arXiv:1105.4592v1 [math.NA] 23 May 2011
A Priori Estimates for Solutions of Boundary Value Problems for Fractional-Order Equations A. A. Alikhanov Kabardino-Balkarian State University, ul. Chernyshevskogo 175, Nalchik, 360004, Russia Institute for Applied Mathematics Research KBSC RAS, ul. Shortanova 89-A, Nalchik, 360000, Russia e-mail:
[email protected] Abstract We consider boundary value problems of the first and third kind for the diffusionwave equation. By using the method of energy inequalities, we find a priori estimates for the solutions of these boundary value problems.
Fractional calculus is used for the description of a large class of physical and chemical processes that occur in media with fractal geometry as well as in the mathematical modeling of economic and social-biological phenomena [1, Chap. 5]. It was proved in [1] that fractional differentiation is a positive operator; this result permits one to obtain a priori estimates for solutions of a wide class of boundary value problems for equations with fractional derivatives. The paper [2] deals with the generalization of the differentiation and integration operations from integer to fractional, real, and complex orders and with applications of fractional integration and differentiation to integral and differential equations and in function theory. In the paper [3], an a priori estimate in terms of a fractional Riemann-Liouville integral of the solution was obtained for the solution of the first initial-boundary value problem for the fractional diffusion equation. The general fractional diffusion equation (0 < α ≤ 1) with regularized fractional derivative was considered in [4]. A more detailed bibliography on fractional partial differential equations, including the diffusion-wave equation, can be found, for example, in [5]. In the present paper, we use the method of energy inequalities to obtain a priori estimates for solutions of boundary value problems for the diffusion-wave equation with Caputo fractional derivative[6].
1
BOUNDARY VALUE PROBLEMS FOR THE FRACTIONAL DIFFUSION EQUATION
¯ T = {(x, t) : 0 ≤ x ≤ 1.1. First Boundary Value Problem. In the rectangle Q l, 0 ≤ t ≤ T }, consider the first boundary value problem
α ∂0t u
∂ = ∂x
∂u k(x, t) ∂x
− q(x, t)u + f (x, t), 1
0 < x < l,
0 < t ≤ T,
(1)
u(0, t) = 0,
u(l, t) = 0,
u(x, 0) = u0 (x), where
1 = Γ(1 − α)
α ∂0t u(x, t)
(2)
0 ≤ t ≤ T,
(3)
0 ≤ x ≤ l, Z
t
0
uτ (x, τ ) dτ (t − τ )α
is the Caputo fractional derivative of order α , 0 < α < 1. Throughout the following, we assume that there exists a solution u(x, t) ∈ 2,1 ¯ ¯ T ) is the class of functions that, together C (QT ) of problem (1)-(3), where C m,n (Q with their partial derivatives of order m with respect to x and order n with respect ¯T . to t, are continuous on Q Let us prove the following assertion. Lemma 1. For any function v(t) absolutely continuous on [0, T ], one has the inequality 1 α 2 α v(t)∂0t v(t) ≥ ∂0t v (t), 0 < α < 1. (4) 2 Proof. Let us rewrite inequality (4) in the form
1 1 α 2 α v (t) = v(t) v(t)∂0t v(t) − ∂0t 2 Γ(1 − α)
Zt
vτ (τ )dτ 1 − α (t − τ ) 2Γ(1 − α)
Zt
vτ (τ )(v(t) − v(τ ))dτ 1 = α (t − τ ) Γ(1 − α)
0
Zt
vτ (τ )dτ (t − τ )α
0
1 = Γ(1 − α)
Zt
vη (η)dη
2v(τ )vτ (τ )dτ = (t − τ )α
0
0
1 = Γ(1 − α)
Zt
Zη
Zt
vη (η)dη =
τ
vτ (τ )dτ ≡ I ≥ 0. (t − τ )α
0
0
Therefore, to prove the lemma, it suffices to show that the integral I is nonnegative. The integral I takes nonnegative values, since
1 I= Γ(1 − α)
Zt
vη (η)dη
Zη
vτ (τ )dτ 1 = α (t − τ ) Γ(1 − α)
0
0
=
1 2Γ(1 − α)
Zt
vη (η)dη (t − η)α (t − η)α
0
Zt 0
η 2 Z ∂ vτ (τ )dτ (t − η)α dη = ∂η (t − τ )α 0
2
Zη 0
vτ (τ )dτ = (t − τ )α
=
α 2Γ(1 − α)
Zt 0
η 2 Z vτ (τ )dτ (t − η)α−1 dη ≥ 0. (t − τ )α 0
The proof of the lemma is complete. We use the following notation: Z l 2 −α kuk0 = u2 (x, t)dx, D0t u(x, t) = 0
1 Γ(α)
Z
0
t
u(x, τ ) dτ (t − τ )1−α
is the fractional Riemann-Liouville integral of order α. ¯ T ), q(x, t), f (x, t) ∈ C(Q ¯ T ), k(x, t) ≥ c1 > 0 and Theorem 1. If k(x, t) ∈ C 1,0 (Q ¯ q(x, t) ≥ 0 everywhere on QT , then the solution u(x, t) of problem (1)-(3) satisfies the a priori estimate −α −α kuk20 + D0t kux k20 ≤ M D0t kf k20 + ku0 (x)k20 .
(5)
Proof. We multiply Eq. (1) by u(x, t) and integrate the resulting relation with respect to x from 0 to l: Zl
α u∂0t udx −
0
Zl
u(kux )x dx +
0
Zl
qu2 dx =
0
Zl
(6)
uf dx.
0
Let us transform the terms occurring in identity (6)
−
Zl
u(kux )x dx =
0
Zl
l Z uf dx ≤ εkuk20 + 1 kf k20 , 4ε
ku2x dx,
ε > 0;
(7)
0
0
by virtue of inequality (4) we obtain Zl 0
α u(x, t)∂0t u(x, t)dx
≥
Zl
1 α 2 1 α ∂0t u (x, t)dx = ∂0t kuk20. 2 2
(8)
0
Identity (6), with regard of the above-performed transformations, implies the inequality 1 1 α ∂0t kuk20 + c1 kux k20 ≤ εkuk20 + kf k20 . (9) 2 4ε By virtue of the inequality kuk20 ≤ (l2 /2)kux k20 for ε = c1 /l2 from (9), we obtain the inequality α ∂0t kuk20 + c1 kux k20 ≤
3
l2 kf k20. 2c1
(10)
−α By applying the fractional differentiation operator D0t to both sides of inequality (10), we obtain the estimate (5) with constant M = max{l2 /(2c1 ), 1}/ min{1, c1}. It follows from the a priori estimate (5) that the solution of problem (1)-(3) is unique and continuously depends on the input data. 1.2. Third Boundary Value Problem. In problem (1)-(3), we replace the boundary conditions (2) by the conditions ( k(0, t)ux (0, t) = β1 (t)u(0, t) − µ1 (t), (11) −k(l, t)ux (l, t) = β2 (t)u(l, t) − µ2 (t), 0 ≤ t ≤ T.
¯ T , consider the third boundary value problem (1), (3), (11). In the rectangle Q To obtain a priori estimates for solutions of various nonstationary problems, we use the well-known Gronwall-Bellman lemma [7, p. 152] , whose generalization is provided by the following assertion. Lemma 2. Let a nonnegative absolutely continuous function y(t) satisfy the inequality α ∂0t y(t) ≤ c1 y(t) + c2 (t), 0 < α ≤ 1, (12) for almost all t in [0, T ], where c1 > 0 and c2 (t) is an integrable nonnegative function on [0, T ]. Then −α y(t) ≤ y(0)Eα(c1 tα ) + Γ(α)Eα,α (c1 tα )D0t c2 (t), (13) P∞ n P∞ n z /Γ(αn + 1) and E (z) = where Eα (z) = α,µ n=0 z /Γ(αn + µ) are the n=0 Mittag-Leffler functions. α Proof. Let ∂0t y(t) − c1 y(t) = g(t), then (e.g., see [5, p. 17]) Z t α y(t) = y(0)Eα (c1 t ) + (t − τ )α−1 Eα,α (c1 (t − τ )α )g(τ )dτ (14) 0
By virtue of the inequality g(t) ≤ c2 (t), the positivity of the Mittag-Leffler function Eα,α (c1 (t−τ )α ) for given parameters, and the growth of the function Eα,α (t), from (14), we obtain the inequality α
y(t) ≤ y(0)Eα(c1 t ) +
Zt
(t − τ )α−1 Eα,α (c1 (t − τ )α )c2 (τ )dτ ≤
0
−α ≤ y(0)Eα(c1 tα ) + Γ(α)Eα,α (c1 tα )D0t c2 (t).
The proof of the lemma is complete. Theorem 2. If , in addition to the assumptions of Theorem 1, βi (t), µi (t) ∈ C[0, T ], |βi (t)| ≤ β, for all t ∈ [0, T ], i = 1, 2, then the solution u(x, t) of problem (1),(3),(11) admits the a priori estimate −α −α −α 2 −α 2 kuk20 + D0t kux k20 ≤ M D0t kf k20 + D0t µ1 (t) + D0t µ2 (t) + ku0 (x)k20 . (15) 4
Proof. Just as in the proof of Theorem 1, we multiply Eq. (1) by u(x, t) and integrate the resulting relation with respect to x from 0 to l. By transforming the terms occurring in identity (7), we obtain relations (8) and (7) with ε = 1/2 and −
Zl
2
2
u(kux )x dx = β1 (t)u (0, t) + β2 (t)u (l, t) − µ1 (t)u(0, t) − µ2 (t)u(l, t) +
0
Zl
ku2x dx,
0
Identity (7), with regard of the above-performed transformations, acquires the form 1 α ∂ kuk20 + c1 kux k20 ≤ 2 0t 1 1 ≤ −β1 (t)u2 (0, t) − β2 (t)u2 (l, t) + µ1 (t)u(0, t) + µ2 (t)u(l, t) + kuk20 + kf k20. (16) 2 2 By virtue of the inequalities 1 1 µ1 (t)u(0, t) ≤ u2 (0, t) + µ21 (t), 2 2
1 1 µ2 (t)u(l, t) ≤ u2 (l, t) + µ22 (t), 2 2
u2 (0, t), u2(l, t) ≤ εkux k20 + (1/ε + 1/l)kuk20 ,
ε > 0,
from (16) with ε = c1 /(4β + 2), we obtain the inequality α ∂0t kuk20 + c1 kux k20 ≤ M1 kuk20 + µ21 (t) + µ22 (t) + kf k20 .
(17)
−α By applying the fractional differentiation operator D0t to both sides of inequality (17), we obtain the inequality −α kuk20 + D0t kux k20 ≤
−α −α 2 −α 2 −α ≤ M2 D0t kuk20 + D0t µ1 (t) + D0t µ2 (t) + D0t kf k20 + ku0 (x)k20 .
(18)
By eliminating the second term from the left-hand side of inequality (18) and by −α α using Lemma 2, where y(t) = D0t ku(x, t)k20 , ∂0t y(t) = ku(x, t)k20 and y(0) = 0, we obtain the inequality
−α D0t kuk20
−2α 2 −2α 2 −2α ≤ M3 D0t µ1 (t) + D0t µ2 (t) + D0t kf k20 +
tα 2 ku0 (x)k0 , (19) Γ(α + 1)
where M3 = Γ(α)Eα,α (M2 T α ). −2α −α Since the inequality D0t h(t) ≤ (tα Γ(α)/Γ(2α))D0t h(t) holds for every nonnegative integrable function h(t) on [0, T ], it follows from (18) and (19) that the a priori estimate (15) is true.
5
2
BOUNDARY VALUE PROBLEMS FOR THE FRACTIONAL WAVE EQUATION
¯ T = {(x, t) : 0 ≤ x ≤ 2.1. First Boundary Value Problem. In the rectangle Q l, 0 ≤ t ≤ T }, consider the first boundary value problem
1+α ∂0t u
∂ = ∂x
∂u k(x, t) − q(x, t)u + f (x, t), ∂x u(0, t) = 0,
u(l, t) = 0,
0 < x < l,
0 < t ≤ T,
(20) (21)
0 ≤ t ≤ T,
u(x, 0) = u0 (x), ut (x, 0) = u1 (x), 0 ≤ x ≤ l, (22) R t 1+α where ∂0t u(x, t) = 0 uτ τ (x, τ )(t−τ )−α dτ /Γ(1−α) is the Caputo fractional derivative of order 1 + α, 0 < α < 1. Throughout the following, we assume that there exists a solution u(x, t) ∈ 2,2 ¯ C (Q) of problem (20)-(22). ¯ T ), q(x, t) ∈ C 0,1 (Q ¯ T ), f (x, t) ∈ C(Q ¯ T ), 0 < Theorem 3. If k(x, t) ∈ C 1,1 (Q c1 ≤ k(x, t) ≤ c2 , 0 < m1 ≤ q(x, t) ≤ m2 and |kt (x, t)|, |qt (x, t)| ≤ c3 everywhere on ¯ T , then the solution u(x, t) of problem (20)-(22) admits the a priori estimate Q
α−1 D0t kuk20 + kuk2W 1(0,l) 2
t Z ≤ M kf k20dτ + ku1(x)k20 + ku0 (x)k2W 1 (0,l) , 2
(23)
0
where kuk2W 1(0,l) = kuk20 + kux k20 . 2 Proof. Let us multiply Eq. (20) by ut (x, t) and integrate the resulting relation with respect to x from 0 to l, Zl
1+α ut ∂0t udx −
Zl
ut (kux )x dx +
quut dx =
0
0
0
Zl
Zl
ut f dx.
(24)
0
Let us transform the terms occurring in identity (24) Zl 0
−
Zl 0
1+α ut ∂0t udx =
Zl
1 α α kut k20 , ut ∂0t ut dx ≥ ∂0t 2
0
1∂ ut (kux )x dx = 2 ∂t
Zl 0
6
1 ku2x dx − 2
Zl 0
kt u2x dx,
(25)
Zl
quut dx =
1∂ 2 ∂t
Zl
qu2dx −
1 2
0
0
Zl
qt u2 dx,
l Z ut f dx ≤ 1 kut k20 + 1 kf k20 . 2 2
(26)
0
0
By taking into account the performed transformations, from identity (24), we obtain the inequality
α ∂0t kut k20
∂ + ∂t
Zl
ku2x dx
∂ + ∂t
0
Zl 0
qu2 dx ≤ M4 kut k20 + kuk2W 1 (0,l) + kf k20 ; 2
(27)
by integrating this relation with respect to τ from 0 to t, we obtain the inequality α−1 D0t kut k20 + kuk2W 1(0,l) ≤ 2
t Z Zt ≤ M5 (kuτ k20 + kuk2W 1(0,l) )dτ + kf k20 dτ + ku1 (x)k20 + ku0 (x)k2W 1 (0,l) . (28) 2
2
0
0
By omitting the first term on the left-hand side inR inequality (28) and by using the t Gronwall-Bellman lemma [7, p. 152], where y(t) = 0 kuk2W 1(0,l) dτ , y ′(t) = kuk2W 1(0,l) 2 2 and y(0) = 0, we obtain t Zt Z kuk2W 1(0,l) dτ ≤ M6 kuτ k20 + kf k20 dτ + ku1(x)k20 + ku0 (x)k2W 1 (0,l) . (29) 2
2
0
0
Then, by omitting the second term on the left-hand side in inequality (28) and by using inequality (29), we obtain the inequality
α−1 kut k20 ≤ M7 D0t
Zt 0
kuτ k20 dτ +
Zt 0
kf k20 dτ + ku1 (x)k20 + ku0(x)k2W 1 (0,l) . 2
(30)
Rt α−1 α By Lemma 2, where y(t) = 0 kuτ (x, τ )k20 dτ , ∂0t y(t) = D0t kut (x, t)k20 and y(0) = 0, from (30) we obtain the inequality Zt 0
−1−α kuτ k20 dτ ≤ M8 D0t kf k20 + ku1 (x)k20 + ku0(x)k2W 1 (0,l) . 2
7
(31)
Rt −1−α By virtue of the inequality D0t kf k20 ≤ (tα /Γ(1 + α)) 0 kf k20 dτ , it follows from inequalities (28), (29) and (31) that the a priori estimate(23) holds. The a priori estimate (23) implies that the solution of problem (20)-(22) exists and continuously depends on the input data. ¯ T , consider the 2.2. Third Boundary Value Problem. In the rectangle Q third boundary value problem (20), (22), (11). Теорема 4. If , in addition to the assumptions of Theorem 3, βi (t), µi (t) ∈ 1 C [0, T ], βi (t) ≥ β > 0 and |βit (t)| ≤ c4 for all t ∈ [0, T ], i = 1, 2, then the solution u(x, t) of problem (20), (22), (11) admits the a priori estimate t Z α−1 D0t kut k20 + kuk2W 1(0,l) ≤ M (kf k20 + µ21τ (τ ) + µ22τ (τ ))dτ + 2
0
+M
kµ1 (t)k2C[0,T ]
+
kµ2 (t)k2C[0,T ]
+
ku1 (x)k20
+
ku0 (x)k2W 1 (0,l) 2
.
(32)
Proof. Just as in Theorem 3, by multiplying Eq. (20) by ut (x, t) and by integrating the resulting relation with respect to x from 0 to l, we obtain identity (24). By transforming the terms occurring in identity (24), we obtain relations (25) and (26) and −
Zl
ut (kux )x dx =
0
=
1∂ β1 (t)u2 (0, t) + β2 (t)u(l, t) − 2µ1 (t)u(0, t) − 2µ2 (t)u(l, t) + 2 ∂t
Zl 0
1 1 1 − β1t (t)u2 (0, t) − β2t (t)u2 (l, t) + µ1t (t)u(0, t) + µ2t (t)u(l, t) − 2 2 2
ku2x dx −
Zl
kt u2x dx.
0
By taking into account the performed transformations and by using the inequalities 1 1 µ1t (t)u(0, t) ≤ µ21t (t) + u2 (0, t), 2 2
1 1 µ2t (t)u(l, t) ≤ µ22t (t) + u2 (l, t), 2 2
u2 (0, t), u2(l, t) ≤ kux k20 + (1 + 1/l)kuk20, from identity (24), we obtain the inequality l Z Zl ∂ α ku2x dx + qu2dx + ∂0t kut k20 + ∂t 0
0
8
+
∂ (β1 (t)u2 (0, t) + β2 (t)u(l, t) − 2µ1 (t)u(0, t) − 2µ2 (t)u(l, t)) ≤ ∂t ≤ M9 kut k20 + kuk2W 1(0,l) + kf k20 + µ21t (t) + µ22t (t) . 2
(33)
By integrating inequality (33) with respect to τ from 0 to t and by taking into account the inequalities 2µ1 (t)u(0, t) ≤ εu2(0, t) + (1/ε)µ21(t),
2µ2 (t)u(l, t) ≤ εu2 (l, t) + (1/ε)µ22(t),
for ε = β, u20 (0), u20 (l) ≤ (1 + 1/l)ku0 (x)k2W 1 (0,l) , 2
we obtain α−1 D0t kut k20 + kuk2W 1(0,l) ≤ 2
t Zt Z ≤ M10 (kuτ k20 + kuk2W 1 (0,l) )dτ + (kf k20 + µ21τ (τ ) + µ22τ (τ ))dτ + 2
0
0
+ M10 kµ1 k2C[0,T ] + kµ2 k2C[0,T ] + ku1 (x)k20 + ku0(x)k2W 1 (0,l) . 2
(34)
By analogy with the first boundary value problem, by using first the GronwallBellman lemma [7, p. 152] and then Lemma 2, from inequality (34), we obtain the a priori estimate (32).
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[5] Pskhu, A.V., Uravneniya v chastnykh proizvodnykh drobnogo poryadka (Fractional Partial Differential Equations), Moscow: Nauka, 2005. [6] Caputo, M., Elasticita e Dissipazione, Zanichelli; Bologna, 1969. [7] Ladyzhenskaya, O.A., Kraevye zadachi matematicheskoi fiziki (Boundary Value Problems of Mathematical Physics), Moscow: Nauka, 1973.
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