Int. J. Appl. Math. Comput. Sci., 2010, Vol. 20, No. 2, 261–269DOI: 10.2478/v10006-010-0019-1

NUMERICAL SOLUTIONS TO INTEGRAL EQUATIONS EQUIVALENT TODIFFERENTIAL EQUATIONS WITH FRACTIONAL TIME

BARTOSZ BANDROWSKI ∗, ANNA KARCZEWSKA ∗, PIOTR ROZMEJ ∗∗

∗ Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona Góra, ul. Szafrana 4a, 65–516 Zielona Góra, Polandemail: {B.Bandrowski, A.Karczewska}@wmie.uz.zgora.pl

∗∗Institute of PhysicsUniversity of Zielona Góra, ul. Szafrana 4a, 65–516 Zielona Góra, Poland

email: P.Rozmej@if.uz.zgora.pl

This paper presents an approximate method of solving the fractional (in the time variable) equation which describes theprocesses lying between heat and wave behavior. The approximation consists in the application of a finite subspace of aninfinite basis in the time variable (Galerkin method) and discretization in space variables. In the final step, a large-scalesystem of linear equations with a non-symmetric matrix is solved with the use of the iterative GMRES method.

Keywords: fractional equations, Galerkin method, anomalous diffusion.

1. Introduction

Several physical phenomena show anomalous transportproperties. Generally, diffusive properties are classified asnormal if their variance grows linearly in time and as ano-malous if the variance growth is different from the line-ar one. In recent years, there has been an increasing in-terest in dynamical processes occuring in systems exhi-biting anomalous diffusive behavior. Such systems ran-ge from physics and chemistry to biology and medici-ne (Meltzer and Klafter, 2000). Dispersion in complexplasmas (Ratynskaia et al., 2006), self-diffusion of sur-factant molecules (Gambin et al., 2005), light in a coldatomic cloud (Labeyrie et al., 2003) and donor-acceptorelectron pairs within a protein (Kou and Sunney Xie,2004) are examples of the more recent experimental evi-dences. Several papers (Goychuk et al., 2006; Heinsaluet al., 2006; Heinsalu et al., 2009) consider the fractionalFokker-Planck equation both in an analytical and a nume-rical approach. The fractional time approach is conside-red also in control theory, see, e.g., (Kaczorek, 2008; Gu-ermah et al., 2008). An interesting numerical approachto solving partial differential equations of fractional or-ders in space was presented in (Ciesielski and Leszczyn-ski, 2006).

In most of applications related to fractional differen-tial or fractional integro-differential equations, the nume-rical methods are limited to 1+1 (time+space) dimensions.Our paper presents a different method for solving a frac-tional integro-differential equation which can handle moredimensional cases within a good approximation. The me-thod is limited to cases when the initial condition u(x, 0)is smooth enough with respect to space variables x. Insuch cases it works also for (1+2) and (1+3) dimensions.

We consider the following Volterra equation:

u(x, t) = u(x, 0) +

t∫

0

a(t − s)Δu(x, s) ds, (1)

where x ∈ Rd, t > 0, a(t) = tα−1/Γ(α), Γ is the gamma

function, α ∈ [1, 2] and Δ is the Laplace operator.Because of the form of the kernel function a, the inte-

gral in Eqn. (1) is a Riemann-Liouville fractional integraloperator (Bazhlekova, 2001),

Jαt f(t) :=

t∫

0

1Γ(α)

(t − s)α−1f(s) ds.

262 B. Bandrowski et al.

Applying the Caputo fractional derivative operator to (1),one obtains a fractional differential equation,

∂α

∂tαu(x, t) = Δu(x, t), (2)

where α ∈ [1, 2], x ∈ Rd, t > 0 and Δ is the Laplace

operator.For particular cases when α = 1 and α = 2, taking

appropriate initial conditions, (2) is the heat and the waveequation, respectively. For α ∈ (1, 2), (2) interpolates theheat and the wave equations.

Equation (1) was discussed by Fujita (1990) andSchneider and Wyss (1989). Fujita found the analyticalsolution u(x, t) to (1) in terms of fundamental solutionsusing Mittag-Leffler functions. Schneider and Wyss ap-plied the Green function method and obtained the ana-lytical form of the solution in terms of Wright functions(which are related to Mittag-Leffler functions). Both ap-proaches are limited to a (1 + 1)-dimensional case andpractically non-computable.

2. Construction of the numerical solution

The numerical approach described below follows in gene-ral that described in (Rozmej and Karczewska, 2005).

Let there be given a set of real orthonormal functions{φj : j = 1, 2, . . . ,∞} on the interval [0, t], spanning aHilbert space H with an inner product

〈f, g〉 :=

t∫

0

f(τ)g(τ)W (τ) dτ,

where W is a weight function.An approximate solution to (1) is considered an ele-

ment of the subspace Hn spanned by the first n basis func-tions {φk : k = 1, 2, . . . , n},

un(x, t) =n∑

k=1

ck(x)φk(t). (3)

Inserting (3) into (1), one obtains

un(x, t)

= u(x, 0)+

t∫

0

a(t−s)Δun(x, s) ds+εn(x, t), (4)

where εn denotes the approximation error. From (3) and(4), we get the following form of the error function:

εn(x, t) =n∑

k=1

ck(x)φk(t) − u(x, 0)

−t∫

0

a(t − s)n∑

k=1

d2

dx2ck(x)φk(s) ds.

(5)

Definition 1. The Galerkin approximation of the solutionto Eqn. (1) is a function un ∈ Hn such that εn ⊥ Hn, i.e.,

〈εn(x, t), φk(t)〉 = 0, k = 1, 2, . . . , n.

From Definition 1 and (5), we obtain

0 =

t∫

0

[n∑

k=1

ck(x)φk(τ)

]φj(τ)W (τ) dτ

−t∫

0

u(x, 0)φj(τ)W (τ) dτ

−t∫

0

⎡⎣

τ∫

0

a(τ−s)n∑

k=1

d2

dx2ck(x)φk(s) ds

⎤⎦

× φj(τ)W (τ) dτ.

for j = 1, 2, . . . , n, which means that

t∫

0

u(x, 0)φj(τ)W (τ) dτ

=

t∫

0

[n∑

k=1

ck(x)φk(τ)

]φj(τ)W (τ) dτ

−t∫

0

⎡⎣

τ∫

0

a(τ−s)n∑

k=1

d2

dx2ck(x)φk(s) ds

⎤⎦

×φj(τ)W (τ)dτ.

The above equations may be written in a shortened form,

gj(x) = cj(x) −n∑

k=1

ajkd2

dx2ck(x), (6)

where

ajk =

t∫

0

⎡⎣

τ∫

0

a(τ − s)φk(s) ds

⎤⎦ φj(τ)W (τ) dτ

and

gj(x) = u(x, 0)

t∫

0

φj(τ)W (τ) dτ,

j = 1, 2, . . . , n. In general, ajk �= akj .Equations (6) describe a set of coupled differen-

tial equations for the coefficient functions ck(x), k =1, 2, . . . , n determining the approximate solution (3).

To solve the set (6), we use standard, centered three-point finite difference approximation to the second deriva-tive (Laplacian).

Numerical solutions to integral equations equivalent to differential equations with fractional time 263

In one spatial dimension one obtains (6) as

gj(xi) = cj(xi) +1h2

n∑k=1

ajk

×[− ck(xi−1) + 2ck(xi) − ck(xi+1)

],

(7)

where there holds xi − xi−1 = h, j = 1, 2, . . . , n, andi = 1, 2, . . . , m.

For two spatial dimensions and a homogenous grid,(6) can be written as

gj(xi, yl)=cj(xi, yl) +1h2

n∑k=1

ajk

×[− ck(xi−1, yl) − ck(xi, yl−1)

+ 4ck(xi, yl) − ck(xi+1, yl)

− ck(xi, yl+1)],

(8)

where xi − xi−1 = yl − yl−1 = h and j = 1, 2, . . . , n,i = 1, 2, . . . , m, l = 1, 2, . . . , m.

The sets (7) and (8) can be presented in the matrixform

g = Ac, (9)

where the vectors g, c and the matrix A have a block struc-ture,

g =

⎛⎜⎜⎜⎝

G1

G2

...Gn

⎞⎟⎟⎟⎠ , c =

⎛⎜⎜⎜⎝

C1

C2

...Cn

⎞⎟⎟⎟⎠ ,

A =

⎛⎜⎜⎜⎝

[A11] · · · [A1n][A21] · · · [A2n]

.... . .

...[An1] · · · [Ann]

⎞⎟⎟⎟⎠ . (10)

In the one-dimensional case (7), for (10) we have

GTj = (gj(x1), gj(x2), . . . , gj(xm))

andCT

j = (cj(x1), cj(x2), . . . , cj(xm)) ,

where GTj and CT

j stand for the transposes of vectorsGj and Cj , respectively. Blocks [Ajk] have the followingstructure:

[Ajk] =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

μjk ηjk 0 · · · 0 θjk

ηjk μjk ηjk · · · 0 00 ηjk μjk · · · 0 0...

......

. . ....

...0 0 0 · · · μjk ηjk

θjk 0 0 · · · ηjk μjk

⎞⎟⎟⎟⎟⎟⎟⎟⎠

m×m

,

where

μjk = δjk +2h2

ajk,

ηjk = − 1h2

ajk,

and

θjk ={ − 1

h2 ajk for periodic boundary conditions,0 for closed boundary conditions.

The vectors g and c are nm-dimensional, and the ma-trix A is nm × nm-dimensional. Moreover, the matrix Ais sparse with at most n2(3m − 2) (closed boundary con-ditions) or 3n2m (periodic boundary conditions) non-zeroelements. In the two-dimensional case (8), we have

GTj =

(gj(x1, y1), gj(x1, y2), . . . , gj(x1, ym),

gj(x2, y1), gj(x2, y2), . . . , gj(xm, ym))

and

CTj =

(cj(x1, y1), cj(x1, y2), . . . , cj(x1, ym),

cj(x2, y1), cj(x2, y2), . . . , cj(xm, ym)).

Each of blocks [Ajk] is a matrix in the form of

[Ajk]=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(αjk)(βjk) (0) · · · (0) (0) (γjk)(βjk)(αjk)(βjk)· · · (0) (0) (0)(0) (βjk)(αjk)· · · (0) (0) (0)

......

.... . .

......

...(0) (0) (0) · · ·(αjk)(βjk) (0)(0) (0) (0) · · ·(βjk)(αjk) (βjk)

(γjk) (0) (0) · · · (0) (βjk) (αjk)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

m×m

,

which consists of smaller blocks, where

(γjk) ={

(βjk) for periodic boundary conditions,(0) for closed boundary conditions.

Block (0) is an m × m-dimensional zero matrix,block (αjk) is in the form of

(αjk) =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

μjk ηjk 0 · · · 0 θjk

ηjk μjk ηjk · · · 0 00 ηjk μjk · · · 0 0...

......

. . ....

...0 0 0 · · · μjk ηjk

θjk 0 0 · · · ηjk μjk

⎞⎟⎟⎟⎟⎟⎟⎟⎠

m×m

,

where

μjk = δjk +4h2

ajk,

ηjk = − 1h2

ajk,

264 B. Bandrowski et al.

θjk ={ − 1

h2 ajk for periodic boundary conditions,0 for closed boundary conditions,

Block (βjk) is in the diagonal form,

(βjk)=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−1h2 ajk 0 0 · · · 0 0 0

0 −1h2 ajk 0 · · · 0 0 0

0 0 −1h2 ajk· · · 0 0 0

......

.... . .

......

...0 0 0 · · ·−1

h2 ajk 0 00 0 0 · · · 0 −1

h2 ajk 00 0 0 · · · 0 0 −1

h2 ajk

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

m×m.

The vectors g and c are nm2-dimensional, and the ma-trix A is nm2 × nm2-dimensional. As in the previous ca-se, the matrix A is sparse. It has at most n2(5m − 4)m(closed boundary conditions) or 5n2m2 (periodic boun-dary conditions) non-zero elements. As basis functions{φj : j = 1, 2, . . . ,∞} one can choose any polynomialsorthogonal with respect to some weight function W .

In our numerical scheme, we use Legendre polyno-mials, which are orthogonal on the interval [−1, 1] withrespect to the weight function W ≡ 1. We obtain ortho-normality on the interval [0, t] by scaling the argument ofthe function and using a normalisation factor, i.e.,

φj(x) =

√2k − 1

tPk−1

(2x

t− 1

), (11)

where {Pk : k = 0, 1, 2, . . . ,∞} are Legendre polyno-mials.

The matrix A of the linear system (9) is sparse,which, taken in conjunction with the large matrix size,suggests the use of iterative methods for solving linearsystems. Morover, the matrix A is non-symmetric. Ournumerical results were obtained with two different iterati-ve methods. The first one, GMRES (Generalized MinimalResiudal method) (Saad and Schultz, 1986), approxima-tes the solution by the vector with a minimal residual ina Krylov subspace found with the use of the Arnoldi ite-ration. The second one, Bi-CGSTAB (BiConjugate Gra-dient Stabilized method) (Van der Vorst, 1992), was deve-loped to solve non-symmetric linear systems. It avoids ir-regular convergence patterns of CGS (Conjugate GradientSquared method) (Barrett et al., 1994). In both methodsa suitable preconditioning is necessary. For our purposes,GMRES appeared more efficient. It was converging fasterand usually required fewer iterations than the Bi-CGSTABmethod.

For three spatial dimensions, vectors g and c in (9)become nm3-dimensional. The matrix A preserves itsblock nested structure (10) but with one more level of ne-sting. The number of the non-zero elements of the matrixA reaches the order of 7n2m3. Therefore, the numericalsolution for the d = 3 case requires much more computerpower than for one and two-dimensional cases.

0

2

4

6

t

-9 -6 -3 0 3 6 9

x

0.25 0.5

0.75 1

Fig. 1. Numerical solutions to the Volterra equation (1) in onespatial dimension: α = 1.5, t ∈ [0, 6], closed boundaryconditions.

0

2

4

6

t

-9 -6 -3 0 3 6 9

x

0.25 0.5

0.75 1

Fig. 2. Numerical solutions to the Volterra equation (1) in onespatial dimension: α = 1.85, t ∈ [0, 6], closed boundaryconditions.

3. Examples of numerical results

Analytical solutions to fractional equations are of greatimportance but they are hardly computable. In most ca-ses, to obtain a solution to a particular problem one hasto apply suitable numerical methods. In (Rozmej and Kar-czewska, 2005) we showed that for the cases when ana-lytical solutions are known in terms of elementary func-tions (1+1-dimensional cases with α = 1 and α = 2), ourapproximate numerical solution reproduces the analyticalones with high accuracy.

We present numerical results to the Volterra equation(1) in three cases: in one spatial dimension with closedboundary conditions (Figs. 1 and 2) and in two spatial di-mensions with both closed (Figs. 3 and 4) and periodicboundary conditions (Figs. 5 and 6).

For the presentation of the results, we choose as ini-tial conditions the Fermi distribution:

u(�r, 0) =1

1 + exp(r − r0

a

) , (12)

Numerical solutions to integral equations equivalent to differential equations with fractional time 265

where r = (∑d

i=1 x2i )

1/2 and d is the space dimension.The constants in (12) are taken as r0 = 3, a = 0.3.

In Table 1, parameters m (the number of grid pointsin space variables) and n (the number of basis functions)used to obtain numerical results are presented. In princi-ple, when the basis of orthonormal functions is richer, thenumerical results are more precise. A similar argument ap-plies to the number of grid points taken in the problem di-scretization. However, the growing size of matrices (as theresult of the increase in the number of grid points and ba-sis functions) results in an incrementation in the number ofcomputer operations and in the accumulation of round-offerrors. Comparing Figs. 3 and 4 or Figs. 5 and 6, one cannotice that for the 1+2-dimensional case the increase in αchanges the time evolution from diffussion-like behaviorto wave-like one. The comparison of the results obtainedwith the same α (for instance, Figs. 3 and 5 or Figs. 4 and6) shows the influence of ‘waves’ incoming to the givenspace cell from the neighbours.

The method used in the paper has some limitations.Numerical approximation of the second derivatives (La-placian) can be good enough when the initial conditionu(x, 0) is a relatively smooth function, i.e., it varies unra-pidly with respect to space variables within a range of gridpoints.

Figures 1 and 2 show the time evolution for the frac-tional time derivative of the order α = 1.5 and α = 1.85,respectively. One can clearly see that the system evolves ina way which exhibits wave motion with a diffusive charac-ter. When α varies towards 1, one obtains more diffusion-like evolution, whereas for α closer to 2, wave-like evo-lution becomes dominant. The closed (or free) bounda-ry conditions mean that the initial system can in princi-ple evolve to infinity in the space variable, where initiallyu(x, 0) = 0. In the presented figures, however, the solu-tion is cut off to the range of the space variable where thesolutions differs from zero substantially.

4. Approximation errors

As mentioned earlier, the analytic form of the solutionu(x, t) is known only for one space dimension (Fujita,1990; Schneider and Wyss, 1989). However, the analyti-cal solutions for cases different from α = 1 and α = 2,given in terms of Mittag-Leffler functions, are practicallynon-computable. To estimate the quality of the numericalsolutions, several tests were performed. For d = 1 and

Table 1. Parameters.

Case Grid size Basis functions

d = 1 121 21d = 2, closed b. c. 101 × 101 21

d = 2, periodic b. c. 71 × 71 21

t=2.4

-8-4 0 4

8x -8-4

0 4

8

y

0.5

1

t=1.2

-8-4 0 4

8x -8-4

0 4

8

y

0.5

1

t=0.0

-8-4 0 4

8x -8-4

0 4

8

y

0.5

1

Fig. 3. Numerical solutions to the Volterra equation (1) in twospatial dimensions at chosen time steps: α = 1.5, closedboundary conditions.

α = 1 and α = 1, we can compare the analytical solu-tion with the numerical one obtained from (3) and (9). Asan error estimate, we define the maximum absolute valueof the difference between the exact analytical solution andthe approximate numerical one,

Δun,m(t) = max∣∣uanal

n,m(xi, t) − unumn,m (xi, t)

∣∣mi=1

,

where the maximum is taken over all grid points xi. Ford = 1 and α = 1 and 2, n > 20, m > 100, t ≤ 6, the errorestimate Δun,m(t) was always less than 10−5 − 10−6.

For cases when we have to rely only on the appro-ximate numerical solutions (1 < α < 2), we procede ina different way. One can expect that when the number ofbasis functions (n in (3)) or the size of grid (m) increases,better approximation of the solution is obtained. In orderto show that trend, we define the absolute value of the dif-ference between the two approximate solutions taken for

266 B. Bandrowski et al.

t=2.4

-8-4 0 4

8x -8-4

0 4

8

y

0.5

1

t=1.2

-8-4 0 4

8x -8-4

0 4

8

y

0.5

1

t=0.0

-8-4 0 4

8x -8-4

0 4

8

y

0.5

1

Fig. 4. Numerical solutions to the Volterra equation (1) in twospatial dimensions at chosen time steps: α = 1.85, clo-sed boundary conditions.

different basis subspaces and the same grid size,

Δun,M (t)

= max∣∣unum

n,M (xi, t)−unumn−2,M(xi, t)

∣∣Mi=1

, (13)

where M = md. In Fig. 7, the error estimate Δun,M (t) ispresented for the one-dimensional case (d = 1), α = 1.5,grid size m = 121 and t = 1.8 in a semilogarithmic plot.The two-dimensional case (d = 2), m = 101 and thesame values of α and t is displayed in Fig. 8. In Figs. 7and 8, the error estimates exhibit almost an exponentialdecrease with the growth of the basis subspace dimensionn in (3).

Comparing the approximate solutions for differentgrid sizes is a little more difficult because their values aregiven at different points. Therefore, in order to comparethem, we define the error estimate in the following way:

Δun,m(t)

= max∣∣unum

n,m (xi, t)−unumn,m′(xi, t)

∣∣Mi=1

. (14)

t=2.4

-4 -2 0 2 4x -4-2

0 2

4

y

0.5

1

t=1.2

-4 -2 0 2 4x -4-2

0 2

4

y

0.5

1

t=0.0

-4 -2 0 2 4x -4-2

0 2

4

y

0.5

1

Fig. 5. Numerical solutions to the Volterra equation (1) in twospatial dimensions at chosen time steps: α = 1.5, perio-dic boundary conditions.

In (14), unumn,m′(xi, t) stands for the value at the point of

the bigger grid m obtained from the solution on the smal-ler grid m′ by cubic spline interpolation. Figure 9 showsan example of the decrease in the error estimate Δun,m(t)for the dimension d = 2. For the one-dimensional calcula-tions (not shown here), the results exhibit the same beha-viour. We see that for the fixed basis subspace the increasein the grid size leads to a substantial decrease in the errorestimate. That decrease is, for larger values of m, close toan exponential decrease.

That almost exponential decrease in the error esti-mate as the function of the number of the basis functions(with a fixed grid) or as the the function of the grid size(with a fixed basis) suggests that in principle one can ob-tain the numerical solution with arbitrary precision cho-osing the appropriate basis and grid sizes. However, thebigger the basis and the grid size, the larger the numberof the computer operations necessary to obtain the solu-tion. To avoid the accumulation of round-off errors withan increase in computer operations, one needs to apply hi-gher and higher precision (longer computer word), which

Numerical solutions to integral equations equivalent to differential equations with fractional time 267

t=2.4

-4 -2 0 2 4x -4-2

0 2

4

y

0.5

1

t=1.2

-4 -2 0 2 4x -4-2

0 2

4

y

0.5

1

t=0.0

-4 -2 0 2 4x -4-2

0 2

4

y

0.5

1

Fig. 6. Numerical solutions to the Volterra equation (1) in twospatial dimensions at chosen time steps: α = 1.85, pe-riodic boundary conditions.

1e-06

1e-05

0.0001

0.001

0.01

8 10 12 14 16 18 20 22

Δun,

M(t)

n

Fig. 7. Error estimate (13), Δun,M (t): the maximum differencebetween approximate solutions for which the number ofbasis functions differs by 2 as a function of n for onespatial dimension d = 1. The case m = 121, α = 1.5and t = 1.8 is displayed.

1e-06

1e-05

0.0001

0.001

0.01

8 10 12 14 16 18 20 22

Δun,

M(t)

nFig. 8. Same as in Fig. 7 but for two spatial dimensions d = 2

and grid size m = 101.

0.0001

0.001

0.01

0.1

20 30 40 50 60 70 80 90 100

Δun,

m(t)

m

Fig. 9. Error estimate (14), Δun,m(t): the maximum differen-ce between consecutive results obtained for grid sizes mand m′ = m − 10 in two spatial dimensions. The casen = 21 basis functions, α = 1.5 and t = 1.8 is presen-ted.

results in much longer execution time and demands muchmore computer power. In practice, a reasonable compro-mise between the precision and computational effort ischosen for a given problem.

The results for the 3-dimensional case show similarqualitative behaviour. We do not present those here becau-se the computing time grows substantialy. The case d = 3and m = 21 requires the same order of computer opera-tions as the case d = 2, m = 101 with the same n. Thetest computations made for grid sizes up to m = 21 andthe number of basis functions up to n = 20 gave trendswhich qualitatively agree with those for d = 2.

5. Conclusions

We presented a succesful numerical method of solving aclass of Volterra equations (1) which are equivalent to dif-

268 B. Bandrowski et al.

ferential equations with fractional time (2). From our re-sults, the following conclusions can be drawn:

• For initial conditions spatially smooth enough and areasonable choice of the (n, m) parameters, the er-rors of the approximate numerical solution may bekept on a desired level.

• Our method works well for d = 1 and d = 2 spatialdimensions. However, the larger d, the more compu-ter power is necessary. Test calculations indicate thatthe metod should work for d = 3, too.

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Bartosz Bandrowski is a Ph.D. student at theFaculty of Mathematics, Computer Science andEconometrics, University of Zielona Góra. He re-ceived an M.Sc. degree in mathematics from theUniversity of Zielona Góra in 2008. His maininterests include mathematical informatics, nu-merical methods in Volterra equations, fractionalcalculus and stochastic equations.

Anna Karczewska is an associate professorat the Faculty of Mathematics, Computer Scien-ce and Econometrics, University of Zielona Gó-ra. She holds an M.Sc. degree from the WarsawTechnical University, a Ph.D. degree from Jagiel-lonian University, Cracow, and habilitation fromthe University of Zielona Góra. Her fields of inte-rest include deterministic and stochastic integralequations, in particular Volterra equations of in-tegral and fractional order, and fluid dynamics.

Numerical solutions to integral equations equivalent to differential equations with fractional time 269

Piotr Rozmej is a professor at the Faculty ofPhysics and Astronomy, University of ZielonaGóra, the head of the Division of Mathemati-cal Methods in Physics. He holds an M.Sc. de-gree from Maria Curie-Skłodowska University,Lublin, a Ph.D. degree from Warsaw Universi-ty, and habilitation from UMCS. He became aprofessor in 1992 and a full professor in 1997.He has spent several years working as a seniorresearcher at the Joint Institute for Nuclear Re-

search (Dubna, USSR), Gesellschaft für Schwerionenforschung (Darm-stadt, Germany) and Institut des Sciences Nucléaires (Grenoble, France).He works in the fields of nuclear structure, nuclear reactions, propertiesof heavy and superheavy nuclei. Other areas of interest include quantumoptics, in particular wave packet dynamics, and nonlinear dynamics.

Received: 24 May 2009Revised: 28 September 2009