Variable order differential equations and diffusion processes with changing modes
Abstract
In this paper diffusion processes with changing modes are studied involving the variable order partial differential equations. We prove the existence and uniqueness theorem of a solution of the Cauchy problem for fractional variable order (with respect to the time derivative) pseudodifferential equations. Depending on the parameters of variable order derivatives short or long range memories may appear when diffusion modes change. These memory effects are classified and studied in detail. Processes that have distinctive regimes of different types of diffusion depending on time are ubiquitous in the nature. Examples include diffusion in a heterogeneous media and protein movement in cell biology.
Department of Mathematics, Tufts University, Medford, MA, USA
Mathematics and Statistics Department, University of New Mexico, Albuquerque, NM, USA
Keywords Variable order differential equations, short memory, long memory, diffusion with changing modes, Cauchy problem, MittagLeffler function
1 Introduction
Diffusion processes can be classified according to the asymptotic behaviour of their mean square displacement (MSD) as a function of time. If the dependence of the MSD on time is linear then the process is classified as normal, otherwise as anomalous. For many processes, the MSD satisfies
(1) 
where is a constant. If the diffusion is normal, if the process is superdiffusive, while if the process is subdiffusive [21, 36]. The ultraslow diffusion processes studied in [5, 20, 22] lead to logarithmic behaviour of MSD for large . The MSD of more complex processes with retardation (see [5, 12]) behaves like for small, and for large, where . Subdiffusive motion with was recorded in [10], and with in [29], depending on macromolecules and cells. In [16, 17] protein movement is studied in the cell membrane with a few types of compartments and made a conclusion that depends on time scales. Our models are subdiffusive, but of variable order with order function .
It is well known that simple homogeneous subdiffusive processes can be modeled using a fractional order partial differential equation where only the time derivative has a constant fractional order [21]. Variable fractional order derivatives and operators were studied by N. Jacob, at al. [15], S. G. Samko, at al. [27, 28], W. Hoh [14]. Recently A. V. Chechkin, at al. [6] used a version of variable order derivatives to describe kinetic diffusion in heterogeneous media. In the recent paper [19], Lorenzo and Hartley introduced several types of fractional variable order derivatives and applied them to engineering problems. We will modify these operators, restrict them to order functions that are piecewise constant and then apply the resulting variable order partial differential equations (VOPDE) to diffusion processes with changing diffusion modes. An important aspect of the modeling is that the VOPDEs provides a description of memory effects arising from a change of diffusion modes that are distinct from the “long range memory” connected with the nonMarkovian character of diffusion. Thus, in the VOPDE based description of anomalous diffusion models, both nonMarkovian long range memory and new type of memory may be present simultaneously.
The paper is organized as follows. In Section 2 we introduce background material. In Section 3 we study the memory effects arising in connection with a change of diffusion modes. In Sections 4 and 5 we study the mathematical model of diffusion processes with changing modes in terms of an initial value problem for VOPDE. Namely, we prove the theorem on the existence and uniqueness of a solution of the initial value problems for variable order differential equations and study some properties of a solution. The theorems are proved under the assumption that diffusion mode change times are known.
2 LHparallelogram and variable order derivatives
Recently Lorenzo and Hartley [19] introduced three types of derivatives of variable fractional order , , all of which are special case of a more general fractional order derivative
(2) 
where and are real parameters, , and
(3) 
For convenience in studying of initial value problems, we prefer to use the closely related Caputo type operator
(4) 
To describe the properties of the kernel (3) and the fractional derivative operators (2) and (4) we introduce the LorenzoHartley (LH) causality parallelogram [19] The kernel (3), and thus, both the operators (2) and (4) are weakly singular for . Further, denote
(5) 
where ^{1}^{1}1if then we agree is a given function, which is called an order function.
Our main goal is to model problems where for different time intervals there are different modes^{2}^{2}2see definition in Section 2.3. of diffusion. To this end, let be a partition of the interval into subintervals , where . Then let be the piecewise constant function
(6) 
where is the indicator of the interval and are constants. Under these conditions, the function (5) becomes
(7) 
and the kernel of the fractional order operator (4) becomes
(8) 
with defined in (7).
We think of the input to our model as the triplet , while the output of our model is determined by the kernel (8). Correspondingly, we say that the triplet determines the diffusion mode in the time interval The output is determined by which values of are used to compute the variable order derivative, that is, by which interval the point belongs to. We always assume that and then note that yields and that . This means that the operators and use information taken in the time subinterval if is positive and from the subinterval if is negative. In both cases, the length of this interval is . The condition predetermines the causality, since for all and
2.1 Generalized function spaces ,
Let , , be two conjugate numbers. The generalized functions space , which we are going to introduce is distinct from the classical spaces of generalized functions. In the particular case of this space was first used by Yu.A.Dubinskii [8] in the course of initialvalue problems for pseudodifferential equations with analytic symbols. Later, the general case for was studied in [31, 32]. Here we briefly recall some basic facts related to these spaces, referring the interested reader to [31, 11] for details.
Let be an open domain and a system of open sets be a locally finite covering of , i.e., , . This means that any compact set has a nonempty intersection with a finite number of sets . Denote by a smooth partition of unity for . We set and It is clear that and for . Further, by (or ) for a given function we denote its Fourier transform, formally setting and by the inverse Fourier transform, i.e. The support of a given we denote by .
Definition 2.1
Let . Denote by the set of functions satisfying the conditions (1)–(3):

;

for ;

.
Lemma 2.1
For the relations
hold, where denote the operation of continuous embedding.
It follows from Lemma 2.1 that form an increasing sequence of Banach spaces. Its limit with the inductive topology we denote by
Definition 2.2
The inductive limit topology of is equivalent to the following convergence.
Definition 2.3
A sequence of functions is said to converge to an element iff:

there exists a compact set such that for all ;

for .
Remark 2.1
According to the PaleyWienerSchwartz theorem, elements of are entire functions of exponential type which, restricted to , are in the space .
The space topologically dual to , which is the projective limit of the sequence of spaces conjugate to , is denoted by
Definition 2.4
In other words, is the space of all linear bounded functionals defined on the space endowed with the weak topology. Namely, a sequence of generalized functions converges to an element in the weak sense if for all the sequence of numbers converges to as . We recall that the notation means the value of on an element .
2.2 Pseudodifferential operators with constant symbols.
Now we recall some properties of pseudodifferential operators with symbols defined and continuous in a domain . Outside of or on its boundary the symbol may have singularities of arbitrary type. For a function the operator corresponding to is defined by the formula
(9) 
Generally speaking, does not make sense even for functions in the space . In fact, let be a nonintegrable singular point of and denote by some neighborhood of . Let us take a function with for and . Then it is easy to verify that . On the other hand, for the integral in Eq. (9) is convergent due to the compactness of . We define the operator acting in the space by the duality formula
(10) 
Lemma 2.2
The spaces and are invariant with respect to the action of an arbitrary pseudodifferential operator whose symbol is continuous in . Moreover, if is a multiplier on for every , then this operator acts continuously.
Remark 2.2
In the case an arbitrary pseudodifferential operator whose symbol is continuous in acts continuously without the additional condition for to be multiplier in for every .
2.3 Subdiffusion processes.
As is known [11, 21], a (sub)diffusion process is governed by the fractional order partial differential equation
(11) 
where is the Caputo fractional derivative of order and
Many diffusion processes driven by a Brownian motion can be described by equation (11) with a second order elliptic differential operator and Lévy stochastic processes (which include jumps) also connected with (11) and an elliptic pseudodifferential operator (see, e.g. [2]). In particular, if particle jumps are given by a symmetric Lévy stable distribution with infinite mean square displacement then is a hypersingular integral, defined as the inverse to the RieszFeller fractional order () operator (for details see [26]). A wide variety of nonGaussian stochastic (subdiffusive) processes lead to equation (11) with (see [21, 22]). For diffusion governed by distributed order differential equations see [1, 34]. The parameter determines the subdiffusive mode, which is slower than the classical free diffusion.
Generalizing this approach we will say that the diffusion mode in the time interval is governed by the equation
(12) 
The entire process then can be described by the equation
(13) 
where is the variable fractional order operator with the kernel in (8).
In Section 4 we will prove the existence of solutions to the initial value problem defined by the differential equation (13).
3 Changing of modes: ’shortrange’ and ’longrange’ memories
We call the triplet admissible if and . Diffusion in complex heterogeneous media is accompanied by frequent changes of diffusion modes. It is known that a particle undergoing nonMarkovian movement possesses a memory of past (see [21, 36]). Protein diffusion in cell membrane, as is recorded in [24, 25] is anomalous diffusion. Descriptions of this process using random walks also shows the presence of nonMarkovian type memory [1, 13, 18]. It turns out, there is another type of memory noticed first by Lorenzo and Hartley in their paper [19] in some particular cases of and . This kind of memory arises when the diffusion mode changes.
In this section we study a special case of this phenomenon where there is a single change of diffusion mode, that is, a subdiffusion mode given by an admissible triplet changes to a subdiffusion mode corresponding to another admissible triplet at some particular time .
Definition 3.1
Let and be two admissible triplets. Assume the diffusion mode is changed at time from mode to mode. Then the process is said to have a ’shortrange’ (or short) memory, if there is a finite such that for all mode holds. Otherwise, the process is said to have a ’longrange’ (or long) memory.
Remark 3.1
According to definition (3.1), a diffusion mode has a long memory if the influence of the old diffusion mode never vanishes, even though the diffusion mode is changed, i.e. the particle does not forget its past. In the case of short memory, the particle remembers the old mode for some critical time, and then forgets it fully, recognizing the new mode.
Theorem 3.1
Let and . Assume the diffusion mode is changed at time to the diffusion mode. Let and Then the process has a short memory. Moreover,
(i) diffusion mode holds for all
(ii) diffusion mode holds for all
(iii) a mix of both and diffusion modes holds for all
Proof. Let for and for Assume . Denote So, the diffusion mode holds if . Let . Then for every we have . This means that the order operator in takes the value giving (i). If then for all , . Hence, obtaining (ii). Now assume Denote . Obviously It follows from dividing by that , i.e. It is easy to check that if then giving while if then giving Hence, in this case the mix of both and diffusion modes is present.
Theorem 3.2
Let and . Assume the diffusion mode is changed at time to the diffusion mode. Let and Then the process has a short memory. Moreover,
diffusion mode for all
diffusion mode holds for all
a mix of both and diffusion modes holds for all
Proof. Let . Assume again for and for As in the previous theorem, denote First we notice that if then , which implies , giving (). Now let be any number. Then for we have , which yields So, we get (). Now assume Again denote . Obviously It follows from dividing by that , i.e. It is easy to check that if then giving while if then giving Hence, in this case the mix of both and diffusion modes is present, obtaining ().
Corollary 3.1
Let and . Assume the diffusion mode is changed at time to the diffusion mode. Let . Then the process has a short memory. Moreover,
(a) for all there holds diffusion mode;
(b) for all there holds diffusion mode.
Proof. If then we have for and for .
Corollary 3.2
Let or . Assume the diffusion mode is changed at time to the diffusion mode. Then the process has the long memory.
Proof. According to the structure of LHparallelogram implies In this case . If then and In both cases we a have long memory effect.
Remark 3.2
Notice, that if then there is no intervals of mix of modes. Moreover, if then In this sense we say that a process has no memory. For all points except and the operator has a short memory. The memory is stronger in the region and weaker in . On the line we have The lines and identify the long range memory.
4 The Cauchy problem for variable order differential equations
In this section we study the Cauchy problem for variable order differential equations with a piecewise constant order function , where is the indicator function of We assume that the diffusion mode change times are known, and set We assume that the solution of the initial value problem for the VOPDE (13) is continuous^{3}^{3}3in the topology of (or ). when the diffusion mode changes.
Thus, the Cauchy problem is formulated in the form
(14) 
(15) 
(16) 
where is a pseudodifferential operator with a continuous symbol and are actual mode change times. It follows from Theorems 3.1 and 3.2 that are defined through and
We note that, since the integration operator order depends on the variable , a variable order analog of the integration operator becomes
(17) 
which we call a variable order integration operator.
For further purpose we recall the definition of the MittagLeffler function [7, 23] in the power series form
Obviously, if For all is an entire function of type 1 and order Note that is completely monotone [23], and has asymptotics
Lemma 4.1
Assume and is the integer part of Let be a function continuous in . Then for arbitrary and every the estimate
(18) 
holds with
Proof. Let be a function continuous in . For large enough, so that we have . Taking this into account, for all such and for all we obtain the estimate
and hence, the estimate in Eq. (18).
Let be critical points corresponding to the mode change times We accept the conventions Let be the MittagLeffler function with parameter . Now we introduce the symbols which play an important role in the representation of a solution. Let
(19) 
and
(20) 
Further, we define recurrently the symbols
and if
is defined for and for all then for
(21) 
for
4.0.1 The case .
Theorem 4.1
Proof. It is not hard to verify that
(23) 
where
Multiplying both sides of equation (14) by and applying the formula (23), we obtain
(24) 
Let Then and In this case taking into account the initial condition (15), we can rewrite equation (24) in the form
(25) 
The obtained equation can be solved by using the iteration method. Determine the sequence of functions in the following way. Let and by iteration
(26) 
We show that this sequence is convergent in the topology of the space and its limit is a solution to the Cauchy problem (14),(15). Moreover, this solution can be represented in the form of functional series
(27) 
Indeed, it follows from the iteration process (26) that
(28) 
Now we estimate applying Lemma 18 term by term in the right hand side of (28). Indeed, let Then taking into account the fact that the Fourier transform in commutes with , we have
Further, since is continuous on there exists a constant , such that