Exploration of nonlinear traveling wave phenomena in quintic conformable Benney-Lin equation within a liquid film

Noorah Mshary; Noorah Mshary

doi:10.3934/math.2024542

AIMS Mathematics

2024, Volume 9, Issue 5: 11051-11075. doi: 10.3934/math.2024542

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Exploration of nonlinear traveling wave phenomena in quintic conformable Benney-Lin equation within a liquid film

Noorah Mshary ^,

Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Saudi Arabia

Received: 20 January 2024 Revised: 29 February 2024 Accepted: 04 March 2024 Published: 20 March 2024
MSC : 33B15, 34A34, 35A20, 35A22, 44A10

In this article, we use the modified extended direct algebraic method (mEDAM) to explore and analyze the traveling wave phenomena embedded in the quintic conformable Benney-Lin equation (CBLE) that regulates liquid film dynamics. The proposed transformation-based approach developed for nonlinear partial differential equations (PDEs) and fractional PDEs (FPDEs), efficiently produces a plethora of traveling wave solutions for the targeted CBLE, capturing the system's nuanced dynamics. The methodically determined traveling wave solutions are in the form of rational, exponential, hyperbolic and trigonometric functions which include periodic waves, bell-shaped kink waves and signal and double shock waves. To accurately depict the wave phenomena linked to these solutions, we generate 2D, 3D, and contour graphs. These visualizations not only improve understanding of the CBLE model's dynamics, but also provide a detailed way to examine its behavior. Moreover, the use of the proposed techniques contributes to a better understanding of the other FPDEs' distinct characteristics, enhancing our comprehension of their underpinning dynamics.

Keywords:

Citation: Noorah Mshary. Exploration of nonlinear traveling wave phenomena in quintic conformable Benney-Lin equation within a liquid film[J]. AIMS Mathematics, 2024, 9(5): 11051-11075. doi: 10.3934/math.2024542

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Abstract

1. Introduction

In structural mechanics, particularly in the field of elasticity, there are equations that describe the behavior of thin plates under loads. These equations are often partial differential equations that govern the displacement of a thin plate. The plate equation depends on factors like material properties, geometry, and boundary conditions. In the context of geophysics, "plate equation" could refer to the equations that describe the movement and interaction of tectonic plates on the Earth's surface. Plate tectonics is a theory that explains the movement of the Earth's lithosphere (the rigid outer layer of the Earth) on the more fluid asthenosphere beneath it. In mathematics, specifically in the field of differential equations, the term "plate equation" might be used to refer to certain types of equations. For instance, in polar coordinates, Laplace's equation takes a specific form that is sometimes informally referred to as the "plate equation". Plate equation models are hyperbolic systems that arise in several areas in real-life problems, (see, for instance, Kizilova et al.^[1], Lasiecka et al. ^[2] and Huang et al.^[3]). The theory of plates is the mathematical formulation of the mechanics of flat plates. It is defined as flat structural components with a low thickness compared to plane dimensions. The advantage of the theory of plates comes from the disparity of the length scale to reduce the problem of the mechanics of three-dimensional solids to a two-dimensional problem. The purpose of this theory is to compute the stresses and deformation in a loaded plate. The equation of plates results from the composition of different subsets of plates: The equilibrium equations, constitutive, kinematic, and force resultant, ^[4,5,6].

Following this, there are a wide number of works devoted to the analysis and control of the academic model of hyperbolic systems, the so-called plate equations, For example, the exact and the approximate controllability of thermoelastic plates given by Eller et al. ^[7] and Lagnese and Lions in ^[8] treated the control of thin plates and Lasiecka in ^[9] considered the controllability of the Kirchoff plate. Zuazua ^[10] treated the exact controllability for semi-linear wave equations. Recently many problems involving a plate equations were considered by researchers. Let us cite as examples the stabilization of the damped plate equation under general boundary conditions by Rousseau an Zongo ^[11]; the null controllability for a structurally damped stochastic plate equation studied by Zhao ^[12], Huang et al. ^[13] considerrd a thermal equation of state for zoisite: Implications for the transportation of water into the upper mantle and the high-velocity anomaly in the Farallon plate. Kaplunov et al. ^[14] discussed the asymptotic derivation of 2D dynamic equations of motion for transversely inhomogeneous elastic plates. Hyperbolic systems have recently continued to be of interest to researchers and many results have been obtained. We mention here the work of Fu et al. ^[15] which discusses a class of mixed hyperbolic systems using iterative learning control. Otherwise, for a class of one-dimension linear wave equations, Hamidaoui et al. stated in ^[16] an iterative learning control. Without forgetting that for a class of second-order nonlinear systems Tao et al. proposed an adaptive control based on an disturbance observer in ^[17] to improve the tracking performance and compensation. In addition to these works, the optimal control of the Kirchoff plate using bilinear control was considered by Bradly and Lenhart in ^[18], and Bradly et al. in ^[19]. In fact, in this work we will talk about a bilinear plate equation and we must cite the paper of Zine ^[20] which considers a bilinear hyperbolic system using the Riccati equation. Zine and Ould Sidi ^[19,22] that introduced the notion of partial optimal control of bilinear hyperbolic systems. Li et al. ^[23] give an iterative method for a class of bilinear systems. Liu, et al. ^[24] extended a gradient-based iterative algorithm for bilinear state-space systems with moving average noises by using the filtering technique. Furthermore, flow analysis of hyperbolic systems refers to the problems dealing with the analysis of the flow state on the system domain. We can refer to the work of Benhadid et al. on the flow observability of linear and semilinear systems ^[25], Bourray et al. on treating the controllability flow of linear hyperbolic systems ^[26] and the flow optimal control of bilinear parabolic systems are considered by Ould Sidi and Ould Beinane on ^[27,28].

For the motivation the results proposed in this paper open a wide range of applications. We cite the problem of iterative identification methods for plate bilinear systems ^[23], as well as the problem of the extended flow-based iterative algorithm for a plate systems ^[24].

This paper studies the optimal control problem governed by an infinite dimensional bilinear plate equation. The objective is to command the flow state of the bilinear plate equation to the desired flow using different types of bounded feedback. We show how one can transfer the flow of a plate equation close to the desired profile using optimization techniques and adjoint problems. As an application, we solve the partial flow control problem governed by a plate equation. The results open a wide way of applications in fractional systems. We began in section two by the well-posedness of our problem. In section three, we prove the existence of an optimal control solution of (2.3). In section four, we state the characterization of the optimal control. In section five we debate the case of time bilinear optimal control. Section six, proposes a method for solving the flow partial optimal control problem governed by a plate equation.

2. Well-posedness of the problem

Consider $\Theta$ an open bounded domain of $I\!\!R^2$ with $C^2$ boundary, for a time $m$ , and $\Gamma = \partial\Theta \times (0, m)$ . The control space time set is such that

$\begin{equation} Q \in U_{p} = \{ Q\in L^{\infty}([0,m];L^{\infty}(\Theta)){\mbox{ such that }} -p\leq Q(t) \leq p\}, \end{equation}$

(2.1)

with $p$ as a positive constant. Let the plate bilinear equation be described by the following system

$\begin{equation} \left \{ \begin{array}{lll} {\frac{{\partial }^2 {u} }{{\partial t}^2} } + \Delta^{2} u = Q(t)u_{t}, & & (0,m) \times \Theta, \\ u(x,0) = u_0(x) ,\; {\frac{{\partial } u }{{\partial t}} } (x,0) = u_1(x), & & \Theta,\\ u = \frac{\partial u}{\partial \nu} = 0, & & \Gamma, \end{array} \right. \end{equation}$

(2.2)

where $u_{t} = \frac{\partial u}{\partial t}$ is the velocity. The state space is $H_{0}^{2}(\Theta)\times L^{2}(\Theta)$ , (see Lions and Magenes ^[29] and Brezis ^[30]). We deduce the existence and uniqueness of the solution for (2.2) using the classical results of Pazy ^[31]. For $\lambda > 0$ , we define $\nabla u$ as the flow control problem governed by the bilinear plate equation (2.2) as the following:

$\begin{equation} \min\limits_{Q\ \in U_{p} }C_{\lambda}(Q), \end{equation}$

(2.3)

where $C_{\lambda}$ , is the flow penalizing cost defined by

$\begin{align} C_{\lambda}(Q) = \frac{1}{2}\Big\|\nabla u-u^{d}\Big\|^{2}_{(L^2(0,m;L^2(\Theta)))^{n}} + \frac{\lambda}{2}\int_{0}^{m} \int_{\Theta} Q^2(x,t)dxdt\\ = \frac{1}{2}\sum\limits_{i = 1}^{n} \Big\|\frac{\partial u}{\partial x_{i}} - u_{i}^{d}\Big\|^{2}_{L^2(0,m;L^2(\Theta))} + \frac{\lambda}{2}\int_{0}^{m} \int_{\Theta} Q^2(x,t)dxdt, \end{align}$

(2.4)

where $u^d = (u_{1}^{d}, ....u_{n}^{d})$ is the flow target in $L^2(0, m;L^2(\Theta))$ . One of the important motivations when considering the problem (2.3) is the isolation problems, where the control is maintained to reduce the flow temperature on the surface of a thin plate (see El Jai et al. ^[32]).

3. Existence of solution

Lemma 3.1. If $(u_{0}, u_{1}) \in H_{0}^{2}(\Theta)\times L^{2}(\Theta)$ and $Q \in U_{p}$ , then the solution $(u, u_{t})$ of (2.2) satisfies the following estimate:

$\Big\| (u, u_{t}) \Big\|_{{\cal C}(0,m; H_{0}^{2}(\Theta)\times L^{2}(\Theta))} \leq T (1+ \eta m)e^{\eta K m},$

where $T = \Big\| (u_{0}, u_{1}) \Big\|_{ H_{0}^{2}(\Theta)\times L^{2}(\Theta)}$ and $K$ is a positive constant ^[18,19].

Using the above Lemma 3.1, we prove the existence of an optimal control solution of (2.3).

Theorem 3.1. $(u^{*}, Q^*)\in {\cal C}([0, m];\; \; H_{0}^{2}(\Theta)\times U_{p})$ , is the solution of (2.3), where $u^{*}$ is the output of (2.2) and $Q^*$ is the optimal control function.

Proof. Consider the minimizing sequence $(Q_{n})_{n}$ in $U_{p}$ verifying

$C^{*} = \lim\limits_{n\rightarrow +\infty}C_{\lambda}(Q_n) = \inf\limits_{Q\in L^{\infty}(0,m; L^{\infty}(\Theta))} C_{\lambda}(Q).$

We choose $\bar{u}^{n} = (u^{n}, \frac{\partial u^{n}}{\partial t})$ to be the corresponding state of Eq (2.2). Using Lemma 3.1, we deduce

$\begin{equation} \Big\| u^{n}(x,t) \Big\|_{H_{0}^{2}(\Theta)}^{2} + \Big\| u_{t}^{n} (x,t) \Big\|_{ L^{2}(\Theta)}^{2} \leq T_{1} e^{\eta K m} {\mbox{ for }} 0\leq t\leq m {\mbox{ and }} T_{1}\in I\!\!R^{+}. \end{equation}$

(3.1)

Furthermore, system (2.2) gives

$\Big\| u^{n}_{tt}(x,t) \Big\|_{H^{-2}(\Theta)}^{2} \leq T_{2} \Big\| u_{t}^{n} (x,t) \Big\|_{ L^{2}(\Theta)}^{2} {\mbox{ with }} T_{2}\in I\!\!R^{+}.$

Then easily from (3.1), we have

$\begin{equation} \Big\| u^{n}_{tt}(x,t) \Big\|_{H^{-2}(\Theta)}^{2} \leq T_{3} e^{\eta K m} {\mbox{ for }} 0\leq t\leq m {\mbox{ and }} T_{3}\in I\!\!R^{+}. \end{equation}$

(3.2)

Using (3.1) and (3.2), we have the following weak convergence:

$\begin{equation} \begin{array}{lll} Q_{n} \rightharpoonup Q^{*}, & & L^{2}(0,m ; L^{2}(\Theta)),\\ u^{n} \rightharpoonup u^{*}, & & L^{\infty}(0,m ; H_{0}^{2}(\Theta)),\\ u_{t}^{n} \rightharpoonup u^{*}_{t}, & & L^{\infty}(0,m ; L^{2}(\Theta)),\\ u_{tt}^{n} \rightharpoonup u^{*}_{tt}, & & L^{\infty}(0,m ; H^{-2}(\Theta)),\\ \end{array} \end{equation}$

(3.3)

From the first convergence property of (3.3) with a control sequence $Q_{n} \in U_{p}$ , easily one can deduce that $Q^{*} \in U_{p}$ ^[20].

In addition, the mild solution of (2.2) verifies

$\begin{align} \int_{0}^{m} u^{n}_{tt}f(t)dt+ \int_{0}^{m} \int_{\Theta}\Delta u^{n} \Delta f(t) dx dt = \int_{0}^{m} Q^{*}u_{t}^{n} f(t)dt, \forall f \in H_{0}^{2}(\Theta). \end{align}$

(3.4)

Using (3.3) and (3.4), we deduce that

$\begin{align} \int_{0}^{m} u^{*}_{tt}f(t)dt+ \int_{0}^{m} \int_{\Theta}\Delta u^{*} \Delta f(t) dx dt = \int_{0}^{m} Q^{*}u_{t}^{*} f(t)dt, \forall f \in H_{0}^{2}(\Theta), \end{align}$

(3.5)

which implies that $u^{*} = u(Q_{*})$ is the output of (2.2) with command function $Q^{*}$ .

Fatou's lemma and the lower semi-continuous property of the cost $C_{\lambda}$ show that

$\begin{align} C_{\lambda}(Q^{*}) \leq \frac{1}{2}\lim\limits_{k\rightarrow +\infty}\Big\|\nabla u^{k}-u^{d}\Big\|^{2}_{(L^2(0,m;L^2(\Theta)))^{n}} + \frac{\lambda}{2}\lim\limits_{k\rightarrow +\infty} \int_{0}^{m} \int_{\Theta} Q_{k}^2(x,t)dxdt\\ \leq \liminf\limits_{k\rightarrow +\infty} C_{\lambda}(Q_{n})\\ = \inf\limits_{Q \in U_{p}} C_{\lambda}(Q), \end{align}$

(3.6)

which allows us to conclude that $Q^{*}$ is the solution of problem (2.3). □

4. Characterization of solution

We devote this section to establish a characterization of solutions to the flow optimal control problem (2.3).

Let the system

$\begin{equation} \left \{ \begin{array}{lll} {\frac{{\partial^{2} } v }{{\partial t^{2}}} } = -\Delta^{2}v(x,t)+Q(x,t) v_{t} + d(x,t) v_{t}, & & (0, m)\times \Theta,\\ v(x,0) = v_{t}(x,0) = v_0(x) = 0, & & \Theta, \\ v = \frac{\partial v}{\partial \nu } = 0, & & \Gamma, \end{array} \right. \end{equation}$

(4.1)

with $d \in L^{\infty}(0, m; L^{\infty}(\Theta))$ verify $Q+\delta d \in U_{p}, \quad \forall \delta > 0$ is a small constant. The functional defined by $Q \in U_{p}\mapsto \bar{u}(Q) = (u, u_{t}) \in {\cal C}(0, m; H_{0}^{2}(\Theta)\times L^{2}(\Theta))$ is differentiable and its differential $\overline{v} = (v, v_{t})$ is the solution of (4.1) ^[21].

The next lemma characterizes the differential of our flow cost functional $C_{\lambda}(Q)$ with respect to the control function $Q$ .

Lemma 4.1. Let $Q \in U_{p}$ and the differential of $C_{\lambda}(Q)$ can be written as the following:

$\begin{align} \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k} = \sum\limits_{i = 1}^{n} \int _{\Theta}\int_{0}^{m} \frac{\partial v(x,t)}{\partial x_{i}}(\frac{\partial u}{\partial x_{i}} - u_{i}^{d} )dtdx+ \varepsilon\int_{\Theta}\int_{0}^{m} d Q dt dx.\\ \end{align}$

(4.2)

Proof. Consider the cost $C_{\lambda}(Q)$ defined by (2.4), which is

$\begin{align} C_{\lambda}(Q) = \frac{1}{2} \sum\limits_{i = 1}^{n}\int_{\Theta}\int_{0}^{m}(\frac{\partial u}{\partial x_{i}} - u_{i}^{d} )^2 dt dx + \frac{\lambda}{2} \int_{\Theta}\int^{m}_{0} Q^2(t) dt dx. \end{align}$

(4.3)

Put $u_{k} = z(Q+k d)$ , $u = u(Q)$ , and using (4.3), we have

$\begin{align} \lim\limits_{k \longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k} = \lim\limits_{\beta\longrightarrow 0}\sum\limits_{i = 1}^{n} \frac{1}{2}\int_{\Theta}\int_{0}^{m}\frac{( \frac{\partial u_{k}}{\partial x_{i}} - u_i^d)^{2}-( \frac{\partial u}{\partial x_{i}}-u_i^d)^{2}}{k}dt dx\\ + \lim\limits_{k \longrightarrow 0}\frac{\lambda}{2}\int_{\Theta} \int_{0}^{m}\frac{(Q+k d)^2-Q^2}{k}(t)dt dx. \end{align}$

(4.4)

Consequently

$\begin{align} \lim\limits_{k \longrightarrow 0} \frac{C_{\lambda}(Q +kd)-C_{\lambda}(Q )}{k} \\ = \lim\limits_{k \longrightarrow 0}\sum\limits_{i = 1}^{n}\frac{1}{2}\int_{\Theta}\int^{m}_{0}\frac{( \frac{\partial u_{k}}{\partial x_{i}}- \frac{\partial u}{\partial x_{i}})}{k}( \frac{\partial u_{k} }{\partial x_{i}}+ \frac{\partial u}{\partial x_{i}}-2u_i^d) dt dx \\ + \lim\limits_{k \longrightarrow 0} \int_{\Theta}\int_{0}^{m}(\lambda d Q+ k \lambda d^2)dt dx\\ = \sum\limits_{i = 1}^{n}\int_\Theta \int_{0}^{m}\frac{\partial v(x,t)}{\partial x_{i}}( \frac{\partial u(x,t)}{\partial x_{i}}-u_i^d)dt dx+\int_\Theta \int_{0}^{m}\lambda d Q dt dx. \end{align}$

(4.5)

□ We define the following family of adjoint equations for system (4.1)

$\begin{equation} \left \{ \begin{array}{lll} {\frac{{\partial^{2} } {w_{i}}}{{\partial t^{2}}} } +\Delta^{2} w_{i} = Q^*(x,t) (w_{i})_{t}+( \frac{\partial u}{\partial x_{i}} - u_{i}^{d}), & & (0, m)\times\Theta, \\ w_{i}(x,m) = (w_{i})_{t}(x,m) = 0, & & \Theta, \\ w_{i} = \frac{\partial w_{i}}{\partial \nu } = 0, & & \Gamma. \end{array} \right. \end{equation}$

(4.6)

Such systems allow us to characterize the optimal control solution of (2.3).

Theorem 4.1. Consider $Q \in U_{p}$ , and $u = u (Q)$ its corresponding state space solution of (2.2), then the control solution of (2.3) is

$\begin{equation} Q(x,t) = max(-p, min(\frac{-1}{\lambda}(u_{t})(\sum\limits_{i = 1}^{n}\frac{\partial w_{i}}{\partial x_{i}}),p)), \end{equation}$

(4.7)

where $w = (w_{1}....w_{n})$ with $w_{i} \in C ([ 0, T ]; H^{2}_{0}(\Theta))$ is the unique solution of (4.6).

Proof. Choose $d\in U_{p}$ such that $Q + k d \in U_{p}$ with $k > 0$ . The minimum of $C_{\lambda}$ is realized when the control $Q$ , verifies the following condition:

$\begin{equation} 0 \leq \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q )}{k}. \end{equation}$

(4.8)

Consequently, Lemma 4.1 gives

$\begin{align} 0 \leq \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k} \\ = \sum\limits_{i = 1}^{n}\int_\Theta \int_{0}^{m}\frac{\partial v(x,t)}{\partial x_{i}}( \frac{\partial u(x,t)}{\partial x_{i}}-u_i^d)dt dx+\int_\Theta \int_{0}^{m}\lambda d Q dt dx.\\\\ \end{align}$

(4.9)

Substitute by equation (4.6) and we find

$\begin{align} 0 \leq \sum\limits_{i = 1}^{n}\int_\Theta \int_{0}^{m}\frac{\partial v(x,t)}{\partial x_{i}}( {\frac{{\partial^{2} } {w_{i}}(x,t) }{{\partial t^{2}}} } +\Delta^{2} w_{i}(x,t)-Q(x,t)( w_{i})_{t}(x,t))dt dx+\int_\Theta \int_{0}^{m}\lambda d Q dt dx\\ \\ = \sum\limits_{i = 1}^{n}\int_\Theta \int_{0}^{m}\frac{\partial }{\partial x_{i}}( {\frac{{\partial^{2} } v}{{\partial t^{2}}} } +\Delta^{2} v-Q(x,t) v_{t})w_{i}(x,t)dt dx+\int_\Theta \int_{0}^{m}\lambda d Qdt dx\\ \\ = \sum\limits_{i = 1}^{n}\int_\Theta \int_{0}^{m}\frac{\partial }{\partial x_{i}}(d(x,t) u_{t})w_{i}dt dx+\int_\Theta \int_{0}^{m}\lambda dQ dt dx\\ \\ = \int_\Theta \int_{0}^{m}d(x,t)[u_{t}(\sum\limits_{i = 1}^{n}\frac{\partial w_{i}(x,t)}{\partial x_{i}})+\lambda Q dt dx].\\ \\ \end{align}$

(4.10)

It is known that if $d = d(t)$ in a chosen function with $Q+k d \in U_{p}$ , using Bang-Bang control properties, one can conclude that

$\begin{equation} Q(x,t) = max(-p, min(\frac{-u_{t}}{\lambda}(\sum\limits_{i = 1}^{n}\frac{\partial w_{i}}{\partial x_{i}}),p)) = max(-p, min( \frac{-u_{t}}{\lambda}Div(w), p)), \end{equation}$

(4.11)

with $Div(w) = \sum\limits_{i = 1}^{n}\frac{\partial w_{i}}{\partial x_{i}}$ . □

5. Flow time bilinear control problem

Now, we are able to discuss the case of bilinear time control of the type $Q = Q(t)$ . We want to reach a flow spatial state target prescribed on the whole domain $\Theta$ at a fixed time $m$ .

In such case, the set of controls (2.1) becomes

$\begin{equation} Q \in U_{p} = \{ Q\in L^{\infty}([0,m]){\mbox{ such that }} -p\leq Q(t) \leq p {\mbox{ for }} t \in (0, m)\}, \end{equation}$

(5.1)

with $p$ as a positive constant.

The cost to minimize is

$\begin{align} C_{\lambda}(Q) = \frac{1}{2}\Big\|\nabla u(x,m)-u^{d}\Big\|^{2}_{(L^2(\Theta))^{n}} + \frac{\lambda}{2}\int_{0}^{m} Q^2(t) dt\\ = \frac{1}{2}\sum\limits_{i = 1}^{n} \Big\|\frac{\partial u}{\partial x_{i}}(x,m) - u_{i}^{d}\Big\|^{2}_{L^2(\Theta)} + \frac{\lambda}{2}\int_{0}^{m} Q^2(t)dt, \end{align}$

(5.2)

where $u^d = (u_{1}^{d}, ....u_{n}^{d})$ is the flow spatial target in $L^2(\Theta)$ . The flow control problem is

$\begin{equation} \min\limits_{Q\ \in U_{p} }C_{\lambda}(Q), \end{equation}$

(5.3)

where $C_{\lambda}$ is the flow penalizing cost defined by (5.2), and $U_{p}$ is defined by (5.1).

Corollary 5.1. The solution of the flow time control problem (5.3) is

$\begin{align} Q(t) = max(-p, min(\int_{\Theta}\frac{-u_{t}}{\lambda} (\sum\limits_{i = 1}^{n}\frac{\partial w_{i}}{\partial x_{i}})dx,p)) \end{align}$

(5.4)

with $u$ as the solution of (2.2) perturbed by $Q(t)$ and $w_{i}$ as the solution of

$\begin{equation} \left \{ \begin{array}{lll} {\frac{{\partial^{2} } {w_{i}}}{{\partial t^{2}}} } +\Delta^{2} w_{i} = Q(t) (w_{i})_{t}, & & (0, m)\times\Theta, \\ w_{i}(x,m) = ( \frac{\partial u}{\partial x_{i}}(x,m) - u_{i}^{d}), & & \Theta, \\ (w_{i})_{t}(x,m) = 0, & & \Theta, \\ w_{i} = \frac{\partial w_{i}}{\partial \nu } = 0, & & \Gamma. \end{array} \right. \end{equation}$

(5.5)

Proof. Similar to the approach used in the proof of Theorem 4.1, we deduce that

$\begin{align} 0\leq \int_{0}^{m}d(t)[\int_\Theta u_{t}(\sum\limits_{i = 1}^{n}\frac{\partial w_{i}(x,t)}{\partial x_{i}})dx+\lambda Q ]dt, \end{align}$

(5.6)

where $d(t)\in L^{\infty}(0, m)$ , a control function such that $Q+k d \in U_{p}$ with a small positive constant $k$ . □

Remark 5.1. (1) In the case of spatiotemporal target, we remark that the error $(\frac{\partial u}{\partial x_{i}}(x, t) - u_{i}^{d})$ between the state and the desired one becomes a the change of velocity induced by the known forces acting on system (4.6).

(2) In the case of a prescribed time $m$ targets, we remark that the error $(\frac{\partial u}{\partial x_{i}}(x, m) - u_{i}^{d})$ between the state and the desired one becomes a Dirichlet boundary condition in the adjoint equation (5.5).

6. Partial flow optimal control problem

This section establishes the flow partial optimal control problem governed by the plate equation (2.2). Afterward we characterize the solution. Let $\theta\subset \Theta$ be an open subregion of $\Theta$ and we define

$\begin{array}{lll} \widetilde{P_{\theta}}: (L^2(\Theta)) & \longrightarrow & (L^2(\theta)) \\ \qquad \qquad u & \longrightarrow & \widetilde{P}_{\theta} u = u|_{\theta}, \end{array}$

and

$\begin{array}{lll} P_{\theta}: (L^2(\Theta))^{n} & \longrightarrow &(L^2(\theta))^{n} \\ \qquad \qquad u & \longrightarrow & P_{\theta} u = u|_{\theta}. \end{array}$

We define the adjoint of $P_{\theta}$ by

$P_{\theta}^{*}u = \left\{\begin{array}{lll} u \, \, in \, \, \Theta, \\ 0 \in \Theta \setminus \theta. \end{array} \right.$

Definition 6.1. The plate equation (2.2) is said to flow weakly partially controllable on $\theta \subset \Theta$ , if for $\forall \, \, \beta > 0$ , one can find an optimal control $Q\in L^{2}(0, m)$ such that

$|P_\theta \nabla u_Q(m)-u^d||_{(L^2(\theta))^{n}}\leq \beta,$

where $u^d = (z_{1}^{d}, ...., u_{n}^{d})$ is the desired flow in $(L^2(\theta))^{n}$ .

For $U_{p}$ defined by (5.1), we take the partial flow optimal control problem

$\begin{equation} \min\limits_{Q \in U_{p}} C_{\lambda}(Q), \end{equation}$

(6.1)

and the partial flow cost $C_{\lambda}$ is

$\begin{align} C_{\lambda}(Q) = \frac{1}{2}\Big\|P_\theta\nabla u(m)-u^{d}\Big\|^{2}_{(L^2(\theta))^{n}} + \frac{\lambda}{2} \int_{0}^{m} Q^2(t) dt\\ = \frac{1}{2}\sum\limits_{i = 1}^{n} \Big\|\widetilde{P}_{\theta}\frac{\partial u(T)}{\partial x_{i}} - u_{i}^{d}\Big\|^{2}_{(L^2(\theta))} + \frac{\lambda}{2} \int_{0}^{m} Q^2(t) dt. \end{align}$

(6.2)

Next, we consider the family of optimality systems

$\begin{equation} \left \{ \begin{array}{lll} {\frac{{\partial^{2} } {w_{i}} }{{\partial t^{2}}} } = \Delta^{2} w_{i}+Q(t) (w_{i})_{t},& & (0, m)\times\Theta,\\ w_{i}(x,m) = ( \frac{\partial u(m)}{\partial x_{i}} - \widetilde{P}_{\theta}^{*}u_{i}^{d}), & & \Theta, \\ (w_{i})_{t}(x,m) = 0, & & \Theta, \\ w_{i}(x,t) = \frac{\partial w_{i}(x,t)}{\partial \nu } = 0, & & \Gamma. \end{array} \right. \end{equation}$

(6.3)

Lemma 6.1. Let the cost $C_{\lambda}(Q)$ defined by (6.2) and the control $Q(t) \in U_{p}$ be the solution of (6.1). We can write

$\begin{align} \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q )}{k} = \sum\limits_{i = 1}^{n} \int_\theta \widetilde{P}^*_{\theta}\widetilde{P}_{\theta}\left[ \int_{0}^{m}\frac{\partial^{2} w_{i}}{\partial t^{2}} \frac{\partial v(x,t)}{\partial x_{i}} dt+ \int_{0}^{m}w_{i} \frac{\partial }{\partial x_{i}}( \frac{\partial^{2} v}{\partial t^{2}}) dt\right]dx\\ + \int_{0}^{m}\lambda d Q dt, \end{align}$

(6.4)

where the solution of (4.1) is $v$ , and the solution of (6.3) is $w_{i}$ .

Proof. The functional $C_{\lambda}(Q)$ given by (6.2), is of the form:

$\begin{align} C_{\lambda}(Q) = \frac{1}{2} \sum\limits_{i = 1}^{n}\int_{\theta}(\widetilde{P}_{\theta}\frac{\partial u}{\partial x_{i}} - u_{i}^{d} )^2 dx + \frac{\lambda}{2} \int^{m}_{0} Q^2(t) dt. \end{align}$

(6.5)

Choose $u_{k} = u(Q+k d)$ and $u = u(Q)$ . By (6.5), we deduce

$\begin{align} \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k} = \lim\limits_{k\longrightarrow 0}\sum\limits_{i = 1}^{n} \frac{1}{2}\int_{\theta}\frac{(\widetilde{P}_\theta \frac{\partial u_{k}}{\partial x_{i}}-u_i^d)^{2}-(\widetilde{P}_\theta \frac{\partial u}{\partial x_{i}}-u_i^d)^{2}}{k}dx\\ + \lim\limits_{k\longrightarrow 0}\frac{\lambda}{2}\int_{0}^{m}\frac{(Q+k d)^2-Q^2}{k}dt. \end{align}$

(6.6)

Furthermore,

$\begin{align} \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k} \\ = \lim\limits_{k\longrightarrow 0}\sum\limits_{i = 1}^{n}\frac{1}{2}\int_{\theta}\widetilde{P}_{\theta}\frac{( \frac{\partial u_{k}}{\partial x_{i}}- \frac{\partial u}{\partial x_{i}})}{k}(\widetilde{P}_{\theta} \frac{\partial u_{k} }{\partial x_{i}}+ \widetilde{P}_{\theta} \frac{\partial u}{\partial x_{i}}-2u_i^d)dx \\ + \lim\limits_{k\longrightarrow 0}\frac{1}{2}\int_{0}^{m}(2\lambda d Q+k \lambda d^2)dt\\ = \sum\limits_{i = 1}^{n}\int_\theta \widetilde{P}_{\theta} \frac{\partial v(x,m)}{\partial x_{i}} \widetilde{P}_{\theta}( \frac{\partial u(x,m)}{\partial x_{i}}-\widetilde{P}_{\theta}^{*}u_i^d)dx+\int_{0}^{m}\lambda dQ dt \\ = \sum\limits_{i = 1}^{n} \int_\theta \widetilde{P}_{\theta} \frac{\partial v(x,m)}{\partial x_{i}} \widetilde{P}_{\theta}w_{i}(x,m)dx+\lambda \int_{0}^{m}dQ dt. \end{align}$

(6.7)

Using (6.3) to (6.7), we conclude

$\begin{align} \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k} = \sum\limits_{i = 1}^{n} \int_\theta \widetilde{P}^*_{\theta}\widetilde{P}_{\theta}\left[ \int_{0}^{m}\frac{\partial^{2} w_{i}}{\partial t^{2}} \frac{\partial v(x,t)}{\partial x_{i}} dt+ \int_{0}^{m}w_{i} \frac{\partial }{\partial x_{i}}( \frac{\partial^{2} v}{\partial t^{2}}) dt\right]dx\\\\ + \int_{0}^{m}\lambda d Qdt.\\ \end{align}$

(6.8)

□

Theorem 6.1. Consider the set $U_{p}$ , of partial admissible control defined as (5.1) and $u = u (Q)$ is its associate solution of (2.2), then the solution of (6.1) is

$\begin{equation} Q_{\varepsilon}(t) = max(-p,min( \frac{-1}{\lambda}\widetilde{P}_{\theta}(u_{t})(\widetilde{P}_{\theta} Div(w)),p)), \end{equation}$

(6.9)

where $Div(w) = \sum\limits_{i = 1}^{n}\frac{\partial w_{i}}{\partial x_{i}}$ .

Proof. Let $d \in U_{p}$ and $Q + k d \in U_{p}$ for $k > 0$ . The cost $C_{\lambda}$ at its minimum $Q$ , verifies

$\begin{equation} 0 \leq \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k}. \end{equation}$

(6.10)

From Lemma 6.1, substituting $\frac{\partial^{2} v}{\partial t^{2}}$ , by its value in system (4.1), we deduce that

$\begin{align} 0 \leq \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k}\\ = \sum\limits_{i = 1}^{n}\int_\theta \widetilde{P}^*_{\theta}\widetilde{P}_{\theta}\left[\int_{0}^{m} \frac{\partial v}{\partial x_{i}}\frac{\partial^{2} w_{i} }{\partial t^{2}}dt+ \int_{0}^{m}(-\Delta^{2} \frac{\partial v}{\partial x_{i}}+Q(t) \frac{\partial }{\partial x_{i}}(v_{t}) + d(t)\frac{\partial }{ \partial x_{i}}(u_{t}))w_{i} dt\right]dx\\ + \int_{0}^{m}\lambda d Q dt, \end{align}$

(6.11)

and system (6.3) gives

$\begin{align} 0 \leq \sum\limits_{i = 1}^{n} \int_\theta \widetilde{P}^*_{\theta}\widetilde{P}_{\theta}\left[\int_{0}^{m} \frac{\partial v}{\partial x_{i}}(\frac{\partial^{2} w_{i} }{\partial t^{2}}-\Delta^{2} w_{i}-Q(t) (w_{i})_{t})dt+d(t) \frac{\partial }{\partial x_{i}}(u_{t})w_{i} dt\right]dx+\int_{0}^{m}\lambda d Q dt.\\ \\ = \sum\limits_{i = 1}^{n} \int_\theta \widetilde{P}^*_{\theta}\widetilde{P}_{\theta} \int_{0}^{m} (h(t)u_{t})\frac{\partial w_{i}}{\partial x_{i}} dt+ \int_{0}^{m}\lambda d Q dt.\\ \\ = \int_{0}^{m} h(t)\int_\theta\left[ u_{t}\widetilde{P}^*_{\theta}\widetilde{P}_{\theta} \sum\limits_{i = 1}^{n}\frac{\partial w_{i}}{\partial x_{i}} + \lambda Q\right] dx dt, \end{align}$

(6.12)

which gives the optimal control

$\begin{equation} Q_{\varepsilon}(t) = max(-p,min( \frac{-1}{\lambda}\widetilde{P}_{\theta}(u_{t})(\widetilde{P}_{\theta} Div(w)),p)). \end{equation}$

(6.13)

□

7. Conclusions

This paper studied the optimal control problem governed by an infinite dimensional bilinear plate equation. The objective was to command the flow state of the bilinear plate equation to the desired flow using different types of bounded feedback. The problem flow optimal control governed by a bilinear plate equation was considered and solved in two cases using the adjoint method. The first case considered a spatiotemporal control function and looked to reach a flow target on the whole domain. The second case considered a time control function and looks to reach a prescribed target at a fixed final time. As an application, the partial flow control problem was established and solved using the proposed method. More applications can be examined, for example. the case of fractional hyperbolic systems.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was funded by the Deanship of Scientific Research at Jouf University under Grant Number (DSR2022-RG-0119).

Conflict of interest

The authors affirm that they have no conflicts of interest to disclose.

References

[1]	D. G. Prakasha, P. Veeresha, H. M. Baskonus, Two novel computational techniques for fractional Gardner and Cahn‐Hilliard equations, Comput. Math. Methods, 1 (2019), e1021.
[2]	B. Bonilla, M. Rivero, L. Rodriguez-Germa, J. J. Trujillo, Fractional differential equations as alternative models to nonlinear differential equations, Appl. Math. Comput., 187 (2007), 79–88.
[3]	P. Veeresha, D. G. Prakasha, Solution for fractional generalized Zakharov equations with Mittag-Leffler function, Results Eng., 5 (2020), 100085.
[4]	J. Liouville, Memoire sur quelques questions de geometrie et de mecanique et sur un nouveau genre de calcul pour rsoudre ces quations, Ecole Polytech., 13 (1832), 71–162.
[5]	M. Caputo, Elasticita e dissipazione, Bologna: Zanichelli, 1969.
[6]	S. Meng, F. Meng, F. Zhang, Q. Li, Y. Zhang, A. Zemouche, Observer design method for nonlinear generalized systems with nonlinear algebraic constraints with applications, Automatica, 162 (2024), 111512. https://doi.org/10.1016/j.automatica.2024.111512 doi: 10.1016/j.automatica.2024.111512
[7]	B. Li, T. Guan, L. Dai, G. Duan, Distributionally Robust Model Predictive Control with Output Feedback, IEEE Trans. Autom. Control, 2023. http://doi.org/10.1109/TAC.2023.3321375 doi: 10.1109/TAC.2023.3321375
[8]	Y. Shi, C. Song, Y. Chen, H. Rao, T. Yang, Complex Standard Eigenvalue Problem Derivative Computation for Laminar-Turbulent Transition Prediction, AIAA J., 61 (2023), 3404–3418. http://doi.org/10.2514/1.J062212 doi: 10.2514/1.J062212
[9]	H. M. He, J. G. Peng, H. Y. Li, Iterative approximation of fixed point problems and variational inequality problems on Hadamard manifolds, UPB Bull. Ser. A, 84 (2022), 25–36.
[10]	R. Subashini, C. Ravichandran, K. Jothimani, H. M. Baskonus, Existence results of Hilfer integro-differential equations with fractional order, Discrete Contin. Dyn. Syst.-Ser. S, 13 (2020), 911–923.
[11]	M. Alqhtani, K. M. Saad, R. Shah, W. M. Hamanah, Discovering novel soliton solutions for (3+1)-modified fractional Zakharov-Kuznetsov equation in electrical engineering through an analytical approach, Opt. Quant. Electron., 55 (2023), 1149.
[12]	C. Ravichandran, K. Jothimani, H. M. Baskonus, N. Valliammal, New results on nondensely characterized integrodifferential equations with fractional order, Eur. Phys. J. Plus, 133 (2018), 109.
[13]	N. Valliammal, C. Ravichandran, Z. Hammouch, H. Mehmet Baskonus, A new investigation on fractional-ordered neutral differential systems with state-dependent delay, Int. J. Nonlinear Sci. Numer. Simul., 20 (2019), 803–809.
[14]	S. Noor, A. S. Alshehry, N. H. Aljahdaly, H. M. Dutt, I. Khan, R. Shah, Investigating the impact of fractional non-linearity in the Klein-Fock-Gordon equation on quantum dynamics, Symmetry, 15 (2023), 881.
[15]	T. Abdeljawad, Q. M. Al-Mdallal, F. Jarad, Fractional logistic models in the frame of fractional operators generated by conformable derivatives, Chaos, Solitons Fractals, 119 (2019), 94–101.
[16]	C. Cattani, Connection coefficients of Shannon wavelets, Math. Modell. Anal., 11 (2006), 117–132.
[17]	A. Jajarmi, B. Ghanbari, D. Baleanu, A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence, Chaos: Interdiscip. J. Nonlinear Sci., 29 (2019), 093111.
[18]	M. Caputo, M. Fabrizio, On the singular kernels for fractional derivatives. Some applications to partial differential equations, Progr. Fract. Differ. Appl., 7 (2021), 79–82.
[19]	X. Cai, R. Tang, H. Zhou, Q. Li, S. Ma, D. Wang, et al., Dynamically controlling terahertz wavefronts with cascaded metasurfaces, Adv. Photonics, 3 (2021), 036003. http://doi.org/10.1117/1.AP.3.3.036003 doi: 10.1117/1.AP.3.3.036003
[20]	C. Guo, J. Hu, Y. Wu, S. Celikovsky, Non-Singular Fixed-Time Tracking Control of Uncertain Nonlinear Pure-Feedback Systems With Practical State Constraints, IEEE Trans. Circuits Syst. I, 70 (2023), 3746–3758. http://doi.org/10.1109/TCSI.2023.3291700 doi: 10.1109/TCSI.2023.3291700
[21]	C. Guo, J. Hu, J. Hao, S. Celikovsky, X. Hu, Fixed-time safe tracking control of uncertain high-order nonlinear pure-feedback systems via unified transformation functions, Kybernetika, 59 (2023), 342–364. http://doi.org/10.14736/kyb-2023-3-0342 doi: 10.14736/kyb-2023-3-0342
[22]	X. Bai, Y. He, M. Xu, Low-Thrust Reconfiguration Strategy and Optimization for Formation Flying Using Jordan Normal Form, IEEE Trans. Aerosp. Electron. Syst., 57 (2021), 3279–3295. http://doi.org/10.1109/TAES.2021.3074204 doi: 10.1109/TAES.2021.3074204
[23]	Y. Kai, J. Ji, Z. Yin, Study of the generalization of regularized long-wave equation, Nonlinear Dyn., 107 (2022), 2745–2752. http://doi.org/10.1007/s11071-021-07115-6 doi: 10.1007/s11071-021-07115-6
[24]	Y. Kai, Z. Yin, On the Gaussian traveling wave solution to a special kind of Schrodinger equation with logarithmic nonlinearity, Mod. Phys. Lett. B, 36 (2021), 2150543. http://doi.org/10.1142/S0217984921505436 doi: 10.1142/S0217984921505436
[25]	X. Zhou, X. Liu, G. Zhang, L. Jia, X. Wang, Z. Zhao, An Iterative Threshold Algorithm of Log-Sum Regularization for Sparse Problem, IEEE Trans. Circuits Syst. Video Technol., 33 (2023), 4728–4740. http://doi.org/10.1109/TCSVT.2023.3247944 doi: 10.1109/TCSVT.2023.3247944
[26]	D. Benney, Long waves on liquid films, J. Math. Phys., 45 (1966), 150–155.
[27]	S. P. Lin, Finite amplitude side-band stability of a viscous film, J. Fluid Mech., 63 (1974), 417–429.
[28]	W. Gao, P. Veeresha, D. G. Prakasha, H. M. Baskonus, New numerical simulation for fractional Benney-Lin equation arising in falling film problems using two novel techniques, Numer. Methods Partial Differ. Eq., 37 (2021), 210–243.
[29]	N. G. Berloff, L. N. Howard, Solitary and periodic solutions of nonlinear nonintegrable equations, Stud. Appl. Math., 99 (1997), 1–24.
[30]	H. A. Biagioni, F. Linares, On the Benney–Lin and Kawahara equations, J. Math. Anal. Appl., 211 (1997), 131–152.
[31]	S. B. Cui, D. G. Deng, S. P. Tao, Global existence of solutions for the Cauchy problem of the Kawahara equation with L 2 initial data, Acta Math. Sin., 22 (2006), 1457–1466.
[32]	H. Tariq, G. Akram, Residual power series method for solving time-space-fractional Benney-Lin equation arising in falling film problems, J. Appl. Math. Comput., 55 (2017), 683–708.
[33]	P. K. Gupta, Approximate analytical solutions of fractional Benney–Lin equation by reduced differential transform method and the homotopy perturbation method, Comput. Math. Appl., 61 (2011), 2829–2842.
[34]	Y. X. Xie, New explicit and exact solutions of the Benney-Kawahara-Lin equation, Chin. Phys. B, 18 (2009), 4094.
[35]	K. K. Ali, R. Yilmazer, H. M. Baskonus, H. Bulut, Modulation instability analysis and analytical solutions to the system of equations for the ion sound and Langmuir waves, Phys. Scr., 95 (2020), 065602.
[36]	H. Qin, R. A. Attia, M. Khater, D. Lu, Ample soliton waves for the crystal lattice formation of the conformable time-fractional (N+ 1) Sinh-Gordon equation by the modified Khater method and the Painleve property, J. Intell. Fuzzy Syst., 38 (2020), 2745–2752.
[37]	A. Gaber, H. Ahmad, Solitary wave solutions for space-time fractional coupled integrable dispersionless system via generalized kudryashov method, Facta Univ. Ser.: Math. Inf., 35 (2021), 1439–1449.
[38]	H. Yasmin, A. S. Alshehry, A. M. Saeed, R. Shah, K. Nonlaopon, Application of the q-Homotopy Analysis Transform Method to Fractional-Order Kolmogorov and Rosenau-Hyman Models within the Atangana-Baleanu Operator, Symmetry, 15 (2023), 671.
[39]	J. H. He, Exp-function method for fractional differential equations, Int. J. Nonlinear Sci. Numer. Simul., 14 (2013), 363–366.
[40]	H. Khan, R. Shah, J. F. Goómez-Aguilar, D. Baleanu, P. Kumam, Traveling waves solution for fractional-order biological population model, Math. Modell. Nat. Phenom., 16 (2021), 32.
[41]	H. Yasmin, N. H. Aljahdaly, A. M. Saeed, R. Shah, Investigating Symmetric Soliton Solutions for the Fractional Coupled Konno-Onno System Using Improved Versions of a Novel Analytical Technique, Mathematics, 11 (2023), 2686.
[42]	M. M. Al-Sawalha, H. Yasmin, R. Shah, A. H. Ganie, K. Moaddy, Unraveling the Dynamics of Singular Stochastic Solitons in Stochastic Fractional Kuramoto–Sivashinsky Equation, Fractal Fract., 7 (2023), 753.
[43]	M. Alquran, Dynamic behavior of explicit elliptic and quasi periodic-wave solutions to the generalized (2+ 1)-dimensional Kundu-Mukherjee-Naskar equation, Optik, 301 (2024), 171697.
[44]	H. Yasmin, A. S. Alshehry, A. H. Ganie, A. M. Mahnashi, Perturbed Gerdjikov-Ivanov equation: Soliton solutions via Backlund transformation, Optik, 298 (2024), 171576.
[45]	I. Jaradat, M. Alquran, A variety of physical structures to the generalized equal-width equation derived from Wazwaz-Benjamin-Bona-Mahony model, J. Ocean Eng. Sci., 7 (2022), 244–247.
[46]	S. A. El-Tantawy, H. A. Alyousef, R. T. Matoog, R. Shah, On the optical soliton solutions to the fractional complex structured (1+ 1)-dimensional perturbed gerdjikov-ivanov equation, Phys. Scr., 99 (2024), 035249.
[47]	M. Alquran, Necessary conditions for convex-periodic, elliptic-periodic, inclined-periodic, and rogue wave-solutions to exist for the multi-dispersions Schrodinger equation, Phys. Scr., 99 (2024), 025248.
[48]	M. Alquran, The amazing fractional Maclaurin series for solving different types of fractional mathematical problems that arise in physics and engineering, Partial Differ. Eq. Appl. Math., 7 (2023), 100506.
[49]	S. Alshammari, K. Moaddy, M. Alshammari, Z. Alsheekhhussain, M. M. Al-Sawalha, M. Yar, Analysis of solitary wave solutions in the fractional-order Kundu-Eckhaus system, Sci. Rep., 14 (2024), 3688.
[50]	S. Mukhtar, S. Noor, The numerical investigation of a fractional-order multi-dimensional Model of Navier-Stokes equation via novel techniques, Symmetry, 14 (2022), 1102.
[51]	P. Sunthrayuth, A. M. Zidan, S. W. Yao, M. Inc, The comparative study for solving fractional-order Fornberg-Whitham equation via $\rho$ -Laplace transform, Symmetry, 13 (2021), 784.
[52]	R. Shah, D. Baleanu, Fractional Whitham-Broer-Kaup equations within modified analytical approaches, Axioms, 8 (2019), 125.
[53]	A. Saad Alshehry, M. Imran, A. Khan, W. Weera, Fractional View Analysis of Kuramoto-Sivashinsky Equations with Non-Singular Kernel Operators, Symmetry, 14 (2022), 1463.
[54]	M. M. Al-Sawalha, A. Khan, O. Y. Ababneh, T. Botmart, Fractional view analysis of Kersten-Krasil'shchik coupled KdV-mKdV systems with non-singular kernel derivatives, AIMS Mathematics, 7 (2022), 18334–18359. https://doi.org/10.3934/math.20221010 doi: 10.3934/math.20221010
[55]	H. Yasmin, A. S. Alshehry, A. H. Ganie, A. Shafee, R. Shah, Noise effect on soliton phenomena in fractional stochastic Kraenkel-Manna-Merle system arising in ferromagnetic materials, Sci. Rep., 14 (2024), 1810.
[56]	J. H. He, S. K. Elagan, Z. B. Li, Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A, 376 (2012), 257–259.
[57]	Sarikaya, M. Zeki, H. Budak, H. Usta, On generalized the conformable fractional calculus, TWMS J. Appl. Eng. Math., 9 (2019), 792–799.

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