A note on generalized fractional diffusion equations on Poincarè half plane
aa r X i v : . [ m a t h - ph ] J u l A NOTE ON GENERALIZED FRACTIONAL DIFFUSION EQUATIONS ONPOINCAR´E HALF PLANE.
R. GARRA, F. MALTESE, AND E. ORSINGHER
Abstract.
In this paper we study generalized time-fractional diffusion equations on the Poincar´ehalf plane H +2 . The time-fractional operators here considered are fractional derivatives of afunction with respect to another function, that can be obtained by starting from the classicalCaputo-derivatives essentially by means of a deterministic change of variable. We obtain an ex-plicit representation of the fundamental solution of the generalized-diffusion equation on H +2 andprovide a probabilistic interpretation related to the time-changed hyperbolic Brownian motion.We finally include an explicit result regarding the non-linear case admitting a separating variablesolution. Keywords:
Generalized time-fractional diffusion equation, Hyperbolic geometry, HyperbolicBrownian motion. Introduction
In this paper we study generalized time-fractional diffusion equations on the hyperbolic Poincar´ehalf-plane H +2 = (cid:26) ( x, y ) ∈ R (cid:12)(cid:12)(cid:12)(cid:12) y > (cid:27) . The generalization here considered is based on the application of time-fractional derivatives of afunction with respect to another function (see [1] for the definition and main properties), an inter-esting approach that permits us to take into account both time-varying coefficients and memoryeffects (see e.g. [3] for a physical discussion about this). In the previous paper [7] the authorsstudied for the first time the time-fractional diffusion equation on the hyperbolic space involvingthe classical Caputo derivative. Moreover, in the more recent paper [4], an interesting probabilisticinterpretation of the fundamental solution of the time-fractional telegraph-type equation on hyper-bolic spaces has been provided. In particular, a relevant connection with time-changed hyperbolicBrownian motions has been proved.The main aim of this paper is to provide a rigorous analysis of the generalized time-fractional dif-fusion equation on the hyperbolic space H +2 . We find an explicit representation of the fundamentalsolution by means of the method of separation of variables. Moreover, we obtain a probabilisticinterpretation of the related stochastic process as a time-changed hyperbolic Brownian motion. Inthe first part of the paper we provide some necessary preliminaries about the Poincar´e half-planeand the definition and basic properties of the fractional operators here considered. We decided toprovide detailed preliminaries about the Poincar´e half-plane since many non-trivial computationsare involved and we think that this short guide can be of help for the reader.Then, we analyze the generalized time-fractional diffusion equation on H +2 providing the represen-tation of the fundamental solution and the related probabilistic meaning. Finally, we also considera nonlinear generalized time-fractional diffusion equation on H +2 admitting a solution obtained bymeans of the method of separation of variables.Few papers are devoted to the analysis of time-fractional diffusive equations on hyperbolic spaces, in our view, together with the previous papers [4] and [7], this can be another step to develop thisnew and interesting topic. 2. Preliminaries
A short survey on hyperbolic geometry.
We here give some necessary mathematicalpreliminaries about the model of the Poincair´e half-plane i.e the set H +2 = (cid:8) ( x, y ) ∈ R | y > (cid:9) with the following metric(2.1) ds = dx + dy y First of all, in order to characterize the geometry of the Poincar´e half-plane, we study the formof the geodesics, by using the variational principle.We consider the family of curves in the hyperbolic plane passing through two given points( x , y ) and ( x , y ) in their parametric representation i.e.(2.2) γ = (cid:26) ( x ( t ) , y ( t )) (cid:12)(cid:12)(cid:12)(cid:12) t ≤ t ≤ t (cid:27) , where t and t are such that ( x ( t ) , y ( t )) = ( x , y ) and ( x ( t ) , y ( t )) = ( x , y ).The length of this curve in the hyperbolic plane is(2.3) L ( γ ) = Z t t p x ′ ( t ) + y ′ ( t ) y ( t ) dt. We can simplify this expression by restricting ourselves to the family of parametric curves of(2.2) to curves with Cartesian parameterization i.e(2.4) γ = { ( x, y ( x )) | x ≤ x ≤ x } . In this case, the integral (2.3) becomes(2.5) L ( γ ) = Z x x p y ′ ( x ) y ( x ) dx. We can consider an arbitrary function w ( x ) as w ( x ) = y ( x ) + ǫh ( x ) with ǫ ≥ h ( x ) is suchthat h ( x ) = h ( x ) = 0 .So the length of curve ( x, w ( x )) applying (2.5) becomes(2.6) L ( γ ) = l ( ǫ ) = Z x x p y ′ ( x ) + ǫh ′ ( x )) y ( x ) + ǫh ( x ) dx. The geodesic curve is associated with a minimum point with respect to the ǫ variable of thefunction l ( ǫ ) for which is satisfied the condition(2.7) dldǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 = 0 . By direct computation we have that dldǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 = Z x x ddǫ p y ′ + ǫh ′ ) y + ǫh ! | ǫ =0 dx == Z x x − h p y ′ + ǫh ′ ) ( y + ǫh ) + h ′ ( y ′ + ǫh ′ )( y + ǫh ) p y ′ + ǫh ′ ) ! | ǫ =0 dx = = Z x x − h p y ′ y + h ′ y ′ y p y ′ ! dx = Z x x − h p y ′ y dx + " hy ′ y p y ′ x x − Z x x h ddx y ′ y p y ′ dx (since the function h ( x ) is such that h ( x ) = h ( x ) = 0)= Z x x − p y ′ y − ddx y ′ y p y ′ ! hdx (2.8)By (2.7), the integral (2.8) must be equal to zero for all functions h and therefore we have that − p y ′ y − ddx y ′ y p y ′ = 0and therefore(2.9) − y p y ′ − y ′′ y (cid:16)p y ′ (cid:17) = 0 . We finally obtain that(2.10) 1 + ddx ( yy ′ ) = 0 . Integrating twice (2.10) we have the following equation(2.11) x + y − cx − d = 0which is in H +2 the equation of the semi-circles with an arbitrary center on the x-axis with arbitraryradius.Other geodesic curves in the Poincair´e half-plane are the half-lines parallel to the y-axis, thatemerge if x = x .In order to derive the expression of the Laplacian in H +2 , let us introduce the hyperbolic co-ordinates. First of all, we need introduce the geometric center of the upper Poincar´e half-planewhich is the point (0 ,
1) and consider an arbitrary point of Cartesian coordinates ( x, y ) in H +2 .The hyperbolic coordinates associated with this point are ( η, α ) where η is the hyperbolic distancebetween (0 ,
1) and ( x, y ), i.e., the length according to the metric (2.1) of the arc of geodesic thatpasses through (0 ,
1) and ( x, y ), which is an arc of semi-circumference if ( x, y ) is not on the y-axis.While α is the angle formed by the tangent to that semi-circumference in (0 ,
1) and passing through( x, y ).As hyperbolic coordinates are defined, important relationships can be obtained from the transitionto hyperbolic coordinates to the Cartesian coordinates, for example from the α coordinate to theCartesian coordinates we can get a relationship that starts from the equation of the geodesic thatpasses through the origin (0 ,
1) and the point ( x, y )(2.12) ( x − tan α ) + y = 1cos α and therefore(2.13) tan α = x + y − x . R. GARRA, F. MALTESE, AND E. ORSINGHER
Finally, the relationship between the η coordinate and the Cartesian coordinates is given by(2.14) cosh η = x + y + 12 y . Starting from the relations (2.13) and (2.1) we can get the change of coordinates from cartesiancoordinates ( x, y ) to hyperbolic coordinates ( η, α ) in H +2 (2.15) x = cos α sinh η cosh η − sinh η sin α , η > , < α < πy = 1cosh η − sinh η sin α We are now able to derive the expression of the Laplacian operator in hyperbolic coordinates.First of all, we observe that the Poincar´e upper half-plane is a Riemannian manifold with thefollowing metric tensor g = (cid:18) y y (cid:19) , In general, on a Riemannian manifold with a metric tensor g , the Laplacian is given by(2.16) ∆ f = 1 p | g | n X i =1 ∂ i ( p | g | n X j =1 g ij ∂ j f ) , where | g | is the determinant of the metric tensor and the elements g ij are the components ofthe inverse matrix of g and n is the dimension of the manifold.By observing that the inverse matrix g − and | g | are respectively given by(2.17) g − = (cid:18) y y (cid:19) , | g | = 1 y , we have that in this case ∆ f becomes∆ f = 1 q y (cid:20) ∂∂x (cid:18)r y y ∂∂x f (cid:19) + ∂∂y (cid:18)r y y ∂∂y f (cid:19)(cid:21) == y (cid:20) ∂∂x (cid:18) ∂∂x f (cid:19) + ∂∂y (cid:18) ∂∂y f (cid:19)(cid:21) = y (cid:18) ∂ ∂x + ∂ ∂y (cid:19) f, x ∈ R , y > η ∂∂η (cid:18) sinh η ∂∂η (cid:19) + 1sinh η ∂ ∂α , < α < π, η > . Fractional derivatives of a function with respect to another function.
Fractionalderivatives of a function with respect to another function have been considered in the classicalmonograph by Kilbas et al. [6] (Section 2.5) and recently studied by Almeida in [1] that hasprovided the Caputo-type regularization of the existing definition and some interesting properties.Starting from this paper, this topic has gained interest both for mathematical reasons (see e.g. [2])and for physical applications (e.g. [3] and the references therein). The utility of these generalizedfractional operators in the applications is represented by the fact that they are essentially obtainedby a deterministic time-change and permits us to take into account both time-variable coefficientsand memory effects. Moreover, this class of operators include as special cases classical well-known time-fractional derivatives (for example, fractional derivatives in the sense of Hadamard, or Erd´elyi-Kober).Here we recall the basic definitions and properties for the reader’s convenience.Let ν > f ∈ C ([ a, t ]) an increasing function such that f ′ ( t ) = 0 in [ a, t ], the fractionalintegral of a function g ( t ) with respect to another function f ( t ) is given by(2.20) (cid:16) I ν,fa + g (cid:17) ( t ) := 1Γ( ν ) Z ta f ′ ( τ )( f ( t ) − f ( τ )) ν − g ( τ ) dτ. Observe that for f ( t ) = t β we recover the definition of Erd´elyi-Kober fractional integral recentlyapplied, for example, in connection with the Generalized Grey Brownian Motion [9]. For simplicityhereafter we will consider a = 0 (as usual) and suitable functions f such that f (0) = 0. All theresults can be simply generalized.The corresponding Caputo-type evolution operator (see [1]) for 0 < ν < (cid:0) O ν,f g (cid:1) ( t ) := 1Γ(1 − ν ) Z t ( f ( t ) − f ( τ )) − ν ddτ g ( τ ) dτ (2.21) = I − ν,f + (cid:18) f ′ ( t ) ddt (cid:19) g ( t ) . (2.22)For the general case ν ∈ R we refer to [1]. In this paper we are interested to the case 0 < ν < O ν,f ( · ) in order to underline the genericintegro-differential nature of the time-evolution operator, depending on the choice of the function f ( t ) and the real order ν .A relevant property of the operator (2.21) is that if g ( t ) = ( f ( t )) β − with β >
1, then (seeLemma 1 of [1])(2.23) (cid:0) O ν,f g (cid:1) ( t ) = Γ( β )Γ( β − ν ) ( f ( t )) β − ν − . Indeed, by direct calculation we have that (cid:0) O ν,f f β − (cid:1) ( t ) = β − − ν ) Z t ( f ( t ) − f ( τ )) − ν f ′ ( τ )( f ( τ )) β − dτ and taking y = f ( τ ) /f ( t ) we have that (cid:0) O ν,f f β − (cid:1) ( t ) = β − f β − − ν Γ(1 − ν ) Z (1 − y ) − ν y β − dy = Γ( β ) f β − − ν ( t )Γ( β − − ν ) Γ(1 − ν )Γ( β − β − ν )= Γ( β )Γ( β − ν ) ( f ( t )) β − ν − . Therefore, the composite Mittag-Leffler function(2.24) g ( t ) = E ν ( λ ( f ( t )) ν )is an eigenfunction of the operator O ν,f , when ν ∈ (0 ,
1) and f is a well-behaved function suchthat f (0) = 0. This means that(2.25) O ν,f E ν ( λ ( f ( t )) ν ) = λE ν ( λ ( f ( t )) ν ) . R. GARRA, F. MALTESE, AND E. ORSINGHER Generalized linear and nonlinear fractional diffusion on Poincar´e half-plane
The linear case.
In a previous paper [7], the authors considered the following diffusion-typeequation on H +2 (3.1) ∂ β ∂t β u ( η, t ) = 1sinh η (cid:18) ∂∂η sinh η ∂∂η (cid:19) u ( η, t ) , β ∈ (0 , , where ∂ β ∂t β is a fractional derivative of order β in the sense of Caputo. We here analyze the moregeneral case involving the fractional derivative w.r.t. another function. First of all, we have thefollowing result Theorem 3.1.
Let be f ∈ L [0 , t ] such that f (0) = 0 , the fundamental solution for the generalizedtime-fractional diffusion equation (3.2) (cid:0) O β,f u (cid:1) ( η, t ) = 1sinh η (cid:18) ∂∂η sinh η ∂∂η (cid:19) u ( η, t ) is given by (3.3) u ( η, t ) = 2 π Z ∞ xE β (cid:18) − f ( t ) β − x f ( t ) β (cid:19) dx Z ∞ η dϕ sin( xϕ ) √ ϕ − η . Proof.
We find the solution by means of the separation of variables and transform the Laplacianoperator by using the change of variable y = cosh η which leads to u ( y, t ) = F ( y ) · T ( t )and therefore we get (cid:0) O β,f T (cid:1) = − ωT, (3.4) ( y − F ′′ + 2 yF ′ + ωF = 0 . (3.5)The solution of the first equation is given by(3.6) T ( t, ω ) = E β, ( − ωf ( t ) β ) . The spatial part of the solution remains the same as in the classical hyperbolic diffusion equationand we refer to [7] for the details. (cid:3)
Remark 3.2.
Observe that for f ( t ) = t and β = 1 we recover the transition function of thehyperbolic Brownian motion, firstly studied by Gertsenshtein and Vasiliev in [5] .Moreover, for f ( t ) = t and β ∈ (0 , we recover the results obtained in [7] . Let us introduce the process T β ( f ( t )) = B hp ( L β ( f ( t ))) , where B hp is the hyperbolic Brownian motion in H +2 independent from L β ( t ) which is the inverseof the stable subordinator H β ( t ), that is L β ( t ) = inf { s > H β ( s ) ≥ t } , β ∈ (0 , . We have the following
Theorem 3.3.
The distribution p ( x, t ) of the process T β ( f ( t )) coincides with the fundamentalsolution of the equation (3.2) .Proof. We observe that, by means of the deterministic time-change f ( t ) → t , we can essentiallygo back to a time-fractional diffusion equation involving the Caputo derivative. Then, by meansof the time-Laplace transform method, it can be proved that the fundamental solution of (3.1)coincides with the distribution of the process T β ( f ( t )). (cid:3) Observe that this paper is devoted to diffusive models in the Poincar´e half-space H +2 but thegeneralizations to H + n can be obtained in a similar way from the probabilistic point of view andwill be the object of a further detailed analysis.Finally, by means of similar methods, we can generalize the recent results obtained in [4] abouttime-fractional telegraph-type equations in H n . In particular, we have that Theorem 3.4.
The distribution of the composition (3.7) T β ( t ) = B hp ( L β ( f ( t ))) , where L β ( t ) = inf { s > H β ( s ) + (2 λ ) /β H β ( s ) ≥ t } , and H β , H β are independent stable subordinators (with β ∈ (0 , / , coincides with the funda-mental solution of the equation (3.8) (cid:0) O β,f u (cid:1) ( η, t ) + 2 λ (cid:0) O β,f u (cid:1) ( η, t ) = 1sinh η (cid:18) ∂∂η sinh η ∂∂η (cid:19) u ( η, t ) , β ∈ (0 , / . The main idea for the proof is essentially the same of the previous theorem. The result can begeneralized to a multi-term fractional equation involving a finite number of fractional derivativesw.r.t. another function of order less than one (see [8]).3.2.
The nonlinear case.
We recall that the construction of the explicit representation of thefundamental solution of the linear diffusion equation is based (also in the fractional case) on theclassical method of separation of variables. Inspired by this, we observe that particular solutionsfor nonlinear equations can be constructed by the generalized method of separation of variables(see [10]). Based on this idea, a final result on non-linear diffusive equation in H is here considered. Theorem 3.5.
The generalized time-fractional nonlinear diffusive equation in H +2 (3.9) (cid:0) O β,f u (cid:1) ( η, t ) = 1sinh η (cid:18) ∂∂η sinh η ∂∂η (cid:19) u n ( η, t ) − u ( η, t ) , n > , ( η, t ) ∈ R + × R + admits as a particular solution (3.10) u ( η, t ) = g ( η ) · E β (cid:0) − ( f ( t )) β (cid:1) , where g ( η ) is such that dg n dη = η . Proof.
We first search a solution by the generalized separation of variables in the simple form u ( η, t ) = r ( t ) · g ( η ) . We observe that if g ( η ) is such that dg n dη = 1sinh η , then 1sinh η (cid:18) ∂∂η sinh η ∂∂η (cid:19) u n ( η, t ) = 0and therefore by substitution we have that g ( η ) (cid:0) O β,f r (cid:1) ( t ) = − g ( η ) r ( t )and therefore r ( t ) = E β (cid:0) − ( f ( t )) β (cid:1) . (cid:3) R. GARRA, F. MALTESE, AND E. ORSINGHER
The study of nonlinear diffusive equations in H +2 is not the main object of this paper, but weobserve that by starting from this simple result, it is possible to construct exact solutions for manydifferent classes of generalized time-fractional nonlinear equations in H +2 , a completly new topic ofresearch. References [1] Almeida, R., A Caputo fractional derivative of a function with respect to another function,
Communicationsin Nonlinear Science and Numerical Simulation , , , 460–481.[2] Almeida, R. (2019). Further properties of Osler’s generalized fractional integrals and derivatives with respectto another function. Rocky Mountain Journal of Mathematics, 49(8), 2459-2493.[3] Colombaro, I., Garra, R., Giusti, A., Mainardi, F. (2018). Scott-Blair models with time-varying viscosity.Applied Mathematics Letters, 86, 57-63.[4] D’Ovidio, M., Orsingher, E., Toaldo, B. (2014). Fractional telegraph-type equations and hyperbolic Brownianmotion. Statistics & Probability Letters, 89, 131-137.[5] Gertsenshtein, M.E., Vasiliev, V.B., 1959. Waveguides with random inhomogeneities and Brownian motion inthe Lobachevsky plane, Theory of Probability & its Applications, 4(4), 391-398.[6] Kilbas, A. A. A., Srivastava, H. M., Trujillo, J. J., Theory and applications of fractional differential equations ,2006