## Abstract

A Cauchy problem for a time distributed-order multi-dimensional diffusion-wave equation containing a forcing term is reinterpreted in the space of tempered distributions, and a distributional diffusion-wave equation is obtained. The distributional equation is solved in the general case of weight function (or distribution). Solutions are given in terms of solution kernels (Green's functions), which are studied separately for two cases. The first case is when the order of the fractional derivative is in the interval [0, 1], while, in the second case, the order of the fractional derivative is in the interval [0, 2]. Solutions of fractional diffusion-wave and fractional telegraph equations are obtained as special cases. Numerical experiments are also performed. An analogue of the maximum principle is also presented.

## 1. Introduction

A one-dimensional diffusion-wave equation with one fractional derivative of the Caputo type was introduced by Mainardi (1996). The equation was solved for the Cauchy and signalling problem by use of a Laplace transformation and Green's functions. An extensive overview of the diffusion-wave equation with a single time- and space-fractional derivative can be found in the paper by Mainardi *et al*. (2001), where solutions are obtained by means of Laplace and Fourier transformations. There is a recent paper on the same subject by Yu & Zhang (2006), introducing a new function of Mittag–Lefler type in order to solve the space–time fractional diffusion-wave equation. Hanyga (2001) dealt with a multi-dimensional space-fractional diffusion equation with a source term, while, in the other two papers, Hanyga (2002*a*,*b*) studied the well-posedness of Cauchy problems for a multi-dimensional space- and time-fractional diffusion-wave equation in the framework of an abstract Volterra equation. Solutions are calculated by the use of Laplace and Fourier transformations in cases of one, two and three space dimensions. We also mention Eidelman & Kochubei (2004), where a fractional diffusion equation was studied with an elliptic second-order differential operator. Cauchy and signalling problems of the diffusion-wave equation with two fractional derivatives of different order (fractional telegraph equation) on an unbounded domain were studied by Atanackovic *et al*. (2007). In the same paper, in the case of a bounded domain, the maximum principle was proved if the orders of the derivatives do not exceed 1. Another paper dealing with the similar equation is that of Langlands (2006). Furthermore, we mention that the time and space distributed-order diffusion equation was studied by Chechkin *et al*. (2002), through the analysis of second moments of certain probability density functions. A similar approach can be found in the paper of Sokolov *et al*. (2004). Time distributed-order diffusion equations were solved by Naber (2004) for the Dirichlet, Neumann and Cauchy problems. Time distributed-order equations of relaxation are studied by Mainardi *et al*. (2007*b*). In Mainardi *et al*. (2007*a*, 2008), solutions are obtained by the use of Mellin–Barnes integrals, Laplace and Fourier transformations. Hanyga (2007) treated (1.1) with supp *ϕ*⊂[*α*, 1], so that *ϕ* is a right-continuous non-decreasing function on the interval [*α*, 1], satisfying *h*(*β*)=0 for *β*<*α*. Asymptotic analysis of solutions was carried out by Hanyga (2007) using regularly varying functions. An equation similar to (1.1) was treated by Kochubei (2008) with *ϕ*∈*C*^{2}[0, 1], where *ϕ*(*α*)≔*α*^{ν}*ϕ*_{1}(*α*), *ϕ*_{1}(*α*)≥*ρ*>0, 0≤*α*≤1, *ν*≥0. The existence was proved by the use of asymptotic properties of Fourier and Laplace transformations and their inversions.

This work deals with the Cauchy problem,(1.1)(1.2)interpreted in as (*H* is the Heaviside function)(1.3)The meaning of *D*_{ϕ}*u* is explained in our previous paper (Atanackovic *et al*. in press). Recall, if , *α*∈[0, 2] and be continuous in [0, 2]. Then, we define(1.4)Let , *α*_{j}∈[0, 2], *j*∈{0, …, *k*}. Then, we define(1.5)We assume and . Equation (1.3), corresponding to the Cauchy problems (1.1) and (1.2), is obtained in §3 (cf. (3.5)). We study (1.3) by the use of Fourier and Laplace transformations analysing the solution kernel (Green's function) in the context of distribution theory. We obtain more general solutions than we were able to obtain in our previous paper (Atanackovic *et al*. in press) by using the theory for the Volterra equation. The regularity properties are obtained through the investigation of the convolution form of solutions. In this paper, the existence is proven for a general *ϕ* satisfying condition 3.1, if supp *ϕ*⊂[0, 1], or condition 3.2, if supp *ϕ*⊂[0, 2]. Several special cases of *ϕ* are also studied, especially the case *ϕ*(*α*)=*τ*^{α}, *α*∈[0, 2], *τ*=const., *μ*=0, *η*=2. It generalizes a recently treated case presented by Mainardi *et al*. (2007*a*), where *τ*=1, *η*=1.

As an additional general result, we prove an analogue of the maximum principle in the case *F*=0, *μ*=0, *η*=1 (see Atanackovic *et al*. (2007) for a special result).

Concerning the notation, we call (1.1)

a distributed diffusion-wave equation if [

*a*,*b*]⊂supp*ϕ*⊂(0, 2], where*a*<1<*b*<2 and =∇^{2};a distributed diffusion equation if supp

*ϕ*⊂(0, 1] and =∇^{2}; anda distributed wave equation if supp

*ϕ*⊂(1, 2] and =∇^{2}.

Since this paper is a continuation of our previous paper (Atanackovic *et al*. in press), first we will recall the results of that paper. We have studied (1.1), where the Laplacian is replaced by an operator and *ϕ*∈′(), supp *ϕ*⊂[0, 2]. We refer to §2 for the interpretation. The Cauchy problems (1.1) and (1.2) are analysed through the corresponding Volterra-type equation and operator techniques of Banach spaces. Using Prüss (1993), we determined continuous solutions to the Volterra equation(1.6)whereif supp *ϕ*⊂[0, 2], andif supp *ϕ*⊂[0, 1]. Here, and we refer to (3.7) for *B*_{u} and *B*_{v}. We have assumed that is an unbounded linear operator defined on a dense subset *D*() of a suitably defined Banach space *X*. In particular, we consider the special case when =∇^{2} and *X*=*L*^{2}(^{n}) (then *D*()=*W*^{2,2}(^{n})). The existence and uniqueness of the solution to the Volterra equation (1.6) is proved forwith appropriate assumptions on *u*_{0}, *v*_{0} and *F*. Solutions are obtained via resolvent operators. By additional assumptions on *u*_{0}, *v*_{0} and *F*, solutions to the Volterra equation (1.6) are also solutions to the Cauchy problems (1.1) and (1.2).

Our paper is organized as follows. Section 2 is devoted to the definitions of the Caputo, Riemann–Liouville and distributed-order fractional derivatives within the space of distributions. The Laplace transformation is also considered. In §3, we present our main results. We reinterpret (1.1) and (1.2) in the sense of distributions and introduce the solution kernel by the use of Fourier and Laplace transformations in order to write the solution via a solution kernel. In §3*a,* we consider the solution kernel of the distributed diffusion equation (supp *ϕ*⊂[0, 1]), while, in §3*b*, we consider the solution kernel of the distributed diffusion-wave equation (supp *ϕ*⊂[0, 2]). Section 4 is devoted to the study of some special cases of *ϕ*. Numerical examples are provided for each of the cases. In §5, the Laplace inversion is obtained by the use of the Post inversion formula and a numerical example is presented. The maximum principle for the distributed diffusion equation is given in §6. Finally, §7 contains concluding remarks and some directions for further research.

## 2. Caputo, Riemann–Liouville and distributed-order fractional derivatives of functions with the values in the space of distributions

Let us first fix the notation of spaces that will be used throughout the paper. _{+} is the set . denotes the space of locally integrable functions in , and is its subspace consisting of functions vanishing on (−∞,0]. *AC*^{m}(*J*), *J*=[0, *T*], *T*>0, *m*∈, denotes the space of functions *f*, such that *f*^{(k)}, *k*∈{0,1, …, *m*−1} are continuous and *f*^{(m−1)} are absolutely continuous, which means that *f*^{(m)}∈*L*^{1}(*J*). We write if *f* ∈*AC*^{m}([0, *T*]) for every *T*>0. () is the space of smooth functions and ′() is the space of compactly supported distributions, i.e. the dual of (). (^{n}) denotes the space of rapidly decreasing smooth functions. ′(^{n}) denotes the space of tempered distributions, i.e. the dual of (^{n}). It is said that if and . Similarly, we define . In particular, if *n*=0, we have and similarly . Recall, and are convolution algebras. If a function or a distribution *f* is restricted to some subinterval *J*⊂*I*, we will use the same notation, *f*=*f*|_{J}, and if *f* has an additional property, we will say that it belongs to the intersection of appropriate spaces. For example, means that and . We say that if , for every *T*>0, which means that, for every . We denote by the space of functions , such that, for every , is in . If *f* defines a distribution in , then we write .

### (a) Caputo and Riemann–Liouville fractional derivatives. Laplace transformation

Recall (Vladimirov 1984) that (*Γ* is the Euler gamma function)is a family of tempered distributions supported by [0, ∞), which satisfy .

The Caputo time-fractional derivative of order *α*∈(*m*−*1*, *m*], *m*∈, applied to a function , is defined as(2.1)The Riemann–Liouville time-fractional derivative of order *α*∈(*m*−1, *m*], *m*∈, applied to a function , is defined as(2.2)

Let . Then, connection between the Caputo and Riemann–Liouville fractional derivatives is given byas can be found in a tutorial paper by Mainardi (1997).

Let and putHere, means that we consider *u* as a regular distribution; is determined by *u* (so is not the complex conjugate).

Let . Then, we introduce the distributional fractional derivative of order *α*∈ as(2.3)

Laplace transformation is applied in the distributional sense as follows. Let . Then,This definition does not depend on the choice of a smooth function *θ*, *θ*(*t*)=1 on *t*∈(−*T*, ∞) and *θ* (*t*)=0 on *t*∈(−∞, −2*T*), for some *T*>0. If , where *X* is a space of functions or distributions, then is an analytic function in _{+} having values in *X*.

Let and let the distributional fractional derivative be given by (2.3), then

### (b) Distributed-order fractional derivative. Laplace transformation

Let *h* be an element of . Then, as in proposition 2.1 of Atanackovic *et al*. (2009), the mappings and are smooth. We cite the following definition from the paper by Atanackovic *et al*. (2009).

Let *ϕ*∈′(), supp *ϕ*⊂[0, 2] and . Then,is defined as an element of bywhere _{D}*D*_{ϕ}*h* is called the distributed-order fractional derivative of *h*.

In proposition 2.2, we apply a Laplace transformation to _{D}*D*_{ϕ}*h*, . This proposition is stated in Atanackovic *et al*. (2009).

*Let ϕ*∈′(), supp *ϕ*⊂[0, 2] *and* . *Then, the following hold*:

*is a linear and continuous mapping from**to**and*(2.4)

*is an analytic function*.*Let ϕ*∈*C*([*μ*,*η*]), [*μ*,*η*]⊂[0, 2]*and ϕ*(*α*)=0*α*∈[0, 2]\[*μ*,*η*].*Then,*(2.5)

## 3. Solution to the Cauchy problem and the corresponding distributional equation (3.5) by the use of a solution kernel

Let us first reinterpret the Cauchy problem (1.1) and (1.2), with =∇^{2}, within the space of distributions. Let , and *ϕ*∈′(), supp *ϕ*⊂[0, 2]. Then, for every *ψ*∈(^{n}), (1.1) reads(3.1)where *D*_{ϕ}*u* is interpreted as in (1.4) or (1.5) and, therefore,or

One can see that terms , and in (3.1) can be interpreted as the functions solely of variable *t*>0. Hence, for every , regular distributions , and are defined by(3.2)Then, , , . With (3.2),(3.3)we reinterpret the Cauchy problems (1.1) and (1.2) in the sense of distributions, rewriting (3.1) as(3.4)Actually, (3.4) (in ) has the form(3.5)and this equation will be the subject of further analysis, as noted in §1.

Applying the Laplace transformation (in the sense explained in §2) to (3.4), one obtains(3.6)where , and are used to denote the Laplace transformations of , and . Let(3.7)These functions are analytic in _{+}. Moreover, assumeThen, equation (3.6) becomes(3.8)

Note that the Fourier transformation of , with respect to the spatial variable, is defined bywhereWe will use the identity , , where . By applying the Laplace transformation to (3.2), (3.3) andone obtains that, for every , *s*∈_{+} and *t*∈,(3.9)(3.10)Now, (3.8) with (3.9) and (3.10), written in , becomesFinally, by solving the resulting equation, we obtain(3.11)Thus, (3.11) represents the Fourier and Laplace transformations of a solution to the distributional equation (3.5).

We introduce the solution kernel via its Fourier and Laplace transformations as(3.12)This formula has to be understood in the sense of distributions, since is an analytic function in _{+} having values in ′(^{n}). After making a Fourier inversion of (3.11), we obtain (in ′(^{n}))(3.13)Note that, in the case when supp *ϕ*⊂[0, 1], the distributed diffusion equation is obtained by (3.13) by setting *v*_{0}≡0, since the initial condition does not appear. After applying a Laplace inversion to (3.13), we obtain a solution for the distributed diffusion equation (*v*_{0}=0) as(3.14)as well as a solution for the distributed diffusion-wave equation as(3.15)where denotes the convolution with respect to ** x** and

*t*.

Distribution *u* that solves the Cauchy problem (3.5) is given by (3.14), or by (3.15), where the explicit form of a weight function *ϕ* has to be given in order to obtain solution kernels *g* and functions *B*_{u} and *B*_{v}. In the following sections, we determine *g* in both the distributed diffusion and distributed diffusion-wave equations.

### (a) Solution kernel of the distributed diffusion equation

Similarly, as in Chechkin *et al*. (2002) and Kochubei (2008), rewrite (3.12) in the formThis leads to (*x*=|** x**|,

**∈**

*x*^{n},

*s*∈

_{+})and (3.16)Note that, for

*p*=0, it yields ,

**∈**

*x*^{n}. Distribution

*I*

_{n}in (3.16) is obtained after reducing the exponent to the binomial square and integrating over

^{n}as(3.17)The inverse Laplace transformation in (3.16) giveswhere , with suitably chosen

*s*

_{0}, is a Bromwich contour. Thus, we obtain(3.18)

Suppose that *ϕ* satisfies

, and

(3.19)has

*s*=0 and ∞ as the only branch points and all its singularities lie in the half-plane Re*s*<0.

Then, integral *I* can be evaluated by the Cauchy residue theoremwhere is the contour given in figure 1. Since for supp *ϕ*⊂[0, 1], in the limit *R*→∞, *ϵ*→0, integrals along contours *Γ*_{1}, *Γ*_{ϵ} and *Γ*_{4} tend to zero, while integrals along contours *Γ*_{2}, *Γ*_{3} and *γ*_{0} give (*t*>0, *p*>0)Thus, *I*, given by (3.18), is of the formwhich can be rewritten as (cf. (3.7) for )(3.20)

The solution kernel *g* given by (3.18), where *I*_{n} and *I* are given by (3.17) and (3.20), respectively, can be explicitly written. Thus, the solution to (3.12) is given by(3.21)After the weight function *ϕ* is specified, the explicit form of *g* follows from (3.21). Note that for fixed *t*>0, *g* is a distribution in .

### (b) Solution kernel of the distributed diffusion-wave equation

Similarly, as was carried out by Hanyga (2002*b*), Kochubei (2008) and Mainardi *et al*. (2008), we use (3.12) to obtain a solution kernel. Three cases can be distinguished, according to the number of spatial dimensions.

#### (i) One-dimensional case

The inverse Fourier transformation of (3.12), for *n*=1, yieldsThe function is evaluated by integration in the complex plane, using the Cauchy residue theorem(3.22)where contour is either for *x*>0 or for *x*<0, as shown in figure 2. In writing (3.22), we assumed that has no zeros on the real axis for any *z*, i.e. if *z*_{1,2} are solutions of , then and . Let be the Bromwich contour in the Laplace transformation. The Cauchy residue theorem (3.22) determines as(3.23)

Suppose that *ϕ* satisfies

, and

function given by (3.23) has

*s*=0 and ∞ as the only branch points and all its singularities lie in the half plane Re*s*<0.

Then, the inverse Laplace transformation of (3.23), or the solution kernel itself, is evaluated similarly as in §3*a*. The Cauchy residue theorem applied to yields(3.24)where is the contour given in figure 1.

In the limit *R*→∞, *ϵ*→0 integrals along contours *Γ*_{1}, *Γ*_{ϵ} and *Γ*_{4} tend to zero for all *α*∈(0, 2), while integrals along contours *Γ*_{2} and *Γ*_{3} give (*x*∈, *t*>0)Sincethe explicit form of solution kernel *g*_{1} follows from (3.24) and can be written as(3.25)Note that for fixed *t*>0, *g*_{1} is a distribution in .

#### (ii) Two-dimensional case

The inverse Fourier transformation of the solution kernel (3.12) for *n*=2, is obtained in polar coordinates. Let ** ω**=

*ω*

_{1}

*e*_{1}+

*ω*

_{2}

*e*_{2}. By pointing unit vector

*e*_{1}along

**, the scalar product**

*x***.**

*x***becomes**

*ω***.**

*x*

*ω**=xω*cos

*φ*, where

*x*=|

**|,**

*x**ω*=|

**| and**

*ω**φ*is a polar angle. Therefore, function is of the form(3.26)Since , where is a Bessel function of the first kind, and and

*K*

_{0}(

*z*) is a modified Bessel function of the second kind (also known as the Macdonald function), relationship (3.26) is transformed into(3.27)Therefore, the solution kernel in two dimensions reads(3.28)where is the Bromwich contour.

#### (iii) Three-dimensional case

The inverse Fourier transformation of the solution kernel (3.12), for *n*=3, is obtained in spherical coordinates. Again, if ** ω**=

*ω*

_{1}

*e*_{1}+

*ω*

_{2}

*e*_{2}+

*ω*

_{3}

*e*_{3}and by choosing

*e*_{3}so that

**=**

*x**x*

*e*_{3}, the scalar product

**.**

*x***becomes**

*ω***.**

*x***=**

*ω**x*

*ω*cos

*θ*, where

*x*=|

**|,**

*x**ω*=|

**| and**

*ω**θ*is the angle between

*e*_{3}and vector

**. Therefore, function is of the formIntegrating over angles**

*ω**φ*and

*θ*and by substituting

*ω*=−

*ω*, the following is obtained:or by noting becomes(3.29)The inverse Laplace transformation of (3.29), or the solution kernel itself, is

By taking into account (3.25), the solution of the kernel finally becomes(3.30)

## 4. Special cases of the derivative weight function

By specifying the form of the weight function *ϕ*, it is possible not only to recover known equations and their solutions, but also to construct new solutions to generalization of previously treated equations.

### (a) Case *ϕ*(*μ*)=*δ*(*μ*−*α*)

Let *α*∈(0, 2). The wave-diffusion equation, which was thoroughly studied in Mainardi (1997) for one-dimensional and in Hanyga (2002*a*,*b*) for *d*-dimensional cases, *d*∈{1, 2, 3}, is obtained by setting *ϕ*(*μ*)=*δ*(*μ*−*α*), *μ*=0, *η*=2, , *v*_{0}=0 and *u*_{0}=1 in (1.1) and (1.2).

First, use (3.7) to obtain function and then calculate the function (for *α*∈(0, 2)) asThe explicit form of the solutions can be constructed as follows.

In the case when *α*∈(0, 1), the solution kernel *g*, given by (3.21), can be written asFrom (3.19), we calculate , , *t*>0, *p*>0. Since it satisfies condition 3.1 (i.e. its only branch points are *s*=0 and ∞ and it has no singularities in the right complex plane), the solution itself is given by (3.14) and it reads(4.1)

Let *α*∈(0, 2). For a one-dimensional case (*n*=1), function , calculated by (3.23), readsIt has neither branch points different from *s*=0 and ∞ nor singularities in the right complex plane, i.e. it satisfies condition 3.2. Therefore, *g*_{1}, given by (3.25), becomes(4.2)Now we will consider the case *α*∈(0, 1). Then, is a smooth function with the values in . Let us write (4.2) for *t*=0 and make a change of variables as . Then, (4.2) reads, for The first time derivative of (4.2) in *t*=0 readsand, with the same change of variables as before, we obtain, for Note that, for *α*∈[1,2], *g*_{1} is a distribution with respect to time.

The solution can now be computed using (3.14) (*x*∈, *t*>0),(4.3)

As an example, we set *x*=1.23, *t*∈(0, 5), *α*=0.45 in (4.3) and plot the resulting expression in figure 3 as a solid curve. Also in figure 3, we present the plot of *u*(1.23,*t*) (denoted by dots), obtained by using the solution given in Mainardi (1996),

As can be seen in figure 3, for the chosen set of parameters, both solutions agree well.

### (b) *Case ϕ*(*μ*)=*τδ*(*μ*−*α*)+*δ*(*μ*−*β*)

The wave-diffusion equation with two fractional derivatives of different order, or the generalized telegraph equation, which was studied in Atanackovic *et al*. (2007), is obtained by setting *μ*=0, *η*=2, *ϕ*(*μ*)=*τδ*(*μ*−*α*)+*δ*(*μ*−*β*), for 0<*β*<*α*<2, , *F*=0, *v*_{0}=0 and *u*_{0}=1 in (1.1) and (1.2).

We use (3.7) to obtain function and then calculate

The explicit form of solutions can also be constructed. In the case when 0<*β*<*α*<1, the solution kernel *g*, given by (3.21), has the formwhere we used that , , *t*>0, *p*≥0, calculated by (3.19), has no singularities in the right complex plane and its only branch points are *s*=0 and ∞, i.e. it satisfies condition 3.1. Thus, using (3.15), we obtain(4.4)

The solution kernel for the same case and *n*=1 is given via (3.25) by (*x*∈, *t*>0)(4.5)Note that the function , *x*∈, *s*∈_{+}, calculated by (3.23), has no singularities in the right complex plane and no branch points when 0<*β*<*α*<1, i.e. it satisfies condition 3.2. The solution can now be computed by using (3.15)(4.6)

For *x*=1.232, *t*∈(0, 8), *α*=0.9, *β*=0.56, *τ*=10^{−3} and the plot of equation (4.6) is presented in figure 4 by the solid curve. In the same figure, we show *u*(1.232, *t*) denoted by dots, obtained by using the solutionwhich is given in Atanackovic *et al*. (2007).

Again, for the selected set of parameters, both solutions agree well.

### (c) Case *ϕ*(*α*)=*τ*^{α}

This is the simplest case guaranteeing dimensional homogeneity. Parameter *τ*>0 can be physically interpreted as the relaxation time.

Functions , for supp *ϕ*⊂(0, 1) and supp *ϕ*⊂(0, 2), and can now be computed by using (3.7) as(4.7)Since (cf. Atanackovic *et al*. 2005), the Laplace inversions of and give(4.8)where we used .

#### (i) Case *ϕ*(*α*)=*τ*^{α}, *μ*=0≤*α*≤*η*=1

The function *J* calculated by (3.19) readsIt satisfies condition 3.1, i.e. its branch points are *s*=0 and ∞, and it has no singularities in the right complex plane, so the solution kernel *g* is given by (3.21). After computing the integrals and , required by (3.21), the solution kernel *g* becomes(4.9)The solution is given by (3.14), where *B*_{u} and *g* are given by (4.8) and (4.9), respectively.

#### (ii) Case *ϕ*(*α*)=*τ*^{a}, *μ*=0≤*α*≤*η*=2

We treat the one-dimensional case only. Since (see (3.23))(4.10)satisfies condition 3.2, (3.25) implies that(4.11)The solution itself is given by (3.15), where *B*_{u}, *B*_{v} and *g*_{1} are given by (4.8) and (4.11), respectively.

## 5. Solution kernel via the Post inversion formula and numerical examples

The inversion of the Laplace transformation can also be obtained by use of the Post inversion formula. Some recent papers on the subject are by Frolov & Kitaev (1998) and Donalto (2002).

The Post inversion formula yields the original function *f* from its Laplace transformation via(5.1)Note that the first approximation of the original is obtained from (5.1) for *n*=0 as . Although the Post inversion formula can be regarded as an analytical result, it can also be used for numerical purposes, where differentiation can be stopped when the required accuracy on values of function *f* is attained.

We apply (5.1) to the Laplace transformation of the solution kernel of the distributed diffusion-wave equation (3.5) in the case when *ϕ*(*α*)=*τ*^{α}, *α*∈(0, 2), for the one- and two-dimensional solution kernels and , given by (4.10) and (3.27), respectively. For these cases, is given by (4.7). Therefore, (5.1) yields(5.2)and(5.3)

Figure 5 presents the plots of *g*_{1} obtained by using the analytical expression (4.11) (denoted by dots) and by using (5.2) (denoted by the solid curve), with *n*=15, for *x*=1, *t*∈(0, 5), *τ*=10^{−3}.

In figure 6, we plot *g*_{2} obtained by (5.3) for *n*=10, for the set of values *x*=1, *t*∈(0, 5), *τ*=10^{−3}.

Derivatives in both (5.2) and (5.3) are calculated symbolically using Mathematica.

## 6. Maximum principle

In this section, we shall extend the results presented in Atanackovic *et al*. (2007) and Luchko (2009) and prove the maximum principle for a distributed diffusion equation. It extends the classical maximum principle for parabolic partial differential equations (e.g. Widder 1975).

We consider a special case of (1.1)(6.1)where *ϕ* is continuous and non-negative for *α*∈[*μ*,*η*], 0≤*μ*<*η*≤1. We use the notation , and *D* is a bounded domain in ^{n}. We denote by ∂*D* the boundary of *D* and .

*Suppose that u satisfies* (*6.1*)*,* *that it is continuous for* *and that* , , *is Hölder continuous with the exponent h*>*η for* . *Moreover, assume that u satisfies initial and boundary conditions*(6.2)*where g and f are continuous*.

*Then, the maximum and minimum values of u over the region* *are assumed either in* *or* .

In order to prove the theorem, lemma 6.2 is stated.

*Suppose that an absolutely continuous positive function y attains a positive maximum in* . *Then, for any α*∈[0, 1], *we have*(6.3)

Starting from the definition of the left fractional derivative *t*>0 and 0<*α*<1, we have (see Samko *et al*. 1993, p. 111; Podlubny 1999, p. 43)(6.4)Similarly for the Caputo derivative, from , *t*>0, we obtain(6.5)Now suppose that *y* is a non-decreasing positive function on the left of the maximum in . Then, *y*(*t*^{*})−*y*(*t*)≥0, *t*∈[0,*t*^{*}]. From (6.4) and (6.5), we conclude that(6.6)The result (6.6) is obtained in Nahushev (2003) by a different method. ▪

It is enough to prove the theorem for a positive maximum. Since *u*(** x**,

*t*) is continuous, it reaches its maximum in . If

*u*reaches its maximum at the point (

*x*^{*},

*t*

^{*})

*x*^{*}∈

*D*, , then, at this point, we must have(6.7)so that(6.8)Also, from lemma 6.2 at the point

*t*

^{*}, where

*u*reaches a positive maximum, the inequality(6.9)holds for each

*α*∈(0,1). Then, since

*ϕ*(

*α*)≥0, , we have(6.10)If at least in one of (6.8) or (6.10) the strict inequality holds, then (6.1) is not satisfied at (

*x*^{*},

*t*

^{*}).

So, suppose that in (6.8) and (6.10), we have equalities. Suppose further that the maximum of *u* in or is *M*, while the maximum of *u* in is attained at the point (*x*^{*}, *t*^{*}) and that , *ϵ*>0. We will show that this, again, leads to a contradiction. ConsiderClearly,Choose *k*>0, so that . Then,Since *U* is continuous, it reaches its maximum at a point (*x*_{1},*t*_{1}), so that *U*(*x*_{1},*t*_{1})≥*U*(** x**,

*t*) and, as in (6.7) or (6.10),We will show that

*U*cannot have a maximum in at (

*x*_{1},

*t*

_{1}). This will imply that the hypothesis is wrong and the assertion will be proved. Since(6.11)it follows that(6.12)Also, since at (

*x*_{1},

*t*

_{1}), we have the maximum of

*U*, we must have (see (6.7), (6.9) and (6.10))(6.13)By putting (6.11) and (6.12) in (6.13), we obtainorThus, at (

*x*_{1},

*t*

_{1}), equation (6.1) is not satisfied and a maximum of

*u*is not attained in . ▪

The condition that *u* is continuous in can be relaxed by requiring that *u* is continuous in and that, for , the limit inferior lim inf *u* exists, i.e. if *ϵ*>0, then for each , there is a *δ*_{0}, such that *u*(** x**,

*t*≥−

*ϵ*) for all within the distance

*δ*

_{0}of (

*x*_{0},

*t*

_{0}). Applying the same argument as in Atanackovic

*et al*. (2007) (see also Widder 1975), we can prove the following assertion.

*If u is continuous in* *and for* lim inf *u exists, then u reaches its maximum and minimum in* .

Suppose that *g*(** x**)≥0,

**∈**

*x**D*, and

*f*(

**,**

*x**t*)≥0, . Then, by theorem 6.1,

*u*(

**,**

*x**t*)≥0, .

## 7. Conclusions

In this paper, the Cauchy problems (1.1) and (1.2) motivated us to study (3.5) in the sense of distributions, i.e. in . Our results extend the results of Hanyga (2002*a*,*b*) and Atanackovic *et al*. (2007). They can be summarized as follows:

Solution

*u*to (3.5) is obtained by inverting the Laplace and Fourier transformations in given by (3.11). In the case of an arbitrary weight distribution (or function), the*ϕ*the functions*B*_{u}and*B*_{v}are given by (3.7). The solution*u*is determined as follows:by (3.14), if

*α*∈(0, 1) and*ϕ*satisfies condition 3.1, then*g*is calculated by (3.21) andby (3.15), if

*α*∈(0, 2) and*ϕ*satisfies condition 3.2, then*g*has three different forms, depending on the spatial dimension, and it is given for*n*=1, 2, 3 by (3.25), (3.28) and (3.30), respectively.

We obtain solutions to (3.5) for several special cases of weight functions

*ϕ*. In the cases when the solutions are also obtained by other authors, we compared our results with theirs and found good agreement.Let

*ϕ*(*μ*)=*δ*(*μ*−*α*). The solution to (3.5) is given by (4.1) if*α*∈(0, 1) and the space dimension is*n*. If*n*=1 and*α*∈(0, 2), then the solution to (3.5) is given by (4.3). The numerical example is presented in figure 3.Let

*ϕ*(*μ*)=*τδ*(*μ*−*α*)+*δ*(*μ*−*β*). The solution to (3.5) is given by (4.4) if*α*∈(0, 1) and the space dimension is*n*. If*n*=1 and*α*∈(0, 2), then the solution to (3.5) is given by (4.6). The numerical example is given in figure 4.Let

*ϕ*(*α*)=*τ*^{α}. The solution to (3.5) is given by (3.14) if*α*∈(0, 1), where the solution kernel*g*and*B*_{u}are calculated as (4.9) and (4.8), respectively. If*α*∈(0, 2), then the solution to (3.5) is given by (3.15), where the one-dimensional solution kernel*g*and*B*_{u},*B*_{v}are given by (4.11) and (4.8), respectively.

The Post inversion formula (5.1) was applied to the Laplace transformation for the one- and two-dimensional solution kernels (4.10) and (obtained by (3.27), with given by (4.7)) for the case of the weight function

*ϕ*(*α*)=*τ*^{α}. The results agree well with the results for*g*_{1}obtained analytically.We have proved a version of the maximum principle for (6.1) in the case

*α*∈(0, 1). The proof is based on a generalized Fermat theorem for the Caputo time-fractional derivative, stated in lemma 6.2.

## Acknowledgments

This research was supported by grant 144019A (T.M.A. and D.Z.) and grant 144016 (S.P.) of the Serbian Ministry of Sciences.

## Footnotes

- Received October 31, 2008.
- Accepted February 12, 2009.

- © 2009 The Royal Society