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Research article Special Issues

Fractional derivative boundary control in coupled Euler-Bernoulli beams: stability and discrete energy decay

  • This paper analyzes an Euler-Bernoulli beam equation in a bounded domain with a boundary control condition involving a fractional derivative. By utilizing the semigroup theory of linear operators and building on the results of Borichev and Tomilov, the stability properties of the system are examined. Additionally, a numerical scheme is developed to reproduce various decay rate behaviors. The numerical simulations confirm the theoretical stability results regarding the energy decay rate and demonstrate exponential decay for specific configurations of initial data.

    Citation: Boumediene Boukhari, Foued Mtiri, Ahmed Bchatnia, Abderrahmane Beniani. Fractional derivative boundary control in coupled Euler-Bernoulli beams: stability and discrete energy decay[J]. AIMS Mathematics, 2024, 9(11): 32102-32123. doi: 10.3934/math.20241541

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  • This paper analyzes an Euler-Bernoulli beam equation in a bounded domain with a boundary control condition involving a fractional derivative. By utilizing the semigroup theory of linear operators and building on the results of Borichev and Tomilov, the stability properties of the system are examined. Additionally, a numerical scheme is developed to reproduce various decay rate behaviors. The numerical simulations confirm the theoretical stability results regarding the energy decay rate and demonstrate exponential decay for specific configurations of initial data.



    This article examines the solutions to the initial boundary value problem of connected Euler-Bernoulli beam equation, expressed as

    {φtt(x,t)+φxxxx(x,t)+β(φψ)(x,t)=0,(x,t)(0,L)×(0,+),ψtt(x,t)+ψxxxx(x,t)+β(ψφ)(x,t)=0,(x,t)(0,L)×(0,+), (1.1)

    accompanied by the following boundary conditions:

    {φ(0,t)=ψ(0,t)=φx(0,t)=ψx(0,t)=0,in(0,+),φxx(L,t)=ψxx(L)=ψxxx(L,t)=0,in(0,+),φxxx(L,t)=γα,ηtφ(L,t),in(0,+), (1.2)

    where γ and β are positive constants, and the following variables have specific engineering meanings: φ and ψ denote the vertical displacements, φt and ψt represent the velocities, φx and ψx signify the rotations, and φxt and ψxt indicate the angular velocities.

    The system's initial conditions are specified as

    φ(x,0)=φ0(x),φt(x,0)=φ1(x),ψ(x,0)=ψ0(x),ψt(x,0)=ψ1(x),x(0,L), (1.3)

    where the initial data (φ0,φ1) and (ψ0,ψ1) belong to an appropriate function space.

    The notation α,ηt denotes the tempered Caputo time-fractional derivative of order α with respect to time t, where 0<α<1, as introduced in [1,2,3,4]. It is defined as

    α,ηtf(t)=1Γ(1α)t0(ts)αeη(ts)dfds(s)ds,η0,

    where Γ() represents the Gamma function.

    The tempered Caputo time-fractional derivative, a generalization of the classical Caputo derivative, incorporates an exponential tempering factor that allows for a finite memory horizon, making it particularly suitable for processes with decaying memory effects. These exponentially modified fractional integro-differential operators were first proposed by Choi and MacCamy in [5], presenting a slightly different approach from the classical Caputo derivative formulated by Michele Caputo in [6]. Unlike the standard Caputo derivative, the tempered version is especially suited to applications where the memory of past states diminishes over time, as observed in fields such as anomalous diffusion and viscoelasticity. This choice provides both improved model stability and greater accuracy in representing transient dynamics, offering a more realistic approach to systems with decaying temporal correlations.

    A basic model that describes the transverse vibration of a system of non-homogeneous connected Euler-Bernoulli beams, as presented in [7], is represented by the following system:

    φtt(x,t)+φxxxx(x,t)=0,(x,t)(0,L)×(0,+).

    While this model provides a foundational description of transverse vibrations in such systems, it primarily addresses the standard beam dynamics without incorporating advanced effects such as memory and complex dissipation. To enhance the model's applicability to real-world scenarios, we extend the classical Euler-Bernoulli framework by introducing fractional derivatives in the boundary conditions. This extension allows for a more accurate representation of systems exhibiting memory effects and intricate dissipative behavior. By integrating these fractional derivatives, we capture more realistic dissipative dynamics and gain deeper insights into the stability properties of the system.

    In recent years, the scientific community has shown growing interest in unraveling the intricate dynamics and practical applications of wave equations. The behavior of waves, whether occurring naturally, such as seismic waves in the Earth's crust, or in engineered systems, such as acoustic waves in materials, captivates both researchers and practitioners alike. Considerable research effort has been devoted to investigating wave equations with diverse damping types and exploring their stability and controllability. These waves emerge when a vibrating source disrupts the surrounding medium. Researchers have shown a keen interest in addressing damping-related challenges, whether local or global, and have illustrated different forms of stability.

    In [4], B. Mbodje investigated the asymptotic behavior of solutions using the system:

    {2tu(x,t)u2x(x,t)=0,(x,t)(0,1)×(0,+),u(0,t)=0,xu(1,t)=kα,ηtu(1,t),α(0,1),η0,k>0,u(x,0)=u0(x),tu(x,0)=v0(x).

    He established that the corresponding semigroup lacks exponential stability, showing only strong asymptotic stability. Additionally, the system's energy diminishes over time, approaching a decay proportional to t1 as time extends to infinity.

    In [8], Akil and Wehbe investigated a multidimensional wave equation featuring boundary fractional damping applied to a portion of the domain's boundary:

    {utt(x,t)Δu(x,t)=0,(x,t)Ω×(0,+),u(x,t)=0,(x,t)Γ0×(0,+),uν(x,t)+γα,ηtu(x,t)=0,(x,t)Γ1×(0,+).

    They demonstrated the system's strong stability while establishing that it lacked uniform stability. In addition, they derived a polynomial energy decay for smooth solutions of the form t11α. This analysis assumed specific geometric conditions for the boundary control region and leveraged the exponential decay of the wave equation with standard damping.

    Recently, in [9], Beniani et al. examined a system comprising coupled wave equations featuring a diffusive internal control of a general nature:

    {ttuΔxu+ζ+ϱ(ω)ϕ(x,ω,t)dω+βv=0,ttvΔxv+ζ+ϱ(ω)φ(x,ω,t)dω+βu=0,u=v=0,onΩ,ϕt(x,ω,t)+(ω2+η)ϕ(x,ω,t)tuϱ(ω)=0,φt(x,ω,t)+(ω2+η)φ(x,ω,t)tvϱ(ω)=0,u(x,0)=u0(x),tu(x,0)=u1(x),v(x,0)=v0(x),tv(x,0)=v1(x),ϕ(x,ω,0)=ϕ0(x,ω),andφ(x,ω,0)=φ0(x,ω).

    They showed the absence of exponential stability and explored the asymptotic stability of the model, establishing a general decay rate that depends on the density function ϱ. See also the references in [7,10,11,12,13] for further existing results related to the stability and numerical analysis of (1.1).

    In this investigation, we extend the traditional Euler-Bernoulli beam model, commonly used to describe deformations under external loads, by incorporating fractional derivatives in the boundary conditions. This extension is essential, as fractional derivatives offer more precise models for systems exhibiting memory effects and complex dissipation phenomena frequently observed in practical applications such as the analysis of material microstructures. Incorporating fractional derivatives into the boundary conditions constitutes a novel advancement within the Euler-Bernoulli framework, enabling a more accurate representation of dissipative effects in diverse materials and structures. The findings from this study provide new insights into the stability characteristics of systems with fractional boundary dissipation, enhancing both theoretical understanding and the numerical experiments used to validate these observations.

    The organization of the paper is outlined below: In Section 2, we transform the system (1.1) into an augmented system by integrating the wave equations with compatible diffusion equations. In Section 3, we derive global solutions to the problem (1.1) using arguments based on semigroup theory. In Section 4, we establish the strong stability of our system, despite the lack of resolvent compactness, by applying general criteria from Arendt and Batty. In Section 5, we propose a numerical scheme capable of replicating various decay rate profiles and confirm numerically the known stability outcomes related to the energy decay through several examples.

    This section focuses on transforming the model (1.1) into an augmented system. The following theorem is required:

    Theorem 2.1. [4] Consider μ be the function given by

    μ(ξ)=|ξ|(2α1)/2,ξR,0<α<1.

    Then the relationship between the "input" U and the "output" O of the system

    tφ(ξ,t)+(|ξ|2+η)φ(ξ,t)U(t)μ(ξ)=0,ξR,η0,t>0,
    φ(ξ,0)=0,

    and

    O(t)=sin(απ)πRμ(ξ)φ(ξ,t)dξ,

    can be described as follows:

    O=I1α,ηU=Dα,ηU,

    where

    [Iα,ηU](t)=1Γ(α)t0(tτ)α1eη(tτ)U(τ)dτ.

    Lemma 2.1. [4] If λD={λC/λ+η>0}{λ0}, then

    F1(λ):=+μ2(ξ)λ+η+ξ2dξ=πsinαπ(λ+η)α1,

    and

    F2(λ):=+μ2(ξ)(λ+η+ξ2)2dξ=(1α)πsinαπ(λ+η)α2.

    Based on the previous theorem, the system (1.1) can be expressed as the following augmented model:

    {φtt(x,t)+φxxxx+β(φψ)(x,t)=0,(x,t)(0,L)×(0,+),ψtt(x,t)+ψxxxx+β(ψφ)(x,t)=0,(x,t)(0,L)×(0,+),tϕ(ξ,t)+(ξ2+η)ϕ(ξ,t)φt(L,t)μ(ξ)=0,(ξ,t)R×(0,+),φxx(L,t)=ψxx(L,t)=ψxxx(L,t)=0,t(0,+),φ(0,t)=ψ(0,t)=φx(0,t)=ψx(0,t)=0,t(0,+),φxxx(L,t)=ζ+μ(ξ)ϕ(ξ,t)dξ,t(0,+), (2.1)

    where ζ=γπ1sin(απ). The energy associated with the solution to the system (2.1) is expressed by

    E(t)=12(φt22+φxx22+ψt22+ψxx22+βψφ22+ζϕ22), (2.2)

    where 2 refers to the L2(0,L) norm.

    Then, we have the following lemma:

    Lemma 2.2. Given that (φ,ψ,ϕ) is a solution to the problem (2.1), the energy functional defined by (2.2) fulfills

    ddtE(t)=ζ+(ξ2+η)ϕ2(ξ,t)dξ0. (2.3)

    Proof. Multiplying Eqs (2.1)1 and (2.1)2 by φt and ψt, respectively, using integration by parts over (0,L) and Eq (2.1)6, and combining the two equations, we derive

    12ddt[φt22+φxx22+ψt22+ψxx22+βψφ22]+ζφt(L)+μ(ξ)ϕ(ξ,t)dξ=0. (2.4)

    Scaling Eq (2.1)3 by ζϕ and integrating over R yields

    ζ2ddtϕ22+ζ+(ξ2+η)ϕ2(ξ,t)dξζφt(L)+μ(ξ)ϕ(ξ,t)dξ=0. (2.5)

    By combining (2.4) and (2.5), we obtain (2.3). The proof of the lemma is thereby finished.

    Within this section, we establish the well-posedness of (2.1). For this purpose, we define the following Hilbert space (the energy space):

    H=(H2L(0,L)×L2(0,L))2×L2(,+),

    where

    H2L(0,L)={uH2(0,L):u(0)=ux(0)=0}.

    For U=(φ,u,ψ,v,ϕ)T and ¯U=(¯φ,¯u,¯ψ,¯v,¯ϕ)T, we introduce the inner product in the space H as follows:

    U,¯UH=L0(u¯u+φxx¯φxx)dx+L0(v¯v+ψxx¯ψxx)dx+ζ+ϕ¯ϕdξ.

    We then reformulate (2.1) in the framework of semigroup theory: By introducing the vector function U=(φ,u,ψ,v,ϕ)T, the system (2.1) can be expressed as

    {U=AU,t>0,U(0)=U0,

    where U0:=(φ0,φ1,ψ0,ψ1,ϕ0)T. The operator A is linear and specified as follows:

    A(φuψvϕ)=(uφxxxxβ(ψφ)vψxxxxβ(φψ)(ξ2+η)ϕ(ξ,t)+u(L,t)μ(ξ)).

    The domain of A is then

    D(A)={(φ,u,ψ,v,ϕ)TH:φ,ψH4(0,L)H2L(0,L),u,vH2L(0,L),|ξ|ϕL2(,+),(ξ2+η)ϕ+u(L)μ(ξ)L2(,+),φxx(L)=ψxx(L)=ψxxx(L)=0,φxxx(L)ζ+μ(ξ)ϕ(ξ)dξ=0}. (3.1)

    We state the following theorem on existence and uniqueness:

    Theorem 3.1. (1) If U0D(A), then there exists a unique, strong solution to the system (2.1):

    UC0(R+,D(A))C1(R+,H).

    (2) If U0H, then system (2.1) admits a unique weak solution UC0(R+,H).

    Proof. Initially, we prove that the operator A exhibits dissipative properties. For any UD(A), where U=(φ,u,ψ,v,ϕ)T, we find

    RAU,UH=ζ+(ξ2+η)|ϕ(ξ,t)|2dξ0. (3.2)

    Thus, A is dissipative.

    Next, we prove that the operator λIA is surjective for λ>0.

    Given F=(f1,f2,f3,f4,f5)H, we verify the existence of UD(A) such that

    {λφu=f1,λu+φxxxx+β(ψφ)=f2,λψv=f3,λv+ψxxxx+β(φψ)=f4,λϕ+(ξ2+η)ϕ(ξ,t)u(L,t)μ(ξ)=f5. (3.3)

    Then, from (3.3)1 and (3.3)3, we find that

    {u=λφf1,v=λψf3. (3.4)

    It is straightforward to see that uH2L(0,L). Furthermore, from (3.3)5, we can identify ϕ as

    ϕ=f5+u(L,t)μ(ξ)λ+ξ2+η. (3.5)

    By inserting (3.4)1 into (3.3)2 and (3.4)2 into (3.3)4, we obtain

    {λ2φ+φxxxx+β(ψφ)=λf1+f2,λ2ψ+ψxxxx+β(φψ)=λf3+f4. (3.6)

    Finding the solution to system (3.6) is the same as identifying φ,ψH4(0,L)H2L(0,L) such that

    {L0(λ2φ¯w+φxxxx¯w+β(ψφ)¯w)dx=L0(λf1+f2)¯wdx,L0(λ2ψ¯χ+ψxxxx¯χ+β(φψ)¯χ)dx=L0(λf3+f4)¯χdx, (3.7)

    for all w,χH2L(0,L). By utilizing Eqs (3.5) and (3.7) and then performing integration by parts, we arrive at

    L0(λ2φ¯w+λ2ψχ+φxx¯wxx+ψxx¯χxx+β(φψ)(¯w¯χ))dx+¯ζu(L)¯w(L)=L0(λf1+f2)¯w+L0(λf3+f4)¯χdxζ¯w(L)+μ(ξ)ξ2+η+λf5(ξ)dξ, (3.8)

    where ¯ζ=ζ+μ2(ξ)ξ2+η+λdξ. By applying (3.4) once more, we infer that

    {u(L)=λφ(L)f1(L),v(L)=λψ(L)f3(L). (3.9)

    Substituting (3.9) into (3.8), we arrive at

    L0(λ2φ¯w+λ2ψ¯χ+φxx¯wxx+ψxx¯χxx+β(φψ)(¯w¯χ))dx+¯ζλφ(L)¯w(L)=L0(λf1+f2)¯wdx+L0(λf3+f4)¯χdxζ¯w(L)+μ(ξ)ξ2+η+λf5(ξ)dξ+¯ζf1(L)¯w(L). (3.10)

    Consequently, the equivalent form of problem (3.10) is given by the following:

    a((φ,ψ),(w,χ))=L(w,χ), (3.11)

    where the sesquilinear form a:[H2L(0,L)×H2L(0,L)]2C and the antilinear form L:[H2L(0,L)]2C are expressed as

    a((φ,ψ),(w,χ))=L0(λ2φ¯w+λ2ψ¯χ+φxx¯wxx+ψxx¯χxx+β(φψ)(¯w¯χ))dx+¯ζλφ(L)¯w(L)

    and

    L(w,χ)=L0(λf1+f2)¯wdx+L0(λf3+f4)¯χdxζ¯w(L)+μ(ξ)ξ2+η+λf5(ξ)dξ+¯ζf1(L)¯w(L).

    One can readily verify that the bilinear form a is continuous and coercive, while L is continuous. Therefore, by applying the Lax-Milgram theorem, we deduce that for all w,χH2L(0,L), the problem (3.11) admits a unique solution (φ,ψ)H2L(0,L)×H2L(0,L). Furthermore, by invoking classical elliptic regularity, it follows from (3.10) that (φ,ψ)[H4(0,L)]2. As a result, the operator λIA is surjective for any λ>0. As a result, the proof of Theorem 3.1 is completed by applying the Hille-Yosida theorem.

    In this part, we make use of a general criterion of Arendt and Batty (see [14] or [15]), which states that a C0-semi-group of contractions eAt on a Banach space is strongly stable if A has no purely imaginary eigenvalues and σ(A)iR comprises only a countable set of elements. We express our primary result in the form of the following theorem.

    Theorem 4.1. [16] The C0-semigroup eAt is strongly stable in H; i.e., for all U0H, the solution of (2.1) satisfies

    limteAtU0H=0.

    To prove Theorem 4.1, we need the lemmas listed below:

    Lemma 4.1. A does not have eigenvalues on iR.

    Proof. We consider two cases: iλ=0 and iλ0.

    Case 1. Solving AU=0 under the boundary conditions given in (3.1) leads to the conclusion that U=0. Therefore, iλ=0 is not an eigenvalue of the operator A.

    Case 2. We proceed by contradiction. Assume there exists λR,λ0, and U=(φ,u,ψ,v,ϕ1,ϕ2)0, for which AU=iλU. This gives the following system:

    {iλφu=0,iλu+φxxxx+β(ψφ)=0,iλψv=0,iλv+ψxxxx+β(φψ)=0,iλϕ+(ξ2+η)ϕ(ξ,t)u(L,t)μ(ξ)=0. (4.1)

    From (3.2), we deduce that ϕ0. Consequently, from (4.1)5, we derive u(L)=0.

    Consequently, from (4.1)1 and (3.1), we obtain

    φ(L)=ψ(L)=φxxx(L)=ψxxx(L)=0.

    Inserting (4.1)1 into (4.1)3 and (4.1)2 into (4.1)4, we obtain

    {λ2φ+φxxxx+β(ψφ)=0,λ2ψ+ψxxxx+β(φψ)=0.

    Then, Φ=φ+ψ and Ψ=φψ satisfy

    {λ2Φ+Φxxxx=0,(λ2+2β)Ψ+Ψxxxx=0. (4.2)

    By applying Lemma 5.2 and Corollary 5.1 from [7], we obtain the following boundary conditions:

    Φ(L)=Φx(L)=Φxx(L)=Φxxx(L)=0. (4.3)

    Similarly, the same conditions hold for Ψ, specifically:

    Ψ(L)=Ψx(L)=Ψxx(L)=Ψxxx(L)=0. (4.4)

    Now, consider X=(Φ,Φx,Φxx,Φxxx,Ψ,Ψx,Ψxx,Ψxxx). We can rewrite (4.2)–(4.4) as the initial value problem:

    {ddxX=BX,X(L)=0, (4.5)

    where

    B=(010000100001λ200000000000010000100001λ2+2β000).

    By virtue of the Picard theorem for ordinary differential equations, the system (4.5) has a unique solution X=0. Hence, Φ=Ψ0. Therefore, φ=0 and ψ=0, i.e., U=0. As a result, A does not have purely imaginary eigenvalues.

    Lemma 4.2. For λ0 or λ=0 and η0, the operator iλIA is surjective.

    Proof. We distinguish the following cases:

    Case 1: λ0.

    Our goal is to prove that the operator iλIA is surjective for λ0. To this end, let F=(f1,f2,f3,f4,f5)H. We aim to identify X=(u,φ,v,ψ,ϕ)D(A) such that the following equation holds:

    (iλIA)X=F.

    Alternatively,

    {iλφu=f1,iλu+φxxxx+β(ψφ)=f2,iλψv=f3,iλv+ψxxxx+β(φψ)=f4,iλϕ+(ξ2+η)ϕ(ξ,t)u(L,t)μ(ξ)=f5. (4.6)

    Inserting (4.6)1 into (4.6)2 and (4.6)3 into (4.6)4, we obtain

    {λ2φ+φxxxx+β(ψφ)=f2+iλf1,λ2ψ+ψxxxx+β(φψ)=f4+iλf3. (4.7)

    Finding a solution to system (4.7) is the same as finding φ,ψH4(0,L)H2L(0,L) such that

    {L0(λ2φ¯w+φxxxx¯w+β(ψφ)¯w)dx=L0(iλf1+f2)¯wdx,L0(λ2ψ¯χ+ψxxxx¯χ+β(φψ)¯χ)dx=L0(iλf3+f4)¯χdx, (4.8)

    for all w,χH2L(0,L). Using (4.8) and (3.5), followed by integration by parts, yields

    L0(λ2φ¯wλ2ψ¯χ+φxx¯wxx+ψxx¯χxxβ(φψ)(¯w¯χ))dx+¯ζλφ(L)¯w(L)=L0(iλf1+f2)¯w+L0(iλf3+f4)¯χdxζ¯w(L)+μ(ξ)ξ2+η+iλf5(ξ)dξ+¯ζf1(L)¯w(L), (4.9)

    where ¯ζ=ζ+μ2(ξ)ξ2+η+λdξ. Consequently, problem (4.9) can be reduced to the problem

    a((φ,ψ),(w,χ))=L(w,χ), (4.10)

    where the sesquilinear form a:[H2L(0,L)×H2L(0,L)]2C and the antilinear form L:[H2L(0,L)]2C are given by

    a((φ,ψ),(w,χ))=L0(λ2φ¯wλ2ψ¯χ+φxx¯wxx+ψxx¯χxxβ(φψ)(¯w¯χ))dx

    and

    L(w,χ)=L0(iλf1+f2)¯w+L0(iλf3+f4)¯χdxζ¯w(L)+μ(ξ)ξ2+η+iλf5(ξ)dξ+¯ζf1(L)¯w(L),

    where

    H2L(0,L)={(φ,ψ)H10(0,L)×H2(0,L)/φxxx(L,t)=γπ1sin(απ)+μ(ξ)ϕ(ξ)dξ,andψxxx(L,t)=0}.

    It is straightforward to confirm that the bilinear form a is continuous and coercive and that the functional L is continuous. By implementing the Lax-Milgram theorem, we gather that for all w,χH2L(0,L), the problem (4.10) admits a unique solution (φ,ψ)H2L(0,L)×H2L(0,L). Furthermore, using classical elliptic regularity, it results from (4.9) that (φ,ψ)[H4(0,L)]2. Accordingly, the operator λIA is surjective for any λ>0. Hence, by invoking the Hille–Yosida theorem, we obtain the desired result.

    Case 2: λ=0 and η0.

    The problem in system (4.6) can be expressed as

    {u=f1,φxxxx+β(ψφ)=f2,v=f3,ψxxxx+β(φψ)=f4,(ξ2+η)ϕ(ξ,t)u(L,t)μ(ξ)=f5.

    Then, from (4.9), we obtain

    L0(φxx¯wxx+ψxx¯χxxβ(φψ)(¯w¯χ))dx+¯ζλφ(L)¯w(L)=L0f2¯wdx+L0f4¯χdxζ¯w(L)+μ(ξ)ξ2+ηf5(ξ)dξ+¯ζf1(L)¯w(L), (4.11)

    for all w,χH2L(0,L).

    Thus, the system (4.11) can be expressed as the problem:

    aη((u,v),(χ,ζ))=Lη(χ,ζ), (4.12)

    where the sesquilinear form aη:[H2L(0,L)×H2L(0,L)]2C and the antilinear form Lη:[H2L(0,L)]2C are given by

    aη((φ,ψ),(w,χ))=L0(φxx¯wxx+ψxx¯χxxβ(φψ)(¯w¯χ))dx

    and

    Lη(w,χ)=L0f2¯w+L0f4¯χdxζ¯w(L)+μ(ξ)ξ2+ηf5(ξ)dξ+¯ζf1(L)¯w(L).

    It is straightforward to confirm that aη is continuous and coercive, and Lη is continuous. Then, by Lax-Milgram's theorem, the variational problem (4.12) admits a unique solution (φ,ψ)H2L(0,L)×H2L(0,L). We then deduce from (4.11) that (φ,ψ)[H4(0,L)]2. Hence, the operator A is surjective.

    The proof of the Lemma is thus complete.

    Proof of Theorem 4.1. According to Lemma 4.1, the operator A lacks purely imaginary eigenvalues. Additionally, Lemma 4.2 establishes that R(iλA)=H for all λR and R(iλA)=H for λ=0 when η>0. Consequently, by the closed graph theorem of Banach, we conclude that σ(A)iR= for η>0, and σ(A)iR={0} when η=0. This completes the proof.

    We will start making use of finite difference methods (FDM) to derive a discrete representation of the solution of (2.1). More precisely, we use the classical finite difference discretization for the time variable and the implicit compact finite difference method of fourth-order for discretization of the space variable. The comparison between different finite difference methods (implicit, explicit, and semi-implicit) will be considered in a subsequent work.

    Consider the discrete domain Ωh of Ω=[0,L], with a uniform grid given by xi=iΔx,i=0,1,,M,(M5) (see Figure 1). The time discretization of the interval I=[0,T] is expressed as tj=jΔt,j=0,1,,N; Δt=tjtj1=TN, where M and N are positive integers. Denote by φ(xi;tj)=φji and ψ(xi;tj)=ψji the value of the functions φ and ψ evaluated at the point xi and the instant tj.

    Now, we define the following approximation of the derivatives of φ and ψ, respectively:

    φtt(xi,tj+1)φj+1i2φji+φj1iΔt2, (5.1)
    ψtt(xi,tj+1)ψj+1i2ψji+ψj1iΔt2, (5.2)
    φxxxx(xi,tj+1)φj+1i24φj+1i1+6φj+1i4φj+1i+1+φj+1i+2Δx4, (5.3)

    and

    ψxxxx(xi,tj+1)ψj+1i24ψj+1i1+6ψj+1i4ψj+1i+1+ψj+1i+2Δx4. (5.4)

    Then, using (5.1)–(5.4), we obtain the discrete formulation of the initial condition of the system (1.1) as follows:

    {φj+1i2φji+φj1iΔt2+φj+1i24φj+1i1+6φj+1i4φj+1i+1+φj+1i+2Δx4+β(φj+1iψj+1i)=0,ψj+1i2ψji+ψj1iΔt2+ψj+1i24ψj+1i1+6ψj+1i4ψj+1i+1+ψj+1i+2Δx4+β(ψj+1iφj+1i)=0, (5.5)

    for j=¯1,N1, and i=¯2,M2. The discrete formulation of the initial conditions (1.3) reads as follows:

    φ0i=φ0(xi),ψ0i=ψ0(xi),φ1iφ0iΔt=φ1(xi),ψ1iψ0iΔt=ψ1(xi), (5.6)

    for i=¯0,M. Next, we seek the discrete formulation of the boundary conditions.

    Figure 1.  Mesh of the domain [0,L]×[0,T] with black bullet points at each (xi,tj).

    Lemma 5.1. The discrete formulation of the boundary conditions (1.2)3 reads as follows:

    φj+1M3φj+1M1+3φj+1M2φj+1M3Δx3=ηα1γΔtΓ(1α)jk=0(φk+1MφkM)(Γ(1α,η(tj+1tk))Γ(1α,η(tj+1tk+1))). (5.7)

    Proof. For j=¯1,N1, we have φxxx(L,tj+1)=γα,ηtφ(L,tj+1). By using finite differences, we arrive at

    φj+1M3φj+1M1+3φj+1M2φj+1M3Δx3=γΓ(1α)jk=0tk+1tk(tj+1r)αeη(tj+1r)φk+1MφkMΔtdr=γΔtΓ(1α)jk=0(φk+1MφkM)tk+1tk(tj+1r)αeη(tj+1r)dr. (5.8)

    On the other hand, using the change of variable u=η(tj+1r), we obtain

    tk+1tk(tj+1r)αeη(tj+1r)dr=ηα1η(tj+1tk)η(tj+1tk+1)u1(1+α)eudu.=ηα1(Γ((1α,η(tj+1tk))Γ(1α,η(tj+1tk+1))). (5.9)

    Inserting (5.9) in (5.8), we obtain (5.7).

    Hence, the discrete formulation of boundary condition in the system (1.1) follows as

    {φj0=0,ψj0=0,φj+11φj+10Δx=0,ψj+11ψj+10Δx=0,φj+1M2φj+1M1+φj+1M2Δx2=0,ψj+1M2ψj+1M1+ψj+1M2Δx2=0,φj+1M3φj+1M1+3φj+1M2φj+1M3Δx3=ηα1γΔtΓ(1α)jk=0(φk+1MφkM)(Γ(1α,η(tj+1tk))Γ(1α,η(tj+1tk+1))),ψj+1M3ψj+1M1+3ψj+1M2ψj+1M3Δx3=0.

    In the following, we reformulate the system (5.5)-(5.6) in abstract form as follows:

    K×W=F,

    where we denoted by W=(φj+10,φj+11,,φj+1M,ψj+10,ψj+11,,ψj+1M)t,

    F=(φj+10,φj+11,2φj2φj12Δt2,,2φjM2φj1M2Δt2,0,λ,ψj+10,ψj+11,2ψj2ψj12Δt2,,2ψjM2ψj1M2Δt2,0,0)t

    with

    λ=ηα1γΔtΓ(1α)j1k=0(φk+1MφkM)(Γ(1α,η(tj+1tk))Γ(1α,η(tj+1tk+1)))ηα1φjMγΓ(1α)Δt(Γ(1α,ηΔt))

    and K=(A+CBBA)M2M(R), where A is an M×M matrix:

    A=[100000000010000000c14c1c24c1c100000c14c1c24c1c100000c14c1c24c1c1000000000c14c1c24c1c1000000c32c3c300000c43c43c4c4]

    with c1=1Δx4,c2=β+1Δt2+6c1,c3=1Δx2,c4=1Δx3 and the matrices B and C are given as follows:

    Bij={β,if i=j and 3iM2,0,otherwise,
    Cij={ηα1γΓ(1α)Δt(Γ(1α,ηΔt)),if i=j=M,0,otherwise.

    Algorithm 1 provides a summary of all the steps for the calculation of φj and ψj using matrix decomposition techniques:

    Algorithm 1. Compute φj,ψj from [A+CBBA][φjψj]=[f1f2]
    Require: Matrices A, C, B, {φ0,φ1}, {ψ0,ψ1}, vectors f1, f2
    Ensure: φj+1, ψj+1
        for j=1 to N1 do
            Update f1 and f2 for the current iteration j+1
            φj+1=(A+CBA1B)1(f1BA1f2)
            ψj+1=(AB(A+C)1B)1(f2B(A+C)1f1)
        end for

    Recalling that the energy E(t) is expressed as

    E(t)=12(φt22+φxx22+ψt22+ψxx22+βψφ22).

    Now, we define the following approximations of the derivatives of φ and ψ, respectively:

    φtφj+1iφjiΔt,ψtψj+1iψjiΔt,φxxφji+12φji+φji1Δx2, and ψxxψji+12ψji+ψji1Δx2.

    The L2 norm of a discretized function is approximated by

    f22ΔxNi=0f(xi,tj)2. (5.10)

    Thus, the discrete energy of the system (1.1)-(1.2) at time tj is as follows:

    E(tj)12ΔxMi=0((φj+1iφjiΔt)2+(φji+12φji+φji1Δx2)2+(ψj+1iψjiΔt)2+(ψji+12ψji+ψji1Δx2)2+β(φjiψji)2).

    Algorithm 2 summarizes all the steps for calculating of the discrete Energy E(t) using the L2 norm define by (5.10).

    Algorithm 2. Calculation of the discrete energy E(tj)
    Require: {φj}, {ψj} for each j, Δt, Δx, β
    Ensure: Ej (calculated energy)
        for j=0 to N1 do
            Compute time derivatives
            φj+1tφj+1φjΔt
            ψj+1tψj+1ψjΔt
            Compute second spatial derivatives
            for i=1 to M1 do
                (φxx)j+1iφj+1i+12φj+1i+φj+1i1Δx2
                (ψxx)j+1iψj+1i+12ψj+1i+ψj+1i1Δx2
            end for
            Compute L2 norms at time step j
            norm_phi_tΔx(φj+1t)2
            norm_phi_xxΔx(φj+1xx)2
            norm_psi_tΔx(ψj+1t)2
            norm_psi_xxΔx(ψj+1xx)2
            norm_diffΔx((ψj+1φj+1)2)
            Compute energy
        Ej+10.5×(norm_phi_t+norm_phi_xx+norm_psi_t+norm_psi_xx+β×norm_diff)
    end for

    To evaluate the asymptotic properties of the solutions to the system (1.1), we consider the following data: α=0.5, β=η=1, Δt=102,Δx=101, L=1, and the initial conditions:

    φ(x,0)=3x2(1x),φt(x,0)=0,ψ(x,0)=x(1x)2,ψt(x,0)=0,x(0,L).

    Figures 24 illustrate the decay of the discrete energy for the chosen initial data. Specifically, Figure 3 demonstrates that when T is sufficiently large, the decay becomes more rapid. Figure 5 demonstrates the exponential decay of the discrete energy for the chosen initial data, as well as for other initial conditions. The conjecture of exponential decay arises from the observation that the plot of the logarithm of the discrete energy forms a straight line with a negative slope, indicating an exponential decay pattern. This behavior has been consistently observed for various initial data, further reinforcing the conjecture. This observation also poses an open question regarding the exponential decay of the energy in the system (1.1).

    Figure 2.  Discrete Energy decay for Δt=102,Δx=101, T=1, L=1.
    Figure 3.  Discrete Energy decay for Δt=102,Δx=101, T=10, L=1.
    Figure 4.  Log Discrete Energy decay for Δt=102,Δx=101, T=10, L=1.
    Figure 5.  Log Discrete Energy decay for Δt=102,Δx=101, T=100, L=1.

    Figures 6 and 7 illustrate the exponential decay of the discrete energy for the chosen initial data and for other initial conditions as well, for different values of α: α=0.75 and α=0.95. We observe a slight increase in the decay rate as α approaches 1.

    Figure 6.  Log Discrete Energy decay for Δt=102,Δx=101, T=100, L=1, and α=0,75.
    Figure 7.  Log Discrete Energy decay for Δt=102,Δx=101, T=100, L=1, and α=0,95.

    Now, we provide an example to test the correctness and effectiveness of the numerical scheme. This example includes a comparison with the exact solution to demonstrate the scheme's convergence rate and validate its accuracy. These results ensure the reliability of the analysis and conclusions presented in Section 5.2.

    We consider the exact solutions to the problem as follows:

    φ(x,t)=(x1)4t3andψ(x,t)=xet.

    The problem is governed by the equations:

    {φtt(x,t)+φxxxx(x,t)+(φψ)(x,t)=f1(x,t),(x,t)(0,1)×(0,+),ψtt(x,t)+ψxxxx(x,t)+(ψφ)(x,t)=f2(x,t),(x,t)(0,1)×(0,+),

    with the following boundary conditions:

    {φ(0,t)=t3,ψ(0,t)=0,φx(0,t)=4t3,ψx(0,t)=et,fort(0,+),φxx(1,t)=ψxx(1,t)=ψxxx(1,t)=0,fort(0,+),φxxx(1,t)=0,fort(0,+),

    and the initial conditions:

    {φ(x,0)=0,φt(x,0)=0,x(0,1),ψ(x,0)=x,ψt(x,0)=x,x(0,1).

    The source terms are given by

    f1(x,t)=t3(x1)4+24t3+6t(x1)4xetandf2(x,t)=t3(x1)4+2xet.

    Figures 8 and 9 show the comparison between the exact solutions and the numerical solutions for φ and ψ.

    Figure 8.  Numerical solution.
    Figure 9.  Exact solution.

    Now, we define the root mean square error (RMSE) as

    RMSE=1Nmi=0nj=0(ujiˆuji)2, (5.11)

    where u and ˆu denote the exact and the numerical solutions, respectively.

    The data presented in Table 1 below were obtained by comparing the exact solutions φ and ψ with their respective numerical approximations ˆφ and ˆψ using different discretization values of Δx and Δt.

    Table 1.  The rate of convergence for different value of Δx and Δt.
    Δx Δt RMSEφ RMSEψ
    0.2 0.1 0.066421 0.000768
    0.1 0.01 0.026582 0.000373
    0.1 0.001 0.026449 0.000434

     | Show Table
    DownLoad: CSV

    Finally, to present the rate of convergence, we begin by defining the pointwise rate of convergence, which corresponds to the opposite of the slope of the logarithmic energy curve. This calculation is performed for the example provided at the beginning of Section 5.2.

    Define the pointwise rate of convergence as

    λj=ln(E(tj+1))ln(E(tj))Δt,for j=0,,N1,

    and the rate of convergence as

    λ1N1N1i=1λi.

    In Figure 10, we observe the behavior of the decay rate over time. We note that the curve becomes horizontal, indicating that the rate is constant, which supports the notion that the energy decay is exponential. In Table 2, we provide the values of the energy decay rate for different values of Δx and Δt.

    Figure 10.  The pointwise rate of convergence for Δt=102,Δx=101, T=100, L=1.
    Table 2.  The rate of convergence for different value of Δx and Δt.
    Δx Δt λ
    0.1 0.1 2.302732
    0.1 0.01 2.425248
    0.01 0.01 2.382123
    0.01 0.001 2.412021

     | Show Table
    DownLoad: CSV

    In the present work, we studied the strong stabilization of a coupled Euler-Bernoulli beam system with one boundary dissipation of fractional derivative type. First, we analyzed the strong stability of the system (1.1)-(1.2). Next, we employed a finite difference scheme to compute the numerical solutions and demonstrated the stability of the discrete energy. Although we achieved exponential energy decay in one example, the question of exponential energy decay remains open.

    In advancing the finite difference methods (FDM) for solving complex fractional and coupled PDEs, this manuscript makes a considerable impact on numerical analysis and applied mathematics, offering a significant resource for researchers and practitioners working on advanced modeling challenges.

    Boumediene Boukhari, Ahmed Bchatnia and Abderrahmane Beniani: Visualization; Foued Mtiri: Investigation; Ahmed Bchatnia: Supervision, Validation; Ahmed Bchatnia and Abderrahmane Beniani: Conceptualization, Methodology; Boumediene Boukhari, Foued Mtiri and Abderrahmane Beniani: Resources; Boumediene Boukhari, Foued Mtiri, Ahmed Bchatnia and Abderrahmane Beniani: Writing–original draft. All authors have read and approved the final version of the manuscript for publication.

    The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khaled University for funding this work through Large Research Project under grant number RGP2/368/45.

    The authors declare that there are no conflicts of interest in this paper.



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