����������� ������ 53 (2001), 151–158

UDK 517.982.4

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GENERALIZED FRACTIONAL CALCULUS WITH APPLICATIONS IN MECHANICS

Ljubica Oparnica

Abstract. In the studies concerned with the stability of the viscoelastic rod of fractional type it is shown that many problems are described by stability of coupled systems of differential equations with fractional derivatives. Here, we are dealing with analysis of such systems in the space of distributions. The main result of this work is stated in Theorem 5.

1. Introduction

Fractional calculus is the field of mathematical analysis which deals with investigation and application of integrals and derivatives of real order. Fractional calculus may be considered as an old and yet novel topic. It is an old topic since its developing started from G.W. Leibniz and L. Euler. Afterwards there is a list of mathematicians who have provided important contribution to the theory. It is a novel topic since it has become an object of specialized conferences and treatises only about 25 years ago. First conference was “The First Conference on Fractional Calculus and its Applications” in 1974, first monograph was written by K.B. Oldham and Spanier who after a joint collaboration starting in 1968 published a book devoted to fractional calculus in 1974.

Nowadays, the list of texts and proceedings devoted solely or partly to fractional calculus and its applications includes about a dozen of titles, among which the encyclopedic treatise by Samko, Kilbas and Marichev [5] is the most prominent.

In recent years the interest for fractional calculus has been stimulated by its wide survey of applications which includes viscoelasticity, fractional capacitor theory, electrical circuits, electroanalytical chemistry, biology, physics, etc.

Viscoelasticity seems to be the field with the most extensive application of fractional operators. Using fractional derivatives for modeling viscoelastic materials is quite natural. Our aim in this note is to formulate and analyse a mathematical model for a viscoelastic rod described by a constitutive equation containing fractional derivatives that impacts against a rigid wall.

AMS Subject Classification: 26 A 33, 34 D 99, 46 F 99, 73 H 99

Communicated at the 5th International Symposium on Mathematical Analysis and its Applications, Niška banja, Yugoslavia, October, 2–6, 2002.

2. The model

We consider light viscoelastic rod with a body B of mass m which is atached to its end. At the moment t_{0} the rod is about to impact a rigid wall. Rod is moving at the speed v_{0} and its length in undeformed state is L. At the moment t > t_{0} the rod impacts the wall, its length gets reduced to l(t), and at the contact point contact force (stress) f appears.

Strain of the rod x(t) is given as

x(t) = \frac{L − l(t)}{L}, t ≥ 0.

The theory of constitutive equations gives relation between f and x through the generalized Zener model τ_{f}f^{(α)} + f = τ_{x}x^{(α)} + x, where τ_{f} and τ_{x} are constants called time of relaxation of stress and strain, and f^{(α)} and x^{(α)} denote the Riemann-Liouville fractional derivative of stress and strain. The Riemann-Liouville fractional derivative for 0 < α < 1 is defined as

u^{(α)}(t) = \frac{1}{Γ(1 − α)} ∫_{0}^{1}(t − τ)^{−α}u(τ) dτ,  t ≥ 0.  (1)

Differential equation of motion or the second Newton’s law for the body B is expressed by mx^{(2)} = −f, where m is the mass of body B, and x^{(2)} is its acceleration.

Initial conditions are x(t_{0}) = 0, x^{(1)}(t_{0}) = v_{0}. Without loss of generality we take m = 1, v_{0} = 1, t_{0} = 0. Also we take τ_{f} = a and τ_{x} = b and we get the system

\begin{aligned}x^{(2)}(t) & = −f(t) \\ af^{(α)} (t) + f (t) & = bx^{(α)} (t) + x (t) \\ x(0) & = 0 \\ x^{(1)}(0) & = 1 \end{aligned}  (2)

We denote by T^{∗} the moment of separation. This is the time instant when contact force between rod and wall is equal to zero, f(T^{∗}) = 0.

The second law of thermodynamics requires that b > a > 0 (see [1]).

Our main goal is to estimate |x′(T^{∗})|. We claim that b > a implies |x′(T^{∗})| ≤ 1, for otherwise the rod would represent a perpetuum mobile of the first kind.

3. Mathematical preliminaries: Spaces D′_{+} and S+^{′}. Operator f_{α}∗

Denote by D and S Schwartz test function spaces of compactly supported and rapidly decreasing smooth functions on the real line ℝ. Their duals are well known spaces of distributions D′ and tempered distributions S′. The space of Schwartz’s distributions vanishing on interval (−∞, 0) is denoted by D′_{+}, i.e. D+^{′} = { f ∈ D′ (ℝ) : supp f ⊂ [0, ∞) }.

If f ∈ D′_{+} and g ∈ D′_{+}, then convolution^{1} f ∗ g exists in D′_{+}. Recall that the operation of convolution is associative and commutative in D′_{+}, so that D′_{+} is a convolution algebra with Dirac delta function as the unit element, since δ ∗ f = f.

1For localy integrable functions on ℝ vanishing on (−∞, 0), f ∗ g(t) = ∫_{0}^{t} g(t − τ)f(τ) dτ, t ∈ ℝ. For f ∈ S′_{+}, the convolution is ⟨f × g, φ(x + y)⟩, φ ∈ S_{+}.

Space S+^{′} is the intersection of D+^{′} and S′. It is a convolution algebra, a subalgebra of D+^{′}.

We consider the equation in D′_{+}

a ∗ u = f,  (3)

where a and f are specified elements of D′_{+} and u is a unknown generalized function in D′_{+}. The solution of (3) for f = δ is called the fundamental solution of convolution operator a ∗ .

The Laplace transform (or Fourier-Laplace transform) of an f ∈ S+^{′} is defined as

Lf(s) = ⟨f(t), η(t)e^{−st}⟩ ,  s ∈ D_{+,}  (4)

where η is a smooth function on ℝ equal to one on (−a, ∞) and zero on (−∞, −a) and D_{+} = {z : ℜz > 0}. Note, Lf is an analytic function on D_{+} not depending on the cutoff function η.

The space S+^{′} is isomorphic, via Laplace transform, to the corresponding space of holomorphic functions in D_{+} [6]. This is the claim of the following theorem.

Theorem 1. [6] Let h(p) be holomorphic in half-plane ℜp > σ_{0}. Then there exists a distribution T ∈ D+^{′} the Laplace transform of which is h(p) if and only if there exist numbers σ_{1} ∈ ℝ, σ_{0} ≤ σ_{1}, C and k ≥ 0, such that

|h(p)| ≤ C(1 + |p|)^{k}, p ∈ ℂ, ℜp > σ_{1}.

Let a ∈ S+^{′} be given. Then by Theorem 1, equation (3) is uniquely solvable in S+^{′} if and only if there exist p, q ∈ ℝ, C > 0 such that

\frac{1}{|La(z)|} ≤ C \frac{(1 + |z|)^{p}}{x^{q}} ,  z = x + iy ∈ D_{+}.  (5)

Next we define a generalized function f_{α}, for arbitrary α ∈ ℝ,

f_{α}(x) = \{ \begin{matrix}θ(x)\frac{x^{α−1}}{Γ(α)} , & α > 0 \\ f_{α+N}^{(N)}, & α ≤ 0, α + N > 0, N ∈ ℕ \end{matrix}  (6)

where (·)^{(N)} is the N-th derivative in the sense of distributions and θ(x) is the Heaviside’s function. Then f_{α} ∈ D′_{+} and the group property

f_{α} ∗ f_{β} = f_{α+β}  α, β ∈ ℝ  (7)

is valid.

Let us consider the convolution operator f_{α}∗ in algebra D′_{+}. Since f_{0} = θ′ = δ,

(7) implies that a fundamental solution to equation f_{α} ∗ u = δ exists and is equal to f_{−α}. Let n ∈ ℤ. For n < 0, f_{n} = δ^{(n)}, so that f_{n} ∗ u = δ^{(n)} ∗ u = u^{(n)}, i.e. the operator f_{n}∗ is the operator of n-folding differentiation. Finally, for n > 0, we have (f_{n} ∗ u)^{(n)} = f_{−n} ∗ (f_{n} ∗ u) = u, which implies that f_{n} is the operator of n-folding integration.

For α < 0 the operator f_{α}∗ is the operator of fractional differentiation, and the operator of fractional integration for α > 0. This is the Riemann-Liouville operator.

Remark 1. For α ∈ (0, 1) we have

f_{−α} ∗ u = f′_{1−α} ∗ u = (f_{1−α} ∗ u)′ = \frac{1}{Γ(1 − α)} ∫_{0}^{t}(t − τ)^{−α}u(τ) dτ.

This is (1).

Let α ∈ ℝ. Then the Laplace transform of f_{α} is given by

L[f_{α}(t)](s) = \frac{1}{s^{α}} ,  ℜs > 0.  (8)

3.1. Special functions of Mittag-Leffler type

Functions of Mittag Lefler’s type have very important role in the theory of differential equations where fractional derivative appears. Here we will recall from [4] some facts concerning this class of functions. One-parameter generalization of exponential function e^{z} is given by

E_{α}(z) = ∑_{k=0}^{∞} \frac{z^{k}}{Γ(αk + 1)}  z ∈ ℂ, α > 0,  (9)

and was introduced by G.M. Mittag-Leffler.

More interesting for us is e_{α}(t, λ) = E_{α}(−λt^{α}), t ≥ 0, α > 0 with Laplace transform

L[e_{α}(t, λ)](s) = \frac{s^{α−1}}{s^{α} + λ},  ℜs > |λ|^{\frac{1}{α}}  (10)

4. Analysis of system (2)

Let u ∈ S+^{′}(ℝ), a > 0, α ∈ (0, 1). We define the operator T_{a,α}

T_{a,α}u = au^{α} + u = af_{−α} ∗ u + u u ∈ S+^{′}.

T_{a,α} is a linear and continuous operator from S+^{′} to S+^{′}. We have

L[T_{a,α}f](s) = (as^{α} + 1)L[f](s), ℜs > 0.

The following proposition is valid.

Proposition 1. Let a > 0 and v ∈ S+^{′}. Then

T_{a,α}u = v  (11)

has a unique solution u ∈ S+^{′}.

Proof. Applying the Laplace transform to (11), we get (1+as^{α})Lu(s) = Lv(s),

ℜs > 0. Let h(s) = \frac{Lv(s)}{1 + as^{α}}, ℜs > 0. Since ℜs > 0 and α ∈ (0, 1) it follows that

ℜs^{α} > 0. Further

|1 + as^{α}| = ℜ^{2}(1 + as^{α}) + ℑ^{2}(1 + as^{α}) > ℜ^{2}(1 + as^{α}) > 1

and this implies that h is holomorphic for ℜs > 0. Also,

|h(s)| = \frac{|Lv(s)|}{|1 + as^{α}|} < |Lv(s)| ≤ C,  ℜs > 0,  (12)

Thus by (5) it follows that there exists u^{∗} ∈ S+^{′}(ℝ) such that h(s) = Lu^{∗}(s).

Let us show that u^{∗} is the unique solution to (11). Indeed, we know that Lf = 0 if and only if f = 0. So L(T_{a,α}u^{∗} − v) = 0 implies T_{a,α}u^{∗} = v and u^{∗} is unique since if we had two solutions u_{1} and u_{2}, then from T_{a,α}u_{1} = T_{a,α}u_{2} we would have T_{a,α}(u_{1} −u_{2}) = 0, i.e. (1+as^{α})L[u_{1} −u_{2}](s) = 0 which yields L[u_{1} −u_{2}](s) = 0 and u_{1} − u_{2} = 0.

Now, we formulate

Theorem 2. Let a, b ∈ ℝ, a > 0, b > 0, α ∈ ℝ. Let f and x be solution to x′′(t) = −f(t), af^{(α)} (t) + f (t) = bx^{(α)} (t) + x (t) in S+^{′} and let f be a continuous function on the interval [0, ∞) and f(0) = 0. Then

(i) x ∈ C^{2}([0, ∞)) and x is the unique solution to given equations.

(ii) With the given assumptions on f, the solutions f and x of system (2) satisfies the following: b > a if and only if f(t) > x(t) in a neighborhood of zero.

Proof. (i) Let us define f = 0 on (−∞, 0). Then f belongs to S+^{′}. (Moreover it is continuous on ℝ).) We can rewrite the second equation of the system

T_{a,α}f = T_{b,α}x  (13)

Due to Proposition 1 equation (13) has a unique solution x ∈ S+^{′}. Since it satisfies x′′(t) = −f(t), x is a function with continuous first and second derivative on [0, ∞). We use the general assertion of Schwartz: if u is a distribution defined by a locally integrable function and if its distributional derivative is a locally integrable function, then the function u has a classical derivative equal to that locally integrable function.

(ii) If we apply Laplace transform to (13) we get Lf(s) = \frac{L[T_{a,α}x](s)}{1 + as^{α}}, ℜs > 0, i.e.

Lf(s) = Lx(s) + (\frac{b}{a} − 1)\frac{s^{α}}{s^{α} + \frac{1}{a}} Lx(s),  ℜs > 0.  (14)

Now we apply inverse Laplace transform to (14). We obtain

f(t) = x (t) + (\frac{b}{a} − 1) (δ + e′_{α} (t, \frac{1}{a})) ∗ x (t) .  (15)

Integration by parts with initial conditions gives

(δ + e′_{α} (t, \frac{1}{a})) ∗ x (t) = e_{α} (t, \frac{1}{a}) ∗ x′(t)

wherefrom it folows

f(t) = x (t) + (\frac{b}{a} − 1)e_{α} (t, \frac{1}{a}) ∗ x′(t).  (16)

The function e_{α}(λ, t) is completely monotone^{2} for λ > 0 and 0 < α < 1 (see [2]). Therefore e_{α} (t, \frac{1}{a}) > 0. Since x′(0) = 1 there is ε > 0 such that x′(t) > 0 for t ∈ [0, ε). Hence from (16) we have that f(t) > x(t) if and only if (\frac{b}{a} −1) > 0 which we wanted to prove.

5. Existence of solution

We solve system (2) by using two methods. Firstly, we use the Laplace transform method and, secondly, we formulate an iterative procedure for obtaining the solution.

5.1. Laplace transform method

Theorem 3. Let a, b ∈ ℝ, a > 0, b > 0, α ∈ (0, 1). System (2) has a unique solution in S+^{′} ∩ C^{2}(ℝ) given by

x(t) = ∑_{i=1}^{2} \frac{(1 + as_{i}^{α})e^{s_{i}t}}{a(α + 2)s_{i}^{α+1} + 2s_{i} + bαs_{i}^{α−1}} +

+ \frac{sin απ}{π} ∫_{0}^{∞} \frac{r^{α} (b − a) e^{−rt} dr}{(1 + r^{2})^{2} + 2 (1 + r^{2}) r^{α}(b + r^{2}a) cos απ + (r^{α}(b + r^{2}a))^{2}} , (17)

f(t) = ∑_{i=1}^{2} \frac{(1 + bs_{i}^{α})e^{s_{i}t}}{a(α + 2)s_{i}^{α+1} + 2s_{i} + bαs_{i}^{α−1}} +

+ \frac{sin απ}{π} ∫_{0}^{∞} \frac{r^{α} (a − b) e^{−rt} dr}{(1 + r^{2})^{2} + 2 (1 + r^{2}) r^{α}(b + r^{2}a) cos απ + (r^{α}(b + r^{2}a))^{2}} , (18)

t ≥ 0, where s_{1} and s_{2} (s_{1} = \overline{s_{2}}) are poles of the first order of the function Lx(t) and Lf (t), and ℜs_{i} < 0.

Remark 2. Initial conditions x(0) = 0 and f(0) = 0 imply

∑_{i=1}^{2} \frac{(1 + as_{i}^{α})}{a(α + 2)s_{i}^{α+1} + 2s_{i} + bαs_{i}^{α−1}}

= −\frac{sin απ}{π} ∫_{0}^{∞} \frac{r^{α} (b − a) dr}{(1 + r^{2})^{2} + 2 (1 + r^{2}) r^{α}(b + r^{2}a) cos απ + (r^{α}(b + r^{2}a))^{2}} ,

∑_{i=1}^{2} \frac{(1 + bs_{i}^{α})}{a(α + 2)s_{i}^{α+1} + 2s_{i} + bαs_{i}^{α−1}}

= −\frac{sin απ}{π} ∫_{0}^{∞} \frac{r^{α} (a − b) dr}{(1 + r^{2})^{2} + 2 (1 + r^{2}) r^{α}(b + r^{2}a) cos απ + (r^{α}(b + r^{2}a))^{2}} .

Remark 3. From equations (17) and (18) we conclude that x and f are continuous. Also we can differentiate (17) and (18) two times and get continuous functions, i.e. f and x belong to C^{2}([0, ∞)), and the claim of Theorem 2 is valid.

2A function f(x) is completely monotone for t > 0 if (−1)^{n}f^{(n)}(x) ≥ 0 for all n = 0, 1, 2,… and all t > 0.

Generalized fractional calculus with application in mechanics 157

Proof of Theorem 3. The first equation in (2) gives

x = t − t ∗ f, t ≥ 0 (19)

Substituting (19) into (13) we get T_{a,α}f = T_{b,α}(t − t ∗ f). Applying the Laplace transform to (11) and (19) we get

(as^{α} + 1) Lf(s) = (bs^{α} + 1)L[t − t ∗ f]  ℜs > 0,
(as^{α} + 1) Lf(s) = (bs^{α} + 1)(\frac{1}{s^{2}} − \frac{1}{s^{2}} Lf(s))  ℜs > 0,
Lf(s) = \frac{bs^{α} + 1}{as^{α+2} + s^{2} + bs^{α} + 1}  ℜs > 0,  (20)

and

Lx(s) = \frac{1}{s^{2}} (1 − Lf(s))  ℜs > 0,  (21)
Lx(s) = \frac{as^{α} + 1}{as^{α+2} + s^{2} + bs^{α} + 1}  ℜs > 0.  (22)

Now we apply the inverse Laplace transform to (20) and (22)

h(t) = \frac{1}{2πi} ∫_{γ} e^{st}Lh(s) ds t ≥ 0,

where γ = {s : ℜs = σ, σ > σ_{0} = 0}.

After the use Cauchy formula on an appropriate conture we obtain

x(t) = ∑_{i=1}^{2} \frac{e^{s_{i}t}(1 + as_{i}^{α})}{a(α + 2)s_{i}^{α+1} + 2s_{i} + bαs_{i}^{α−1}} + \frac{sin απ}{π} ×

× ∫_{0}^{∞} \frac{r^{α} (b − a) dr}{(1 + r^{2})^{2} + 2 (1 + r^{2}) r^{α}(b + r^{2}a) cos απ + (r^{α}(b + r^{2}a))^{2}} ,

t ≥ 0. Similarly we get

f(t) = ∑_{i=1}^{2} \frac{e^{s_{i}t}(1 + bs_{i}^{α})}{a(α + 2)s_{i}^{α+1} + 2s_{i} + bαs_{i}^{α−1}} + \frac{sin απ}{π} ×

× ∫_{0}^{∞} \frac{r^{α} (a − b) e^{−rt} dr}{(1 + r^{2})^{2} + 2 (1 + r^{2}) r^{α}(b + r^{2}a) cos απ + (r^{α}(b + r^{2}a))^{2}} ,

t ≥ 0.

In a forthcoming paper we will show that zeros of Δ(s) = as^{α+2}+s^{2}+bs^{α}+1 = (s^{2} + \frac{b}{a})(1 + as^{α}) − (\frac{b}{a} − 1) exist and lie in the left complex halfplane.

5.2. Iterative procedure

Recall that we have formulas (21) and (15),

Lx(s) = \frac{1}{s^{2}} (1 − Lf(s)),  ℜs > 0,  (23)
Lf(s) = \frac{bs^{α} + 1}{as^{α} + 1}Lx(s),  ℜs > 0.  (24)

Let us denote B = \frac{bs^{α}+1}{as^{α}+1}. If we substitute B and (23) into (24) we get

Lf(s) = \frac{B}{s^{2}} (1 − Lf(s)).

Now we calculate Lx(s) and obtain

Lx(s) = \frac{1}{s^{2}} ∑_{k=0}^{n} (−\frac{B}{s^{2}} )^{k} + (−1)^{n+1} (\frac{B}{s^{2}} )^{n} Lf(s).

One can show that for |s| > \frac{b}{a}, ℜs > 0, |(\frac{B}{s^{2}} )^{n} Lf(s)| tends to zero as n → ∞.

More precisely, for |s| > \frac{b}{a}, ℜs > 0 we have

|(\frac{B}{s^{2}} )^{n} Lf(s)| ≤ (\frac{b}{a})^{n} \frac{|Lf(s)|}{s^{2n}}

Thus Lx is an analitic function, Lx(s) = \frac{1}{s^{2}} ∑_{k=0}^{∞} (−\frac{B}{s^{2}} )^{k}, |s| > \frac{b}{a}, ℜs > 0. Since

Lx belongs to D_{+} and has the given form for |s| > \frac{b}{a}, ℜs > 0, its inverse Laplace transform is given by

x(t) = t ∗ ∑_{k=0}^{∞} (\frac{b}{a}t + (\frac{b}{a} − 1) e′_{α}(t, \frac{1}{a}) ∗ t)^{k}  (25)

where u^{k} = \underbrace{u ∗ u ∗ ⋯ ∗ u}_{k times}. Formula (25) represents one more form of the solution

for system (2). The function f is of the form

f(t) = x ∗ (\frac{b}{a}t + (\frac{b}{a} − 1) e′_{α}(t, \frac{1}{a}) ∗ t) , t ≥ 0.

REFERENCES

[1] Atanackovic, T. M., A modified Zener model of a viscoelastic body, Continuum Mech. Thermodyn. 14 (2002), 137–148.

[2] Gorenflo, R. and Mainardi, F., Fractional Calculus: Integral and Differential Equations of Fractional Order, pp. 273–276, ed. A. Carpinteri & F. Mainardi, Springer, Wien 1997.

[3] Pilipović, S. and Stojanović, M., Volterra type of equations in S′_{+}, to appear.

[4] Podllubny, I, Fractional Differential Equations, Academic Press, San Diego 1999.

[5] Samko, S. G., Kilbas, A. A. and Marichev, O. I., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam 1993.

[6] Vladimirov, V. S., Generalized Function in Mathematical Physics, Mir, Moscow.

(received 31.12.2002)