ d τ τ t d t atangana baleanu fractional integral edit recently generalize mittag leffler function atangana baleanu suggest new formulation fractional derivative nonlocal nonsingular kernel integral define b d t α f t b t α f t α b α f t α b α γ α t t τ α f τ d τ ab α normalization function ab ab fractional derivative edit unlike classical newtonian derivative fractional derivative define fractional integral fractional derivative gaussian interpolate continuously function derivative riemann liouville fractional derivative edit correspond derivative calculate lagrange rule differential operator compute n th order derivative integral order n α α order derivative obtain important remark n small integer great α n α similar definition riemann liouville integral derivative upper low variant d t α f t d n d t n d t n α f t d n d t n t n α f t t d b α f t d n d t n t d b n α f t d n d t n t b n α f t caputo fractional derivative edit option compute fractional derivative caputo fractional derivative introduce michele caputo paper contrast riemann liouville fractional derivative solve differential equation caputo definition necessary define fractional order initial condition caputo definition illustrate follow n α c d t α f t γ n α t f n τ d τ t τ α n caputo fractional derivative define d ν f t γ n ν t t u n ν f n u d u n ν n n advantage zero f t constant laplace transform express mean initial value function derivative caputo fractional derivative distributed order define b d ν f t b ϕ ν d ν f t d ν b ϕ ν γ ν t t u ν f u d u d ν φ ν weight function represent mathematically presence multiple memory formalism caputo fabrizio fractional derivative edit paper caputo fabrizio present definition fractional derivative non singular kernel function f t c give c f d t α f t α t f τ exp α t τ α d τ α atangana baleanu derivative edit like integral fractional derivative general mittag leffler function kernel author introduce version atangana baleanu caputo sense abc derivative convolution local derivative give function generalize mittag leffler function atangana baleanu riemann liouville sense abr derivative derivative convolution give function differentiable generalize mittag leffler function atangana baleanu fractional derivative caputo sense define b c d t α f t b α α t f τ e α α t τ α α d τ atangana baleanu fractional derivative riemann liouville define b r d t α f t b α α d d t t f τ e α α t τ α α d τ riesz derivative edit f α u x α k k α f u k f denote fourier transform type edit classical fractional derivative include grünwald letnikov derivative sonin letnikov derivative liouville derivative caputo derivative hadamard derivative marchaud derivative riesz derivative miller ross derivative weyl derivative erdélyi kober derivative new fractional derivative include coimbra derivative katugampola derivative hilfer derivative davidson derivative chen derivative caputo fabrizio derivative atangana baleanu derivative generalization edit erdélyi kober operator edit erdélyi kober operator integral operator introduce arthur erdélyi hermann kober give x ν α γ α x t x α t α ν f t d t generalize riemann liouville fractional integral weyl integral functional calculus edit context functional analysis function f d general power study functional calculus spectral theory theory pseudo differential operator allow consider power d operator arise example singular integral operator generalisation classical theory high dimension call theory riesz potential number contemporary theory available fractional calculus discuss erdélyi kober operator important special function theory kober erdélyi application edit fractional conservation mass edit describe wheatcraft meerschaert fractional conservation mass equation need model fluid flow control volume large compare scale heterogeneity flux control volume non linear referenced paper fractional conservation mass equation fluid flow ρ α u γ α δ x α ρ β s ϕ β w p t t groundwater flow problem edit atangana et al describe groundwater flow problem concept derivative fractional order work classical darcy law generalize water flow function non integer order derivative piezometric head generalize law law conservation mass derive new equation groundwater flow fractional advection dispersion equation edit equation clarification need show useful model contaminant flow heterogenous porous medium atangana kilicman extend fractional advection dispersion equation variable order equation work hydrodynamic dispersion equation generalize concept variational order derivative modified equation numerically solve crank nicolson method stability convergence numerical simulation show modify equation reliable predict movement pollution deformable aquifer equation constant fractional integer derivative time space fractional diffusion equation model edit anomalous diffusion process complex medium characterize fractional order diffusion equation model time derivative term correspond long time heavy tail decay spatial derivative diffusion nonlocality time space fractional diffusion governing equation write α u t α k δ β u simple extension fractional derivative variable order fractional derivative α β change α x t β x t application anomalous diffusion modeling find reference structural damping model edit fractional derivative model viscoelastic damping certain type material like polymer pid controller edit generalizing pid controller use fractional order increase degree freedom new equation relate control variable u t term measured error value e t write u t k p e t k d t α e t k d d t β e t p d α β positive fractional order k p k k d non negative denote coefficient proportional integral derivative term respectively denote p d acoustical wave equation complex medium edit propagation acoustical wave complex medium biological tissue commonly imply attenuation obey frequency power law kind phenomenon describe causal wave equation incorporate fractional time derivative u c u t τ σ α α t α u τ ϵ β c β u t β holm näsholm reference model link commonly recognize hypothesis multiple relaxation phenomenon rise attenuation measure complex medium link describe näsholm holm 2011b survey paper acoustic attenuation article holm nasholm paper compare fractional wave equation model power law attenuation book power law attenuation cover topic detail pandey holm give physical meaning fractional differential equation derive physical principle interpret fractional order term parameter acoustical medium example fluid saturate granular unconsolidated marine sediment interestingly pandey holm derive lomnitz law seismology nutting law non newtonian rheology framework fractional calculus nutting law model wave propagation marine sediment fractional derivative fractional schrödinger equation quantum theory edit fractional schrödinger equation fundamental equation fractional quantum mechanic follow form ℏ ψ r t t d α ℏ δ α ψ r t v r t ψ r t r r r r solution equation wavefunction ψ r t quantum mechanical probability amplitude particle give position vector r give time t ħ reduced planck constant potential energy function v r t depend system δ r laplace operator d α scale constant physical dimension d α j α m α s α kg α m α s α α d m particle mass m operator ħ δ α fractional quantum riesz derivative define ℏ δ α ψ r t π ℏ d p e ℏ p r p α φ p t r p r p p index α fractional schrödinger equation lévy index α variable order fractional schrödinger equation edit natural generalization fractional schrödinger equation variable order fractional schrödinger equation exploit study fractional quantum phenomena ℏ ψ α r r t t α r ℏ δ β t ψ r t v r t ψ r t r r r r r r t δ r laplace operator operator ħ δ β t variable order fractional quantum riesz derivative edit acoustic attenuation autoregressive fractionally integrate move average differintegral differential equation erdelyi kober operator neopolarogram nonlocal operator riemann liouville integral weyl integral fractional theory edit fractional dynamic fractional fourier transform fractional quantum mechanic note edit symbol j commonly instead intuitive order avoid confusion concept identify similar like glyph identity reference edit daniel zwillinger handbook differential equations elsevier science isbn katugampola udita october new approach generalized fractional derivatives pdf bulletin mathematical analysis applications arxiv bibcode k niels henrik abel oplösne af et par opgaver ved hjelp af bestemte integraler solution de quelques problème à définie solution couple problem mean definite integral pdf magazin naturvidenskaberne kristiania oslo igor podlubny richard magin irina trymorush niel henrik abel birth fractional calculus fractional calculus applied analysis arxiv doi s2cid cs1 maint multiple name author list link history subject thesis french stéphane dugowson les différentielle métaphysique histoire et philosophie de la généralisation de de dérivation thèse université paris nord historical review subject beginning 20th century bertram ross development fractional calculus historia mathematica doi valério duarte machado josé kiryakova virginia pioneer application fractional calculus fractional calculus applied analysis doi s13540 hdl issn s2cid fractional calculus retrieve kilbas srivastava trujillo property bologna mauro short introduction fractional calculus pdf universidad de tarapaca arica chile archive original pdf retrieve hermann richard fractional calculus introduction physicists 2nd ed new jersey world scientific publishing bibcode h doi isbn hadamard essai sur des fonction données par leur développement de taylor pdf journal de mathématiques pures et appliquées b atangana abdon baleanu dumitru new fractional derivatives nonlocal non singular kernel theory application heat transfer model arxiv math gm herrmann richard ed fractional calculus fractional calculus introduction physicists 2nd ed new jersey world 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equation lévy motion water resources research bibcode 2000wrr doi s2cid wheatcraft stephen meerschaert mark schumer rina benson david fractional dispersion lévy motion tracer tests transport porous media citeseerx doi issn s2cid b atangana abdon kilicman adem generalized mass transport equation concept variable fractional derivative mathematical problems engineering doi metzler klafter random walk guide anomalous diffusion fractional dynamic approach phys rep bibcode 2000phr m doi s0370 mainardi luchko pagnini fundamental solution space time fractional diffusion equation fractional calculus applied analysis arxiv cond bibcode m gorenflo rudolf mainardi francesco fractional diffusion process probability distributions continuous time random walk rangarajan ding eds process long range correlations process long range correlations lecture notes physics pp arxiv bibcode 2003lnp g doi isbn s2cid colbrook matthew ma xiangcheng hopkins philip squire jonathan scale law passive scalar diffusion 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society america press isbn b pandey vikash holm sverre connect grain shear mechanism wave propagation marine sediment fractional order wave equation journal acoustical society america arxiv doi issn pmid s2cid pandey vikash holm sverre link fractional derivative lomnitz creep law non newtonian time vary viscosity physical review e doi pmid laskin fractional schrodinger equation phys e arxiv quant bibcode 2002phrve 66e6108l citeseerx doi pmid s2cid laskin nick fractional quantum mechanics citeseerx doi isbn bhrawy zaky improve collocation method multi dimensional space time variable order fractional schrödinger equation applied numerical mathematics doi source edit kilbas anatolii aleksandrovich srivastava hari mohan trujillo juan theory applications fractional differential equations amsterdam netherlands elsevier isbn read edit article history fractional calculus edit ross brief history exposition fundamental theory fractional calculus fractional calculus applications fractional 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calculus theory applications narosa publishing house isbn srivastava hari m special functions fractional calculus related fractional differintegral equations singapore world scientific doi isbn li c p zeng f h numerical methods fractional calcuus usa crc press umarov introduction fractional pseudo differential equations singular symbols development mathematics switzerland springer doi isbn herrmann fractional calculus introduction physicists 3rd ed singapore world scientific doi isbn external link edit eric weisstein fractional differential equation mathworld wolfram web resource mathworld fractional calculus mathworld fractional derivative fractional calculus mathpages specialized journal fractional calculus applied analysis fractional calculus applied analysis specialized journal fractional differential equations fde specialized journal communication fractional calculus issn specialized journal journal fractional calculus applications jfca igor podlubny collection related book article link software etc gigahedron richard herrmann collection book article preprint etc history definition applications engineer pdf adam loverro university notre dame fractional calculus modelling introductory notes fractional calculus power law fractional dynamics crone toolbox matlab simulink toolbox dedicate fractional calculus freely downloadable závada petr operator fractional derivative complex plane communication mathematical physics arxiv funct bibcode 261z doi s002200050299 s2cid závada petr relativistic wave equation fractional derivative pseudodifferential operator journal applied mathematics arxiv hep doi s1110757x02110102 s2cid v t e differential equation classification operations differential operator notation differentiation ordinary partial differential algebraic integro differential fractional linear non linear holonomic attributes variable dependent independent variable homogeneous nonhomogeneous coupled decoupled order degree autonomous exact differential equation complex differential equation relation process difference discrete analogue stochastic stochastic partial delay solutions solution topic picard lindelöf theorem existence uniqueness wronskian phase portrait phase space lyapunov stability asymptotic stability exponential stability rate convergence series solution integral solution numerical integration dirac delta function solution method inspection separation variable method undetermined coefficient variation parameter integrate factor integral transform euler method finite difference method crank nicolson method runge kutta method finite element method finite volume method galerkin method perturbation theory applications list name differential equation mathematicians isaac newton gottfried wilhelm leibniz leonhard euler jacob bernoulli émile picard józef maria hoene wroński ernst lindelöf rudolf lipschitz joseph louis lagrange augustin louis cauchy john crank phyllis nicolson carl david tolmé runge martin kutta sofya 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