Generalised Diffusion and Wave Equations: Recent Advances
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r ✐ ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53 — page 1 ✐✐✐ ✐ ✐✐
1. GENERALISED DIFFUSION AND WAVEEQUATIONS: RECENT ADVANCES
T. Sandev , , R. Metzler and A. Chechkin , Research Center for Computer Science and Information Technologies,Macedonian Academy of Sciences and Arts,Bul. Krste Misirkov 2, 1000 Skopje, Macedonia Institute of Physics, Faculty of Natural Sciences and Mathematics,Ss Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia Institute of Physics & Astronomy, University of Potsdam,D-14776 Potsdam-Golm, Germany Akhiezer Institute for Theoretical Physics, Kharkov 61108, Ukrainee-mail: [email protected]
We present a short overview of the recent results in the theory of diffusion and waveequations with generalised derivative operators. We give generic examples of suchgeneralised diffusion and wave equations, which include time-fractional, distributedorder, and tempered time-fractional diffusion and wave equations. Such equationsexhibit multi-scaling time behaviour, which makes them suitable for the description ofdifferent diffusive regimes and characteristic crossover dynamics in complex systems.KEY WORDS: Memory kernel, mean squared displacement, subordinationMSC (2000): 26A33, 33E12 ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53 — page 2 ✐✐✐ ✐ ✐✐ T. SANDEV, R. METZLER, A. CHECHKIN
1. INTRODUCTIONDiffusion equations with fractional time and space derivativesinstead of the integer ones are widely used to describe anomalousdiffusion processes where the mean squared displacement (MSD)scales as a power of time, (cid:10) x ( t ) (cid:11) ≃ t α . (1.1)Depending on the values of the anomalous diffusion exponent α one distinguishes the cases of subdiffusion for < α < , normalBrownian diffusion for α = 1 , superdiffusion for < α < , ballisticmotion for α = 2 , and superballistic motion for α > . Well-knownexamples of anomalous transport include subdiffusion in artificiallycrowded systems and protein-crowded lipid bilayer membranes [10,11, 33], subiffusive charge carrier motion in semiconductors [29],subdiffusive motion of submicron probes in living biological cells [8],superdiffusive tracer motion in chaotic laminar flows [32], diffusionin porous inhomogeneous media [35], and random search processes[34], to name but a few.Modern microscopic techniques such as fluorescence correlationspectroscopy or advanced single particle tracking methods have ledto the discovery of a multitude of anomalous diffusion processesin living biological cells and complex fluids, see e.g. the reviews[1, 9, 16, 18, 19, 28] and references therein. With the growingnumber of anomalous diffusion phenomena it became clear that awide range of complex systems do not show a unique, mono-scalingbehaviour, Eq. (1.1), but instead demonstrate transitions betweendifferent diffusion regimes in the course of time. Such observationsput forward the idea that in order to capture the multi-scalingdynamics one may generalise the fractional differential operator inthe fractional diffusion equation by more universal operators with — 2 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53 — page 3 ✐✐✐ ✐ ✐✐ GENERALISED DIFFUSION AND WAVE EQUATIONS: RECENT ADVANCES specific memory kernels. Here we analyse in detail the differentversions of such generalised operators and the specific dynamicalcrossovers they effect. Special case of a power-law kernel recoversfractional derivative and respectively, the mono-scaling diffusionregime. 2. GENERALISED DIFFUSION EQUATIONS
We consider generalised diffusion equations in the so-called naturaland modified form as generalizations of the time fractional diffusionequations in the Caputo or Riemann-Liouville sense.The generalised diffusion equation in the natural form is givenby [22] Z t γ ( t − t ′ ) ∂W ( x, t ′ ) ∂t ′ dt ′ = ∂ W ( x, t ) ∂x , (1.2)where the memory kernel γ ( t ) stands on the left hand side ofthe equation. We consider zero boundary conditions at infinity, W ( ±∞ , t ) = 0 , ∂∂x W ( ±∞ , t ) = 0 , and initial conditions of theform W ( x, t = 0) = δ ( x ) . (1.3)In turn, the modified form of the equation is given by [25, 26] ∂W ( x, t ) ∂t = ∂∂t Z t η ( t − t ′ ) ∂ W ( x, t ′ ) ∂x dt ′ , (1.4)with the memory kernel η ( t ) on the right hand side of the equation.As it was shown in [25], these two equations are simply connectedthrough the memory kernels in the form ˆ γ ( s ) → / [ s ˆ η ( s )] , where ˆ γ ( s ) = R ∞ e − st γ ( t ) dt = L [ γ ( t )] and ˆ η ( s ) = L [ η ( t )] are — 3 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53 — page 4 ✐✐✐ ✐ ✐✐ T. SANDEV, R. METZLER, A. CHECHKIN the Laplace transforms of the memory kernels γ ( t ) and η ( t ) ,respectively. These equations have been obtained from continuoustime random walk (CTRW) theory for finite variance of jumplengths and generalised waiting time probability density functions(PDFs) of the forms ˆ ψ ( s ) = s ˆ γ ( s ) and ˆ ψ ( s ) = η ( s )] − ,respectively.In order to have well established stochastic processes encoded inboth equations, we need to prove that their solutions are normalizedand non-negative. The non-negativity of the solutions can be shownby using the subordination approach [4, 14, 15]. We will elaborateon this approach for Eq. (1.2), this can be done for Eq. (1.4) inthe same way. By Fourier ( ˜ f ( k ) = R ∞−∞ f ( x ) e ıkx dx ) and Laplacetransformations of Eq. (1.2) one finds ˜ˆ W ( k, s ) = ˆ γ ( s ) Z ∞ e − u ( s ˆ γ ( s )+ k ) du = Z ∞ e − uk ˆ G ( u, s ) du. (1.5)Here the function G is defined by ˆ G ( u, s ) = ˆ γ ( s ) e − u s ˆ γ ( s ) = − ∂∂u s e − u s ˆ γ ( s ) . (1.6)Therefore, the PDF W ( x, t ) is given by [14, 15] W ( x, t ) = Z ∞ e − x u √ πu G ( u, t ) du, (1.7)which means that the function G ( u, t ) is the PDF providing thesubordination transformation from time scale t (physical time) totime scale u (operational time). The function G ( u, t ) is normalizedwith respect to u for any t , i.e., Z ∞ G ( u, t ) du = L − s (cid:20)Z ∞ ˆ γ ( s ) e − us ˆ γ ( s ) du (cid:21) = L − s (cid:20) s (cid:21) = 1 . (1.8) — 4 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53 — page 5 ✐✐✐ ✐ ✐✐ GENERALISED DIFFUSION AND WAVE EQUATIONS: RECENT ADVANCES
In order to prove the positivity of W ( x, t ) according to theBernstein theorem it is sufficient to show that the function ˆ G ( u, s ) is completely monotone on the positive real axis s [30]. For thatwe only need to show that (i) the function ˆ γ ( s ) is completelymonotone, and (ii) the function s ˆ γ ( s ) is a Bernstein function. If(ii) holds, then the function e − s ˆ γ ( s ) is completely monotone asa composition of completely monotone and a Bernstein function.Moreover, G ( u, s ) is completely monotone, as a product of twocompletely monotone functions, e − s ˆ γ ( s ) and ˆ γ ( s ) . Alternatively, onecan check that s ˆ γ ( s ) is a complete Bernstein function, which is animportant subclass of the Bernstein functions [30]. This condition isenough to ensure the complete monotonicity of ˆ G ( u, s ) due to theproperty of the complete Bernstein function: if f ( s ) is a completeBernstein function, then f ( s ) /s is completely monotone [30]. Theproof of the non-negativity of the solutions to the generaliseddiffusion equations with different memory kernels can be found in[25, 26] along with the list of properties of completely monotone,Bernstein, and complete Bernstein functions.By analogy, the solution of Eq. (1.4) is non-negative if (i) thefunction / [ s ˆ η ( s )] is completely monotone, and (ii) the function / ˆ η ( s ) is a Bernstein function. Alternatively, one can prove the non-negativity of the solution if / ˆ η ( s ) is a complete Bernstein function.By solving both equations (1.2) and (1.4), one can find thecorresponding MSDs, as follows [22, 25, 26] (cid:10) x ( t ) (cid:11) = 2 L − (cid:20) s − ˆ γ ( s ) (cid:21) , (1.9) (cid:10) x ( t ) (cid:11) = 2 L − (cid:2) s − ˆ η ( s ) (cid:3) , (1.10)from which we can analyze the diffusive behaviours for a given formof the memory kernel. — 5 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53 — page 6 ✐✐✐ ✐ ✐✐ T. SANDEV, R. METZLER, A. CHECHKIN
The case with γ ( t ) = δ ( t ) , which means ˆ η ( s ) = 1 / [ s ˆ γ ( s )] = 1 /s , i.e., η ( t ) = 1 , leads us both to the standarddiffusion equation ∂W ( x, t ) ∂t = ∂ W ( x, t ) ∂x , (1.11)for Brownian diffusion with linear time dependence of the MSD, (cid:10) x ( t ) (cid:11) = 2 L − (cid:2) s − (cid:3) = 2 t . The waiting time PDF in thecorresponding CTRW scheme is exponential, ψ ( t ) = L − (cid:2) s (cid:3) = e − t , which in the long time limit ( s → ) can be used as ˆ ψ ( s ) ≃ − s . Another well known example is thecase of a power-law memory kernel γ ( t ) = t − α / Γ (1 − α ) , < α < , from where it follows that ˆ η ( s ) = s − α , i.e., η ( t ) = t α − / Γ( α ) .These memory kernels yield two equivalent formulations of the timefractional diffusion equation, namely, C D α W ( x, t ) = ∂ W ( x, t ) ∂x , (1.12)and ∂W ( x, t ) ∂t = RL D − α ∂ W ( x, t ) ∂x , (1.13)where C D αt f ( t ) = 1Γ( n − α ) Z t f ( n ) ( t ′ )( t − t ′ ) α +1 − n dt ′ , n − < α < n, (1.14)and RL D αt f ( t ) = 1Γ( n − α ) d n dt n Z t f ( t ′ )( t − t ′ ) α +1 − n dt ′ , n − < α < n, (1.15) — 6 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53 — page 7 ✐✐✐ ✐ ✐✐ GENERALISED DIFFUSION AND WAVE EQUATIONS: RECENT ADVANCES are the Caputo and Riemann-Lioville fractional derivatives,respectively [13]. Both derivatives for α = n become ordinaryderivatives, f ( n ) ( t ) . The corresponding MSD shows subdiffusivebehaviour (cid:10) x ( t ) (cid:11) = 2 L − (cid:2) s − α − (cid:3) = 2 t α Γ( α +1) , with the Mittag-Leffler waiting time PDF, ψ ( t ) = L − (cid:2) s α (cid:3) = t α − E α,α ( − t α ) .Here E α,β ( − z ) = P ∞ n =0 ( − z ) n Γ( αn + β ) is the two parameter Mittag-Lefflerfunction [13], which has the following asymptotic E α,β ( − z ) ≃− P ∞ n =1 ( − z ) − n Γ( β − αn ) for z ≫ . Here we note that in the long timelimit the waiting time PDF is of power-law form, ψ ( t ) ≃ t − − α [17]. Now we introduce a memory kernelwith two power-law functions, γ ( t ) = B t − α / Γ(1 − α ) + B t − α / Γ(1 − α ) , < α < α < , B + B = 1 , which gives riseto the bi-fractional diffusion equation in natural form [3], B C D α t W ( x, t ) + B C D α t W ( x, t ) = ∂ W ( x, t ) ∂x . (1.16)Using the relation with the memory kernel η ( t ) , ˆ η ( s ) = 1 / [ s ˆ γ ( s )] =[ B s α + B s α ] − , we find that η ( t ) = B t α − E α − α ,α (cid:16) − B B t α − α (cid:17) ,i.e., the equivalent representation to Eq. (1.16) in the modified formis given by ∂W ( x, t ) ∂t = 1 B ∂∂t Z t ( t − t ′ ) α − × E α − α ,α (cid:18) − B B [ t − t ′ ] α − α (cid:19) ∂ W ( x, t ′ ) ∂x dt ′ . (1.17) — 7 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53 — page 8 ✐✐✐ ✐ ✐✐ T. SANDEV, R. METZLER, A. CHECHKIN
The bi-fractional diffusion equation in the natural form is a model todescribe decelerating subdiffusion, since the MSD is given by [3, 23] (cid:10) x ( t ) (cid:11) = 2 t α B E α − α ,α +1 (cid:18) − B B t α − α (cid:19) ≃ ( B t α Γ(1+ α ) , t ≪ , B t α Γ(1+ α ) , t ≫ . (1.18)The corresponding waiting time PDF is given by [23] ψ ( t ) = L − (cid:20)
11 + B s α + B s α (cid:21) = t α − B ∞ X n =0 ( − n B n t α n E n +1 α − α ,α n + α (cid:18) − B B t α − α (cid:19) . (1.19)Here E δα,β ( z ) = P ∞ n =0 ( δ ) n Γ( αn + β ) z n n ! is the three parameter Mittag-Leffler function [21], where ( δ ) n = Γ( δ + n ) / Γ( δ ) is the Pochhammersymbol. Its asymptotic expansions are given by E δα,β ( − t α ) ≃ β ) − δ t α Γ( α + β ) ≃ β ) exp (cid:16) − δ Γ( β )Γ( α + β ) t α (cid:17) for t ≪ , and E δα,β ( − t α ) = t − αδ Γ( δ ) P ∞ n =0 Γ( δ + n )Γ( β − α ( δ + n )) ( − t α ) − n n ! for < α < and t ≫ [6, 23].In accordance to the previous case, we may introduce thebi-fractional diffusion equation in the modified form, where thememory kernel is given by η ( t ) = B t α − Γ( α ) + B t α − Γ( α ) with < α <α < , B + B = 1 , i.e., [5, 31] ∂W ( x, t ) ∂t = B RL D − α t ∂ W ( x, t ) ∂x + B RL D − α t ∂ W ( x, t ) ∂x . (1.20)Since ˆ γ ( s ) = 1 / [ s ˆ η ( s )] = 1 / [ s ( B s − α + B s − α )] , we have γ ( t ) = B t − α E α − α , − α (cid:16) − B B t α − α (cid:17) , and the equivalent representation — 8 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53 — page 9 ✐✐✐ ✐ ✐✐ GENERALISED DIFFUSION AND WAVE EQUATIONS: RECENT ADVANCES of Eq. (1.20) in the natural form is given by a Mittag-Leffler memorykernel, B Z t ( t − t ′ ) − α E α − α , − α (cid:18) − B B [ t − t ′ ] α − α (cid:19) × ∂W ( x, t ′ ) ∂t ′ dt ′ = ∂ W ( x, t ) ∂x . (1.21)The bi-fractional diffusion equation in the modified form is a usefulmodel for the description of accelerating diffusion since the MSD isgiven by [5, 23] (cid:10) x ( t ) (cid:11) = 2 B t α E − α − α ,α +1 (cid:18) − B B t α − α (cid:19) = 2 B t α Γ(1 + α ) + 2 B t α Γ(1 + α ) ≃ ( B t α Γ(1+ α ) , t ≪ , B t α Γ(1+ α ) , t ≫ . (1.22) At the end of this sectionwe show two other models that describe transitions from one toanother diffusive behaviour. This can be achieved if one introducesan exponential cut-off of the power-law memory kernel of the form γ ( t ) = e − bt t − α / Γ(1 − α ) , < α < , where b > is the truncationparameter. Therefore, we obtain the tempered fractional diffusionequation in the natural form, − α ) Z t e − b ( t − t ′ ) ( t − t ′ ) − α ∂W ( x, t ′ ) ∂t ′ dt ′ = ∂ W ( x, t ) ∂x . (1.23)The corresponding equation in the modified form reads ∂W ( x, t ) ∂t = ∂∂t Z t ( t − t ′ ) α − E − (1 − α )1 ,α ( − b [ t − t ′ ]) ∂ W ( x, t ′ ) ∂x dt ′ , (1.24) — 9 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53page 10 — ✐✐✐ ✐ ✐✐ T. SANDEV, R. METZLER, A. CHECHKIN since η ( t ) = L − h s − ( s + b ) α − i = t α − E α − ,α ( − bt ) . Equation (1.24) isactually the diffusion equation ∂W ( x, t ) ∂t = RL D − α, − α , − b, ∂ W ( x, t ) ∂x . (1.25)with the Prabhakar derivative ( < µ < , ρ > ) [7] RL D γ,µρ,ω, f ( x ) = ddt Z t ( t − t ′ ) − µ E − γρ, − µ ( ω [ t − t ′ ] ρ ) f ( t ′ ) dt ′ . (1.26)The MSD shows a crossover from subdiffusion to normal diffusion, (cid:10) x ( t ) (cid:11) = 2 t α E α − ,α +1 ( − bt ) ≃ (cid:26) t α Γ(1+ α ) , t ≪ , b − α t, t ≫ . (1.27)Consider now the tempered fractional diffusion equation in themodified form with η ( t ) = e − bt t α − / Γ( α ) ( < α < , b > ), i.e., ∂W ( x, t ) ∂t = 1Γ( α ) ∂∂t Z t e − b ( t − t ′ ) ( t − t ′ ) α − ∂ W ( x, t ′ ) ∂x dt ′ . (1.28)From ˆ η ( s ) = ( s + b ) − α it follows that ˆ γ ( s ) = 1 / [ s ( s + b ) − α ] ,and γ ( t ) = t − α E − α , − α ( − bt ) , i.e., the corresponding equation in thenatural form equivalent to (1.28) is given by Z t ( t − t ′ ) − α E − α , − α ( − b [ t − t ′ ]) ∂W ( x, t ′ ) ∂t ′ dt ′ = ∂ W ( x, t ) ∂x , (1.29)which can be presented with the regularized Prabhakar derivative[7] C D γ,µρ,ω, f ( x ) = Z t ( t − t ′ ) − µ E − γρ, − µ ( ω [ t − t ′ ] ρ ) df ( t ′ ) dt ′ dt ′ , (1.30) — 10 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53page 11 — ✐✐✐ ✐ ✐✐ GENERALISED DIFFUSION AND WAVE EQUATIONS: RECENT ADVANCES as C D α,α , − b, W ( x, t ) = ∂ W ( x, t ) ∂x . (1.31)The corresponding MSD shows a crossover from subdiffusion to aplateau value, (cid:10) x ( t ) (cid:11) = 2 t α E α +11 ,α +1 ( − bt ) ≃ (cid:26) t α Γ(1+ α ) , t ≪ , b − α , t ≫ . (1.32)Here we note that different models based on the temperedversions of the generalised Langevin equation and fractionalBrownian motion have been introduced recently, which also givesimilar crossovers from subdiffusion to normal diffusion [20].Moreover, general diffusion equations on two dimensional structureshave been analyzed and different diffusive regimes obtained [24].3. GENERALISED DIFFUSION-WAVE EQUATION In analogy to the generalised diffusion equations in natural andmodified forms, we now consider the generalised diffusion-waveequation Z t ζ ( t − t ′ ) ∂ W ( x, t ) ∂t ′ dt ′ = ∂ W ( x, t ) ∂x , (1.33)in the natural form with non-negative memory kernel ζ ( t ) , andsimilarly ∂ W ( x, t ) ∂t = ∂ ∂t Z t ξ ( t − t ′ ) ∂ W ( x, t ′ ) ∂x dt ′ (1.34) — 11 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53page 12 — ✐✐✐ ✐ ✐✐ T. SANDEV, R. METZLER, A. CHECHKIN in the modified form with non-negative memory kernel ξ ( t ) . Inwhat follows we consider the natural form, only [27]. The boundaryconditions at infinity are W ( ±∞ , t ) = 0 , ∂∂x W ( ±∞ , t ) = 0 , and theinitial conditions are of the form W ( x, t = 0) = δ ( x ) , ∂∂t W ( x, t = 0) = 0 . (1.35)We here refer to [27] for discussion on the choice of the initialconditions.Making the Fourier-Laplace transform of Eq. (1.33), and theninverse Fourier transform, we find ˆ W ( x, s ) = 12 q ˆ ζ ( s ) exp (cid:18) − s q ˆ ζ ( s ) | x | (cid:19) . (1.36)From here one easily concludes that the PDF is normalized to 1,i.e., R ∞−∞ W ( x, t ) dx = 1 , since R ∞−∞ ˆ W ( x, s ) dx = 1 /s . The non-negativity of the solution can be shown by applying the Bernsteintheorem, i.e., by showing that the solution in the Laplace spaceis a completely monotone function [30]. To this end, solution(1.36) can be considered as a product of two functions, q ˆ ζ ( s ) and exp (cid:18) − s q ˆ ζ ( s ) | x | (cid:19) , and it is sufficient to prove that bothfunctions q ˆ ζ ( s ) and exp (cid:18) − s q ˆ ζ ( s ) | x | (cid:19) are completely monotone.Therefore, it is sufficient to show that q ˆ ζ ( s ) is completelymonotone, and s q ˆ ζ ( s ) is a Bernstein function. The non-negativityof the solution can also be shown by proving that the function q ˆ ζ ( s ) is a Stieltjes function, which is again completely monotone,or that s q ˆ ζ ( s ) is a complete Bernstein function. The proof of — 12 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53page 13 — ✐✐✐ ✐ ✐✐ GENERALISED DIFFUSION AND WAVE EQUATIONS: RECENT ADVANCES the non-negativity of the solutions of the generalised diffusion-wave equations with different memory kernels can be found in [27]along with the list of properties of completely monotone, Stieltjes,Bernstein, and complete Bernstein functions.By solving Eq. (1.33) we find the MSD, (cid:10) x ( t ) (cid:11) = (cid:26) − ∂ ∂k L − h ˜ˆ W ( k, s ) i ( k, t ) (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) k =0 = 2 L − " s ˆ ζ ( s ) ( t ) , (1.37)from where we analyze the diffusive regimes depending on thememory kernel ζ ( t ) . The simplest case of Eq. (1.33) is the onewith Dirac delta memory kernel ζ ( t ) = δ ( t ) , which yields theclassical wave equation ∂ W ( x, t ) ∂t = ∂ W ( x, t ) ∂x . (1.38)The MSD (1.37) then becomes (cid:10) x ( t ) (cid:11) = t , (1.39)which reports ballistic motion. The case with the power-lawmemory kernel ζ ( t ) = t − α Γ(2 − α ) , < α < , yields C D αt W ( x, t ) = ∂ W ( x, t ) ∂x , (1.40) — 13 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53page 14 — ✐✐✐ ✐ ✐✐ T. SANDEV, R. METZLER, A. CHECHKIN for < α < , whereas for the case with < α < we get − α ) Z t ( t − t ′ ) − α ∂ W ( x, t ′ ) ∂t ′ dt ′ = ∂ W ( x, t ) ∂x . (1.41)The MSD for the mono-fractional diffusion-wave equation reads (cid:10) x ( t ) (cid:11) = 2 t α Γ(1 + α ) . (1.42)Since < α < , the generalised diffusion-wave equation (1.33)with power-law memory kernel describes both superdiffusive andsubdiffusive processes. The case α = 1 reduces to the classicaldiffusion equation for Brownian motion, i.e., (cid:10) x ( t ) (cid:11) = 2 t , whereasthe case with α = 2 yields ballistic diffusion, (cid:10) x ( t ) (cid:11) = t . The next case we consider is thebi-fractional diffusion-wave equation with the memory kernel of theform η ( t ) = B t − α Γ(2 − α ) + B t − α Γ(2 − α ) , B + B = 1 . The case with < α < α < yields B C D α t W ( x, t ) + B C D α t W ( x, t ) = ∂ W ( x, t ) ∂x , (1.43)where C D α j t is the Caputo fractional derivative (1.14) of the order < α j < ( n = 2 ), whereas the case < α < α < yieldsequation B Γ(2 − α ) Z t ( t − t ′ ) − α ∂ W ( x, t ′ ) ∂t ′ dt ′ + B Γ(2 − α ) Z t ( t − t ′ ) − α ∂ W ( x, t ′ ) ∂t ′ dt ′ = ∂ W ( x, t ) ∂x . (1.44) — 14 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53page 15 — ✐✐✐ ✐ ✐✐ GENERALISED DIFFUSION AND WAVE EQUATIONS: RECENT ADVANCES
The corresponding MSD then becomes (cid:10) x ( t ) (cid:11) = 2 B t α E α − α ,α +1 (cid:18) − B B t α − α (cid:19) , ≃ ( B t α Γ(1+ α ) , t ≪ , B t α Γ(1+ α ) , t ≫ , (1.45)which means decelerating superdiffusion for < α < α < ,including crossover from superdiffusion to normal diffusion in thecase α < α < , and decelerating subdiffusion for < α <α < , including crossover from normal diffusion to subdiffusion forthe case < α < α = 1 . Decelerating superdiffusion has indeedbeen observed, for example, in Hamiltonian systems with long-rangeinteractions [12], and different biological systems [2]. Furthermore, we consider atruncated power-law memory kernel of the form ζ ( t ) = e − bt t − α Γ(2 − α ) ,where b > , and ≤ α < , corresponding to the followingtempered fractional wave equation: − α ) Z t e − b ( t − t ′ ) ( t − t ′ ) − α ∂ W ( x, t ) ∂t ′ dt ′ = ∂ W ( x, t ) ∂x . (1.46)For the MSD we get (cid:10) x ( t ) (cid:11) = 2 RL I t (cid:18) e − bt t − α Γ( − µ ) (cid:19) ≃ (cid:26) t α Γ(1+ α ) , t ≪ ,b − α t , t ≫ , (1.47)where RL I αt f ( t ) = 1Γ( α ) Z t ( t − t ′ ) α − f ( t ′ ) dt ′ , α > , (1.48) — 15 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53page 16 — ✐✐✐ ✐ ✐✐ T. SANDEV, R. METZLER, A. CHECHKIN is the Riemann-Liouville integral [13]. Thus, there is a crossover fromsuperdiffusion to ballistic motion in the case with < α < , andfrom normal diffusion to ballistic motion in the case with α = 1 . Forthe case of the diffusion-wave equation with Prabhakar derivativewe address the reader to [27].4. SUMMARYWe consider different stochastic processes governed by thegeneralised diffusion and diffusion-wave equations which containthe well known time fractional diffusion and wave equations asparticular cases. Such processes demonstrate a rich multi-scalingbehaviour which manifests itself in specific crossovers betweendifferent diffusion regimes in the course of time. We thus obtaina flexible tool which can be applied for the description of diversediffusion phenomena in complex systems demonstrating crossoverbehaviours. ACKNOWLEDGEMENTSThe Authors acknowledge funding from the Deutsche Forschungsge-meinschaft (DFG), project ME 1535/6-1 "Random search processes,L´evy flights, and random walks on complex networks". RM thanksthe Foundation for Polish Science for support within an Alexandervon Humboldt Polish Honorary Research Scholarship. RM andAC also acknowledge support from the DFG project 1535/7-1"Mathematical and physical modeling of single particle tracking -Big Data approach". — 16 — ✐ “generalized_diffusion_and_wave_equations” — 2019/3/5 — 1:53page 17 — ✐✐✐ ✐ ✐✐ GENERALISED DIFFUSION AND WAVE EQUATIONS: RECENT ADVANCES
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