© Hindawi Publishing Corp.
MIXED PROBLEM WITH NONLOCAL BOUNDARY CONDITIONS FOR A THIRD-ORDER PARTIAL DIFFERENTIAL EQUATION
OF MIXED TYPE
M. DENCHE and A. L. MARHOUNE (Received 7 January 2000)
Abstract.We study a mixed problem with integral boundary conditions for a third-order partial differential equation of mixed type. We prove the existence and uniqueness of the solution. The proof is based on two-sided a priori estimates and on the density of the range of the operator generated by the considered problem.
2000 Mathematics Subject Classification. 35B45, 35K20, 35M10.
1. Introduction. In the rectangleΩ=(0,)×(0,T ), we consider the equation ᏸu=∂2u
∂t2− ∂
∂x
a(x,t) ∂2u
∂x∂t
=f (x,t), (1.1)
wherea(x,t)is bounded with 0< a0< a(x,t)≤a1and has bounded partial deriva- tives such that 0< a2≤∂a(x,t)/∂t≤a3and 0< a4≤∂a(x,t)/∂x≤a5for(x,t)∈Ω.
To(1.1) we add the initial conditions l1u=u(x,0)=ϕ(x), l2u=∂u
∂t(x,0)=ψ(x), x∈(0,), (1.2) the Dirichlet condition
u(0,t)=0, t∈(0,T ), (1.3) and the integral condition
0u(ξ,t)dξ=0, t∈(0,T ), (1.4) whereϕandψare known functions which satisfy the compatibility conditions given by (1.3) and (1.4), that is,
ϕ(0)=0,
0ϕ(x)dx=0, ψ(0)=0,
0ψ(x)dx=0. (1.5) Boundary-value problems for parabolic equations with integral boundary condi- tions are investigated by Batten [1], Bouziani and Benouar [2], Cannon [3, 4], Perez Esteva and van der Hoeck [5], Ionkin [8], Kamynin [9], Kartynnik [10], Shi [11], Yurchuk [13], and many references therein. We remark that integral boundary conditions for evolution problems have various applications in chemical engineering, thermoelastic- ity, underground water flow and population dynamics; see for example, [6,7,11,12].
The present paper is devoted to the study of a mixed problem with boundary inte- gral conditions for a third-order partial differential equation of mixed type.
We associate to problem (1.1), (1.2), (1.3), and (1.4) the operatorL=(ᏸ,l1,l2), de- fined fromEintoF, whereEis the Banach space of functionsu∈L2(Ω), satisfying (1.3) and (1.4), with the finite norm
u2E=
Ω(−x)2 ∂2u
∂t2 2+
∂3u
∂x2∂t 2
dx dt + sup
0≤t≤T
0(−x)2 ∂2u
∂x∂t 2+
∂u
∂x 2
dx+ sup
0≤t≤T
0
∂u
∂t
2+|u|2
dx, (1.6) andF is the Hilbert space of vector-valued functionsᏲ=(f ,ϕ,ψ)obtained by com- pletion of the spaceL2(Ω)×W22(0,)×W22(0,)with respect tothe norm
Ᏺ2F=(f ,ϕ,ψ)2
F
=
Ω(−x)2|f|2dx dt+
0(−x)2 dϕ
dx 2+
dψ dx
2 dx+
0
|ϕ|2+|ψ|2 dx.
(1.7) Using the energy inequalities method proposed in [13], we establish two-sided a pri- ori estimates. Then, we prove that the operatorLis a linear homeomorphism between the spacesEandF.
2. Two-sided a priori estimates
Theorem2.1. For any functionu∈E, there is the a priori estimate
LuF≤cuE, (2.1)
where the constantcis independent ofu.
Proof. Using (1.1) and the initial conditions (1.2), we obtain
Ω(−x)2|ᏸu|2dx dt≤3
Ω(−x)2 ∂2u
∂t2 2+a25
∂2u
∂x∂t 2+a21
∂3u
∂x2∂t 2
dx dt,
0(−x)2 dψ
dx 2+
dϕ dx
2
dx≤ sup
0≤t≤T
0(−x)2 ∂2u
∂x∂t 2+
∂u
∂x 2
dx,
0
|ψ|2+|ϕ|2 dx≤ sup
0≤t≤T
0
∂u
∂t
2+|u|2
dx.
(2.2)
Combining the inequalities (2.2), we obtain (2.1) fo ru∈E.
Theorem2.2. For any functionu∈E, there is the a priori estimate
uE≤αLuF, (2.3)
with the constant
α= max
167/10,a1 min
exp(−cT )/20,exp(−cT )a20/15, (2.4)
andcis such that
c≥1, ca0−1≥a3+2a25. (2.5) Before proving this theorem, we first give the following two lemmas.
Lemma2.3. Foru∈Esatisfying the first condition in (1.2), 1
2 τ
0
0(−x)2exp(−ct) ∂2u
∂x∂t
2dx dt+c−1 2
τ
0
0(−x)2exp(−ct) ∂u
∂x
2dx dt
≥1 2
0(−x)2exp(−cτ) ∂u
∂x(x,τ) 2dx−1
2
0(−x)2 dϕ
dx 2dx.
(2.6) Proof. Starting from
τ
0
0(−x)2exp(−ct)∂
∂t ∂u
∂x ∂u
∂xdx dt, (2.7)
then integrating by parts and using elementary inequalities, we obtain (2.6).
Lemma2.4. Foru∈Esatisfying the initial conditions (1.2),
0exp(−cτ)u(x,τ)2dx≤ τ
0
0exp(−ct) ∂u
∂t
2dx dt+
0|ϕ|2dx, (2.8) withc≥1.
Proof. Integrating by parts the expression τ
0
0exp(−ct)u∂u
∂t dx dt (2.9)
and using elementary inequalities yield (2.8).
Remark2.5. We note that Lemmas2.3and2.4hold for weaker conditions onu.
Proof ofTheorem2.2. First, define D(L)=
u∈E| ∂5u
∂x2∂t3∈L2(Ω)
, Mu=(−x)2∂2u
∂t2+2(−x)J∂2u
∂t2, (2.10) where
Ju= x
0 u(ξ,t)dξ. (2.11)
We consider foru∈D(L)the quadratic formula
Re τ
0
0exp(−ct)ᏸuMudx dt, (2.12)
with the constantcsatisfying (2.5), obtained by multiplying (1.1) by exp(−ct)Mu, by
integrating overΩτ, whereΩτ=(0,)×(0,τ), with 0≤τ≤T, and by taking the real part. Integrating by parts (2.12) with the use of boundary conditions (1.3) and (1.4), we obtain
Re τ
0
0exp(−ct)ᏸuMudx dt
= τ
0
0(−x)2exp(−ct) ∂2u
∂t2
2dx dt+1 2
τ
0
0exp(−ct) J∂2u
∂t2
2dx dt +Re
τ
0
0(−x)2exp(−ct)a∂2u
∂x∂t
∂
∂t ∂2u
∂x∂t
dx dt +2Re
τ
0
0exp(−ct)∂u
∂ta∂2u
∂t2dx dt +2Re
τ
0
0exp(−ct)∂a
∂x
∂u
∂tJ∂2u
∂t2dx dt.
(2.13)
On the other hand, by using the elementary inequalities we get Re
τ
0
0exp(−ct)ᏸuMudx dt
≥ τ
0
0(−x)2exp(−ct) ∂2u
∂t2
2dx dt +Re
τ
0
0(−x)2exp(−ct)a ∂2u
∂x∂t
∂
∂t ∂2u
∂x∂t
dx dt +2Re
τ
0
0exp(−ct)∂u
∂ta∂2u
∂t2dx dt
−2Re τ
0
0exp(−ct) ∂a
∂x 2
∂u
∂t
2dx dt.
(2.14)
Again, integrating by parts the second and third terms of the right-hand side of the inequality (2.14) and taking into account the initial conditions (1.2) give
Re τ
0
0exp(−ct)ᏸuMudx dt+
0a(x,0)|ψ|2dx+1 2
0a(x,0)(−x)2 dψ
dx
2dx dt
≥ τ
0
0exp(−ct)(−x)2 ∂2u
∂t2
2dx dt−2 τ
0
0exp(−ct) ∂a
∂x 2
∂u
∂t
2dx dt +1
2
0a(x,τ)exp(−cτ)(−x) ∂2u
∂x∂t(x,τ) 2dx
−1 2
τ
0
0exp(−ct)∂a
∂t(−x)2 ∂2u
∂x∂t
2dx dt
+c 2
τ
0
0exp(−ct)(−x)2a ∂2u
∂x∂t
2dx dt+
0exp(−cτ)a(x,τ) ∂u
∂t(x,τ) 2dx
− τ
0
0exp(−ct)∂a
∂t ∂u
∂t
2dx dt+c τ
0
0exp(−ct)a ∂u
∂t
2dx dt.
(2.15)
By using the elementary inequalities on the first integral in the left-hand side of (2.15), we obtain
33 2
τ
0
0exp(−ct)(−x)2|ᏸu|2dx dt+3 4
τ
0
0exp(−ct)(−x)2 ∂2u
∂t2
2dx dt +
0a(x,0)|ψ|2dx+1 2
0a(x,0)(−x)2 dψ
dx 2dx
≥ τ
0
0exp(−ct)(−x)2 ∂2u
∂t2
2dx dt−2 τ
0
0exp(−ct) ∂a
∂x 2
∂u
∂t
2dx dt +1
2
0exp(−cτ)(−x)2
∂2u(x,τ)
∂x∂t
2dx−1 2
τ
0
0exp(−ct)(−x)2∂a
∂t ∂2u
∂x∂t 2dx dt
+c 2
τ
0
0exp(−ct)(−x)2a ∂2u
∂x∂t
2dx dt+
0exp(−cτ)a(x,τ)
∂u(x,τ)
∂t
2dx
− τ
0
0exp(−ct)∂a
∂t ∂u
∂t
2dx dt+c τ
0
0exp(−ct)a ∂u
∂t
2dx dt.
(2.16) Now, from (1.1) we have
1 5
τ
0
0exp(−ct)(−x)2|ᏸu|2dx dt +1
5 τ
0
0exp(−ct)(−x)2 ∂a
∂x 2
∂2u
∂x∂t
2dx dt +1
5 τ
0
0exp(−ct)(−x)2 ∂2u
∂t2
2dx dt
≥ 1 15
τ
0
0exp(−ct)(−x)2a2 ∂3u
∂x2∂t
2dx dt.
(2.17)
Combining inequalities (2.16), (2.17), and Lemmas2.3and2.4, we get 167
10
Ω(−x)2|ᏸu|2dx dt+a1
2
0(−x)2 dψ
dx 2dx +a1
0|ψ|2dx+1 2
0(−x)2 dϕ
dx 2dx+
0|ϕ|2dx
≥exp(−cT ) 1
20 τ
0
0(−x)2 ∂2u
∂t2
2dx dt+1 2
0(−x)2 ∂2u
∂x∂t(x,τ) 2dx dt +
0|u(x,τ)|2dx+a0
0
∂u
∂t(x,τ) 2dx+1
2
0(−x)2 ∂u
∂x(x,τ) 2dx +a20
15 τ
0
0(−x)2 ∂3u
∂x2∂t
2dx dt
.
(2.18) As the left-hand side of (2.18) is independent ofτ, by replacing the right-hand side by its upper bound with respect toτ in the interval [0,T ], we obtain the desired inequality.
3. Solvability of the problem. From estimates (2.1) and (2.3) it follows that the operatorL:E→F is continuous and its range is closed inF. Therefore, the inverse operatorL−1exists and is continuous from the closed subspaceR(L)ontoE, which means thatL is a homomorphism fromE ontoR(L). To obtain the uniqueness of solution, it remains to show thatR(L)=F. The proof is based on the following lemma.
Lemma3.1. Suppose that∂3a/∂x2∂tis also bounded. LetD0(L)={u∈D(L):l1u=0, l2u=0}. If foru∈D0(L)and someω∈L2(Ω),
Ω(−x)ᏸu" dx dt=0, (3.1) thenω=0.
Proof. From (3.1) we have
Ω(−x)∂2u
∂t2" dx dt=
Ω(−x) ∂
∂x
a∂2u
∂x∂t
" dx dt. (3.2) If we introduce the smoothing operators with respect tot(see [13])Jξ−1=(I+ξ(∂/∂t))−1 and(Jξ−1)∗, then these operators provide the solutions of the respective problems
ξdgξ(t)
dt +gξ(t)=g(t), gξ(t)|t=0=0, (3.3)
−ξdg∗ξ(t)
dt +gξ∗(t)=g(t), gξ∗(t)|t=T=0, (3.4) and also have the following properties: for anyg∈L2(0,T ), the functionsgξ=(Jξ−1)g andgξ∗=(Jξ−1)∗gare inW21(0,T )such thatgξ|t=0=0 andgξ∗|t=T=0. Moreover,Jξ−1 commutes with∂/∂t, soT
0 |gξ−g|2dt→0 andT
0|gξ∗−g|2dt→0 fo rξ→0.
Now, for givenω(x,t), we introduce the function v(x,t)=ω(x,t)−
x
0
ω(ξ,t)
−ξ dξ. (3.5)
Integrating by parts with respect toξ, we obtain x
0 v(ξ,t)dξ= x
0ω(ξ,t)dξ+ x
0
∂
∂ξ(−ξ) ξ
0
ω(η,t) −η dηdξ
=(−x)
ω(x,t)−v(x,t) ,
(3.6)
which implies that
(−x)v+Jv=(−x)w,
0v(x,t)dx=0. (3.7) Then, from equality (3.2) we obtain
−
Ω
∂2u
∂t2Nv dx dt=
ΩA(t)∂u
∂tv dx dt, (3.8)
where
Nv=(−x)v+Jv, A(t)u= − ∂
∂x
(−x)a(x,t)∂u
∂x
. (3.9)
Replace∂u/∂tby the smoothed functionJξ−1(∂u/∂t)in (3.8) and use the relation
A(t)J−1ξ =Jξ−1A(τ)+ξJξ−1∂A(τ)
∂τ Jξ−1. (3.10)
Then, by taking the adjoint of the operator Jξ−1, and by integrating by parts with respect totin the left-hand side, we obtain
Ω
∂u
∂tN∂vξ∗
∂t dx dt=
ΩA(t)∂u
∂tvξ∗dx dt+ξ
Ω
∂A
∂t ∂u
∂t
ξvξ∗dx dt. (3.11) The operatorA(t)has a continuous inverse onL2(0,)defined by the relation
A−1(t)g= − x
0
dξ a(ξ,t)(−ξ)
ξ
0g(η)dη+c x
0
dξ
a(ξ,t)(−ξ), (3.12) where
c=
0
dx/a(x,t)x 0g(ξ)dξ
0
dx/a(x,t) ,
0A−1(t)g dx=0. (3.13) Hence, the function(∂u/∂t)ξ can be represented in the form
∂u
∂t
ξ=Jξ−1A−1(t)A(t)∂u
∂t. (3.14)
Then,(∂A/∂t)(∂u/∂t)ξ=Aξ(t)A(t)(∂u/∂t), where
Aξ(t)= ∂2a
∂x∂tJξ−1−∂a
∂tJξ−1∂a
∂x 1 a
1 a
x
0g(η,t)dη−c
+∂a
∂tJξ−11
ag, (3.15) where the constantcis given by (3.13).
Consequently, equation (3.11) becomes
Ω
∂u
∂tN∂vξ∗
∂t dx dt=
ΩA(t)∂u
∂t
vξ∗+ξA∗ξvξ∗
dx dt, (3.16)
in which the conjugate operatorA∗ξ(t)ofAξ(t)is defined by
A∗ξvξ∗=1 a
Jξ−1∗∂a
∂τvξ∗+ Bvξ∗
(x)−
Bvξ∗ (0)
x
dξ/a(ξ,t)
0
dξ/a(ξ,t), (3.17)
where Bvξ∗
(x)=
x
1 a(ξ,t)
Jξ−1∗ ∂2a
∂ξ∂τ− 1 a(ξ,t)
∂a
∂ξ
Jξ−1∗∂a
∂τ
vξ∗(ξ,τ)dξ. (3.18)
The left-hand side of (3.16) is a continuous linear functional of ∂u/∂t. Hence, the functionhξ=vξ∗+ξA∗ξvξ∗has the derivatives(−x)(∂hξ/∂x)∈L2(Ω),(∂/∂x)((− x)(∂hξ/∂x))∈L2(Ω), and the following conditions are satisfied
hξ|x=0=0, hξ|x==0, (−x)∂hξ
∂x
x==0. (3.19)
From (3.17) we have
(−x)∂hξ
∂x =
I+ξ1 a
Jξ−1∗∂a
∂τ ∂vξ∗
∂x , (3.20)
∂
∂x
(−x)∂hξ
∂x
= I+ξ1
a
Jξ−1∗∂a
∂τ ∂
∂x
(−x)∂vξ∗
∂x
+ξ
−(∂a/∂x)
Jξ−1∗(∂a/∂τ)
a2 +1
a
Jξ−1∗ ∂2a
∂x∂τ
(−x)∂vξ∗
∂x , (3.21)
I+ξ1
a
Jξ−1∗∂a
∂τ
vξ∗
x=0=0, (3.22)
I+ξ1
a
Jξ−1∗∂a
∂τ
vξ∗
x==0, (3.23)
I+ξ1
a
Jξ−1∗∂a
∂τ
(−x)∂vξ∗
∂x
x=
=0. (3.24)
Since ξ(1/a)(Jξ−1)∗(∂a/∂τ)L2(Ω) < 1 for sufficiently small ξ, the operator I+ ξ(1/a)(Jξ−1)∗(∂a/∂τ)has a continuous inverse onL2(Ω). In addition, the derivative of the above operator with respect tox is a bounded operator inL2(Ω). Therefore, from (3.20) and (3.21), the functionvξ∗has derivatives(−x)(∂vξ∗/∂x)∈L2(Ω)and (∂/∂x)((−x)(∂vξ∗/∂x))∈L2(Ω).
In a similar way, we show that for each fixedx∈[0,]and sufficiently smallξ, the operatorI+ξ(1/a)(Jξ−1)∗(∂a/∂τ)has a continuous inverse onL2(0,T ); hence, (3.22), and (3.23), and (3.24) imply that
vξ∗x=0=0, vξ∗x==0, (−x)∂vξ∗
∂x x=
=0. (3.25)
So, forξsufficiently small, the functionvξ∗has the same properties ashξ. In addition, vξ∗satisfies the integral condition in (3.7).
Puttingu=t
0
τ
0exp(cη)vξ∗(η,τ)dηdτin (3.8), where the constantcsatisfiesca0− a3−a23/a0≥0, and using (3.4), we obtain
Ωexp(ct)vξ∗Nv dx dt= −
ΩA(t)∂u
∂t exp(−ct)∂2u
∂t2 dx dt+ξ
ΩA(t)∂u
∂t
∂vξ∗
∂t dx dt.
(3.26)
Integrating by parts each term in the left-hand side of (3.26) and taking the real parts yield
Re
ΩA(t)∂u
∂t exp(−ct)∂2u
∂t2dx dt
≥ c 2
Ω(−x)a(x,t)exp(−ct) ∂2u
∂x∂t
2dx dt
−1 2
Ω(−x)∂a
∂t exp(−ct) ∂2u
∂x∂t
2dx dt, Re
−ξ
ΩA(t)∂u
∂t
∂vξ∗
∂t dx dt
≥−ξa23 2a0
Ω(−x)exp(−ct) ∂2u
∂x∂t
2dx dt.
(3.27)
Now, using (3.27) in (3.26) with the choice ofcindicated above we have 2Re
Ωexp(ct)vξ∗Nv dx dt≤0. (3.28) Then, forξ→0 we obtain 2Re
Ωexp(ct)vNv dx dt≤0, that is, 2Re
Ωexp(ct)(−x)|v|2dx dt+2Re
Ωexp(ct)vJv dx dt≤0. (3.29) Since Re
Ωexp(ct)vJv dx dt=0, we conclude thatv=0; hence,ω=0, which ends the proof of the lemma.
Theorem3.2. The rangeR(L)ofLcoincides withF.
Proof. SinceF is a Hilbert space, we haveR(L)=F if and only if the relation
Ω(−x)2ᏸuf dx dt+
0
(−x)2
dl1u dx
dϕ dx+dl2u
dx dψ dx
dx+
0
l1uϕ+l2uψ dx=0,
(3.30) for arbitraryu∈E and(f ,ϕ,ψ)∈F, implies thatf=0,ϕ=0 andψ=0. Putting u∈D0(L)in (3.30), we conclude fromLemma 3.1that(−x)f =0. Hence,
0
(−x)2
dl1u dx
dϕ dx+dl2u
dx dψ dx
+l1uϕ+l2uψ
dx=0 ∀u∈D(L). (3.31) Setting
D0k(L)=
u∈D(L):u(k)t=0=0, k=0,1
, (3.32)
and takingu∈D01(L)in (3.31) yield
0
(−x)2dl1u dx
dϕ dx +l1uϕ
dx=0. (3.33)
The range of the trace operatorl1is everywhere dense in Hilbert space with the norm [
0((−x)2|dϕ/dx|2+ |ϕ|2)dx]1/2; hence,ϕ=0. Likewise, foru∈D00(L), we get ψ=0.
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M. Denche: Institut de Mathématiques, Université Mentouri Constantine, Constan- tine25000, Algeria
E-mail address:[email protected]
A. L. Marhoune: Institut de Mathématiques, Université Mentouri Constantine, Con- stantine25000, Algeria
Special Issue on
Intelligent Computational Methods for Financial Engineering
Call for Papers
As a multidisciplinary field, financial engineering is becom- ing increasingly important in today’s economic and financial world, especially in areas such as portfolio management, as- set valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the re- cently approved Basel II guidelines advise financial institu- tions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to im- prove the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distribu- tions or statistical models. In such situations, some attempts have also been made to develop financial engineering mod- els using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estima- tion technique which does not make any distributional as- sumptions regarding the underlying asset. Instead, ANN ap- proach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting pa- rameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior perfor- mance of a new intelligent computational method, but also to demonstrate how it can be used e
ffectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should es- pecially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelli- gent, easy-to-use, and/or comprehensible computational sys- tems (e.g., decision support systems, agent-based system, and web-based systems)
This special issue will include (but not be limited to) the following topics:
• Computational methods
: artificial intelligence, neu- ral networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learn- ing, multiagent learning
• Application fields
: asset valuation and prediction, as- set allocation and portfolio selection, bankruptcy pre- diction, fraud detection, credit risk management
• Implementation aspects
: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, imple- mentation
Authors should follow the Journal of Applied Mathemat- ics and Decision Sciences manuscript format described at the journal site
http://www.hindawi.com/journals/jamds/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Track- ing System at
http://mts.hindawi.com/, according to the fol-lowing timetable:
Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009
Guest Editors
Lean Yu,
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong;
[email protected]
Shouyang Wang,
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; [email protected]
K. K. Lai,
Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; [email protected]
Hindawi Publishing Corporation http://www.hindawi.com
