Loading [MathJax]/jax/output/SVG/jax.js
Research article Special Issues

AI-driven adaptive reliable and sustainable approach for internet of things enabled healthcare system


  • Received: 12 October 2021 Revised: 23 January 2022 Accepted: 24 January 2022 Published: 11 February 2022
  • Artificial Intelligence (AI) driven adaptive techniques are viable to optimize the resources in the Internet of Things (IoT) enabled wearable healthcare devices. Due to the miniature size and ability of wireless data transfer, Body Sensor Networks (BSNs) have become the center of attention in current medical media technologies. For a long-term and reliable healthcare system, high energy efficiency, transmission reliability, and longer battery lifetime of wearable sensors devices are required. There is a dire need for empowering sensor-based wearable techniques in BSNs from every aspect i.e., data collection, healthcare monitoring, and diagnosis. The consideration of protocol layers, data routing, and energy optimization strategies improves the efficiency of healthcare delivery. Hence, this work presents some key contributions. Firstly, it proposes a novel avant-garde framework to simultaneously optimize the energy efficiency, battery lifetime, and reliability for smart and connected healthcare. Secondly, in this study, an Adaptive Transmission Data Rate (ATDR) mechanism is proposed, which works on the average constant energy consumption by varying the active time of the sensor node to optimize the energy over the dynamic wireless channel. Moreover, a Self-Adaptive Routing Algorithm (SARA) is developed to adopt a dynamic source routing mechanism with an energy-efficient and shortest possible path, unlike the conventional routing methods. Lastly, real-time datasets are adopted for intensive experimental setup for revealing pervasive and cost-effective healthcare through wearable devices. It is observed and analysed that proposed algorithms outperform in terms of high energy efficiency, better reliability, and longer battery lifetime of portable devices.

    Citation: Noman Zahid, Ali Hassan Sodhro, Usman Rauf Kamboh, Ahmed Alkhayyat, Lei Wang. AI-driven adaptive reliable and sustainable approach for internet of things enabled healthcare system[J]. Mathematical Biosciences and Engineering, 2022, 19(4): 3953-3971. doi: 10.3934/mbe.2022182

    Related Papers:

    [1] Yousef Jawarneh, Humaira Yasmin, Ali M. Mahnashi . A new solitary wave solution of the fractional phenomena Bogoyavlenskii equation via Bäcklund transformation. AIMS Mathematics, 2024, 9(12): 35308-35325. doi: 10.3934/math.20241678
    [2] M. Mossa Al-Sawalha, Safyan Mukhtar, Azzh Saad Alshehry, Mohammad Alqudah, Musaad S. Aldhabani . Kink soliton phenomena of fractional conformable Kairat equations. AIMS Mathematics, 2025, 10(2): 2808-2828. doi: 10.3934/math.2025131
    [3] Hussain Gissy, Abdullah Ali H. Ahmadini, Ali H. Hakami . The travelling wave phenomena of the space-time fractional Whitham-Broer-Kaup equation. AIMS Mathematics, 2025, 10(2): 2492-2508. doi: 10.3934/math.2025116
    [4] M. Ali Akbar, Norhashidah Hj. Mohd. Ali, M. Tarikul Islam . Multiple closed form solutions to some fractional order nonlinear evolution equations in physics and plasma physics. AIMS Mathematics, 2019, 4(3): 397-411. doi: 10.3934/math.2019.3.397
    [5] Khudhayr A. Rashedi, Musawa Yahya Almusawa, Hassan Almusawa, Tariq S. Alshammari, Adel Almarashi . Lump-type kink wave phenomena of the space-time fractional phi-four equation. AIMS Mathematics, 2024, 9(12): 34372-34386. doi: 10.3934/math.20241637
    [6] Humaira Yasmin, Haifa A. Alyousef, Sadia Asad, Imran Khan, R. T. Matoog, S. A. El-Tantawy . The Riccati-Bernoulli sub-optimal differential equation method for analyzing the fractional Dullin-Gottwald-Holm equation and modeling nonlinear waves in fluid mediums. AIMS Mathematics, 2024, 9(6): 16146-16167. doi: 10.3934/math.2024781
    [7] Ali Turab, Hozan Hilmi, Juan L.G. Guirao, Shabaz Jalil, Nejmeddine Chorfi, Pshtiwan Othman Mohammed . The Rishi Transform method for solving multi-high order fractional differential equations with constant coefficients. AIMS Mathematics, 2024, 9(2): 3798-3809. doi: 10.3934/math.2024187
    [8] M. TarikulIslam, M. AliAkbar, M. Abul Kalam Azad . Traveling wave solutions in closed form for some nonlinear fractional evolution equations related to conformable fractional derivative. AIMS Mathematics, 2018, 3(4): 625-646. doi: 10.3934/Math.2018.4.625
    [9] M. Mossa Al-Sawalha, Saima Noor, Saleh Alshammari, Abdul Hamid Ganie, Ahmad Shafee . Analytical insights into solitary wave solutions of the fractional Estevez-Mansfield-Clarkson equation. AIMS Mathematics, 2024, 9(6): 13589-13606. doi: 10.3934/math.2024663
    [10] Waleed Hamali, Abdulah A. Alghamdi . Exact solutions to the fractional nonlinear phenomena in fluid dynamics via the Riccati-Bernoulli sub-ODE method. AIMS Mathematics, 2024, 9(11): 31142-31162. doi: 10.3934/math.20241501
  • Artificial Intelligence (AI) driven adaptive techniques are viable to optimize the resources in the Internet of Things (IoT) enabled wearable healthcare devices. Due to the miniature size and ability of wireless data transfer, Body Sensor Networks (BSNs) have become the center of attention in current medical media technologies. For a long-term and reliable healthcare system, high energy efficiency, transmission reliability, and longer battery lifetime of wearable sensors devices are required. There is a dire need for empowering sensor-based wearable techniques in BSNs from every aspect i.e., data collection, healthcare monitoring, and diagnosis. The consideration of protocol layers, data routing, and energy optimization strategies improves the efficiency of healthcare delivery. Hence, this work presents some key contributions. Firstly, it proposes a novel avant-garde framework to simultaneously optimize the energy efficiency, battery lifetime, and reliability for smart and connected healthcare. Secondly, in this study, an Adaptive Transmission Data Rate (ATDR) mechanism is proposed, which works on the average constant energy consumption by varying the active time of the sensor node to optimize the energy over the dynamic wireless channel. Moreover, a Self-Adaptive Routing Algorithm (SARA) is developed to adopt a dynamic source routing mechanism with an energy-efficient and shortest possible path, unlike the conventional routing methods. Lastly, real-time datasets are adopted for intensive experimental setup for revealing pervasive and cost-effective healthcare through wearable devices. It is observed and analysed that proposed algorithms outperform in terms of high energy efficiency, better reliability, and longer battery lifetime of portable devices.



    Over the last few decades, nonlinear partial differential equations (NLPDEs) have emerged as a major field of interest in the study of mathematical sciences. Since the natural world is highly complex, the interconnection between its two components is fascinating; many authors believe that, while studying the nonlinear science true triumphs of the human mind, one gets the greatest opportunity to understand the basics of specific physical laws. Countless physical behaviors and scientific disciplines including engineering, climatology, applied mathematics, biologic, and chemical reactions are described through the help of NLPDEs [1,2]. In this context of evaluated systems to comprehend these evaluated systems, solving NLPDEs and finding the numerical as well as the analytical solutions proves to be of significant importance. Numerous researchers have developed various methods to obtain solitary wave solutions for the non-linear PDEs [3,4]. These include the lie group method [5], exp(ϕ(x))-expansion method [6], extended direct algebraic method [7], variational iteration method [8], unified method [9], (G/G2)-expansion method [10], sine-Gordon expansion method [11], tanh-coth method [12], homotopy analysis method [13], auxiliary ordinary differential equation (ODE) method [14], and tanh function method [15].

    Fractional derivatives are widely used in mathematics, with the focus being put on the non-integer order derivatives [16,17,18]. These derivatives are crucial when the system under investigation is characterized by power-law processes and memory. In recent years, a large number of scientists have introduced fractional derivatives to analyze the characteristics of the stability of solitons in various fields [19,20,21]. Many researches have been devoted to seeking solitary wave solutions of diverse nonlinear PDEs and these have become an essential part of the development of our thoughts on these systems [22,23,24]. In order to increase the precision of the model, several forms of the fractional have been built. For example, fractional derivatives have been used in signal systems [25], plasma physics models [26], fractional epidemiologic models [27] and different models involving fractional derivatives. In this context, the Riccati-Bernoulli sub-ODE method with Bäcklund transformation [28,29,30] was used on the truncated time-space fractional Oskolkov equation. In this method, one can succeed in the systematic simplification of complicated fractional PDEs into ODEs and obtain explicit exact solutions. This method is very useful in the presence of nonlinear techniques and is very flexible with respect of the type of equation used. Also, it yields precise solutions, and such solutions are good predictors that give additional understanding of the behavior of the system in question relative to numerical or strictly approximate approaches. It is crucial to substitute the conventional Oskolkov equation with the equation of its fractional analogue because the description of phenomena involving some memory and non-locality, which are quite common in numerous applications, is impossible with assistance from the standard approach. The fractional differential parameter (α) provides an opportunity to approximate the intricacy of diffusion and wave propagation that the simple Oskolkov equation cannot consider. This is quite suitable in areas like fluid dynamics and plasma physics since the fractional model is much more detailed in the description of such processes as viscoelastic flow and fractal dispersion. These phenomena can be described more thoroughly using the fractional Oskolkov equation, which provides the researchers and engineers a more accurate approach to investigate complex systems which demonstrate the fractional-order behavior. Hence to ensure that the phenomena under consideration correspond to real-life occurrences, this model was examined in its fractional form [31,32,33]:

    ftβt(2fx2)γ2fx2+f(fx)=0. (1.1)

    This equation is helpful in the calculation of dimensions and geometry of thin-walled pressure vessels like tanks and reactors and is used extensively across the field of chemical and mechanical engineering. It is particularly useful in the design of pressure vessels for high pressure and high-temperature severe service environments. The Oskolkov equation is in fact a major model for viscoelastic, non-Newtonian fluids, capable of approximating major manifestations of the flow behavior such as shear-thinning and viscoelasticity [34]. Hence, incorporating fractional derivatives, the fractional Oskolkov equation also covers such behaviors as memory effects and non-local interactions inherent in non-Newtonian media. This extension expands the applicability of this method to various real-life situations such as polymer solutions and biological fluids. Also, the Oskolkov equation is used in estimating mechanical properties like stress and strain of pressure vessels. Still, many authors studied soliton solutions of similar models by employing significance approaches. For instance, different approaches were used to investigate solitonic wave solutions of the Oskolkov equation. The MSE scheme was used in [35], and another approach was used by the authors of [36]. The modified exp(ϕ(ζ))-expansion function was given in [37] and the (Φ,Ψ) expansion method was also studied in [38]. In addition the Sine-Gordon expansion method was used in [39]. Altogether these investigations have helped in put into place various solitonic wave solutions. Current trends of study also involve optical solitons [40], wave propagation in plasma [41], and nonparaxial solitons [42], including both fractional as well as traditional approaches to vibrations of cross kink solitons [43].

    To construct solitonic wave solutions, several approaches have been employed in previous studies to examine the soliton solutions of models similar to the present one. These investigations span various applications, which include optical solitons, wave propagation in plasma, and discrete as well as fractional approaches. Nevertheless, in this work, we provide a new approach to this problem by using the Riccati-Bernoulli sub-ODE method in combination with the Bäcklund transformation to study the fractional version of the Oskolkov equation. This method offers fresh perspectives on the behavior of solitonic waves and describes some exciting kink characteristics in the fractional Oskolkov model. In this context, the presentation of our analysis concerns the way that fractional parameters affect these solitonic solutions and uses 2D as well as 3D plots to visualize the dependence of these parameters. This contribution stretches the area of application of the fractional Oskolkov equation and provides deeper arguments to further investigate soliton solutions in fractional models.

    Further, the operator integrating α-derivatives of powers agrees exactly to the idea of conformable fractional derivatives [44] for W(ϕ) of order α(0,1) and for t>0 is defined as:

    DαϕW(ϕ)=limi0W(i(ϕ)iαW(ϕ))i,0<α1, (1.2)
    {Dαϕϕp=pϕpα,Dαϕ(p1η(ϕ)±p2t(ϕ))=p1Dαϕ(η(ϕ))±m2Dαϕ(t(ϕ)),Dαϕ[fg]=ϕ1αg(ϕ)Dαϕf(g(ϕ)). (1.3)

    For smooth functions, the derivative simplifies to Dαf(t)=t1αdf(t)dt. One of the major benefits as to the proposed conformable derivative is that it does generalize the classical derivative in a manner as elementary as the concept itself. Indeed, when α=1, it returns the standard derivative, thus making a transition from fractional derivation to classical derivation rather smooth.

    Unlike Caputo and Riemann-Liouville derivatives, the conformable derivative does not use complex integral formulations, which makes it more easy to utilize when finding the derivative of differential equations, yet the conformable derivative keeps a lot of characteristics in fractional calculus. In our analyses, the defined conformable fractional derivative is incorporated to present the time-space fractional Oskolkov equation to describe the fluid flow and other processes with better accuracy as compared to the conventional integer order. This derivative will help us incorporate memory and hereditary properties in the system, which is very important in fractional models.

    Section 2 gives a brief overview of the method that has been used here, which will be explained in detail in Section 3 by providing the solution of the new fractional Oskolkov system. In Section 4, results and discussion are provided and some graphical illustrations are given. Finally, Section 5 provides the conclusion of our work.

    Here, for the clear understanding of the working procedure of the mentioned process, let us explain it broadly.

    Step 1. Consider nonlinear PDEs in the following form:

    P1(R1,Dαt(R1),Dαq1(R1),Dαq2(R1),R1Dαq1(R1),)=0,0<α1, (2.1)

    where R1=R(t,q1,q2,q3,,qk) is a function of (t,q1,q2,q3,,qk) and its partial derivatives.

    Step 2. This transformation changes Eq (2.1) into a nonlinear ODE of the following form:

    Q1(F,F(ϕ),F(ϕ),FF(ϕ),)=0. (2.2)

    Step 3. Let us suppose that Eq (2.2) has the following solution:

    G(ϕ)=nj=nsjg(ϕ)j, (2.3)

    where sj are constants and g(ϕ) is obtained from the Bäcklund transformation,

    g(ϕ)=ζp2+p1Z(ϕ)p1+p2Z(ϕ).

    Here, (ζ), (p1), and (p2) are constants such that p20 and Z(ϕ) is the solution of the following ODE:

    dZdϕ=ζ+Z(ϕ)2. (2.4)

    The Ricatti Eq (2.4) possess the following general solutions [45]:

    Z(ϕ)={ζtanh(ζϕ),as ζ<0,ζcoth(ζϕ),as ζ<0,Z(ϕ)=1ψ,as ζ=0,Z(ϕ)={ζtan(ζϕ),as ζ>0,ζcot(ζϕ),as ζ>0. (2.5)

    Step 4. Solving for the homogeneous balance of the largest nonlinear term and the highest-order derivative in Eq (2.2) gives the positive integer (n) as presented in Eq (2.3). First, it has to be noted that the balance number of a processor can be calculated as follows [46]:

    D[dmfdψm]=n+m,D[fJdmfdψm]w=nJ+w(m+n). (2.6)

    Step 5. Next, we replace the same function with the help of Eq (2.3) into Eq (2.2) or into the expression which appears after integration of Eq (2.2), and collect all terms containing g(ϕ). The coefficients of the polynomial are then set equal to zero and a system of algebraic equations in (si) and other parameters are obtained.

    Step 6. These equations are solved using the Maple computational tool and, in the end, the solitary wave solutions are provided for Eq (2.1).

    By applying the considered model with the Riccati-Bernoulli sub-ODE, we get the wave solutions. The (1+1)-dimensional fractional Oskolkov equation in its fractional form is given by

    Dαt(f)βDαt(D2αx(f))γ(D2αx(f))+f(Dαx(f))=0. (3.1)

    Equation (3.1) encompasses the temporal evolution of viscoelastic fluids where the material memory and the refractive index both play pertinent roles. The term Dαt(f) represents the time fractionality, expressing the fact that the current state of the fluid depends on its prior behavior. The second term, therefore, includes fractional temporal and spatial derivatives and is used to capture dispersive effects that depend on the spatial distribution. The dissipation term (D2αx(f)) is used to express the generalized diffusion processes and the nonlinear interface f(Dαx(f)) depicts how solitons and wave patterns interact and stay preserved while travelling. This equation is very important in modeling this behavior of fluids where conventional models fail to capture the behavior well. Here, β and γ are constants and f(x,t) represents an unknown wave front. Therefore, f(x,t)=F(ψ) and ψ=λxααωtαα are used to change Eq (3.1). It is converted into the following ordinary differential system:

    2λ2ωβd2Fdψ22γλ2dFdψ2ωF+λF2=0. (3.2)

    We use the proposed approach that takes advantage of properties that exist inherently in the system balancing equations to reduce and solve for wave structures. In this way, integrating specific terms, it is possible to extract the individual characteristics of the primary components of wave phenomena. By substituting Eqs (2.3) and (2.4) into Eq (3.2) and then collecting the coefficients of Z(ϕ), we derive the following system of equations:

    12λ2ωβs2p28ζ2λs22p28=0,4λ2ωβs1p28ζ3+2λs2p28s1ζ+4λ2γs2p28ζ2=0,16λ2ωβs2p28ζ3+2ωs2p28ζ2λs12p28ζ22λs2p28s0ζ22λ2γs1p28ζ3=0,2ωs1p28ζ3+4λ2γs2p28ζ3+4λ2ωβs1p28ζ4+2λs1p28s0ζ3+2λs2p28s1ζ3=0,2λs1p28s1ζ44λ2ωβs2p28ζ62λ2γs1p28ζ4+2λ2γs1p28ζ52λs2p28s2ζ4+2ωs0p28ζ4λs02p28ζ44λ2ωβs2p28ζ4=0,2ωs1ζ5p28+2λs0s1ζ5p284λ2γs2p28ζ6+4λ2ωβs1p28ζ6+2λs1p28s2ζ5=0,2ωs2ζ6p28λs12ζ6p2816λ2ωβs2p28ζ72λs0s2ζ6p28+2λ2γs1p28ζ6=0,4λ2ωβs1p28ζ7+2λs1ζ7p28s24λ2γs2p28ζ7=0,λs22ζ8p2812λ2ωβs2p28ζ8=0. (3.3)

    This give us the algebraic equations by setting Z(ϕ)=0. The solutions of this system of algebraic equations obtained from Maple are:

    Set 1.

    s0=3/106γβ,s1=26βω,s1=0,s2=0,s2=106β3/2ω2γ,ζ=1100γ2ω2β2,λ=5/66βωγ,ω=ω. (3.4)

    Set 2.

    s0=1/46γβ,s1=26βω,s1=12006γ2β3/2ω,s2=1160006γ3β5/2ω2,s2=106β3/2ω2γ,ζ=1400γ2ω2β2,λ=5/66βωγ,ω=ω. (3.5)

    Solution Group 1. For Set 1 (ζ<0) provided that ω=110, we obtain the following set of solutions for Eq (3.1), where in this case

    ζ=1100γ2ω2β2,ψ=5/66βωxαγαωtαα,
    f1(x,t)=3/106γβ+26βω(1100γ2p2ω2β2p1ζtanh(ζψ))(p1p2ζtanh(ζψ))1106β3/2ω2(1100γ2p2ω2β2p1ζtanh(ζψ))2γ1(p1p2ζtanh(ζψ))2 (3.6)

    or

    f2(x,t)=3/106γβ+26βω(1100γ2p2ω2β2p1ζcoth(ζψ))(p1p2ζcoth(ζψ))1106β3/2ω2(1100γ2p2ω2β2p1ζcoth(ζψ))2γ1(p1p2ζcoth(ζψ))2. (3.7)

    Solution Group 2. For Set 1 (ζ>0) provided that ω=110, we obtain the following set of solutions for Eq (3.1):

    f3(x,t)=3/106γβ+26βω(1100γ2p2ω2β2+p1ζtan(ζψ))(p1+p2ζtan(ζψ))1106β3/2ω2(1100γ2p2ω2β2+p1ζtan(ζψ))2γ1(p1+p2ζtan(ζψ))2 (3.8)

    or

    f4(x,t)=3/106γβ+26βω(1100γ2p2ω2β2p1ζcot(ζψ))(p1p2ζcot(ζψ))1106β3/2ω2(1100γ2p2ω2β2p1ζcot(ζψ))2γ1(p1p2ζcot(ζψ))2. (3.9)

    Solution Group 3. For Set 1 (ζ=0), we obtain the following set of solutions for Eq (3.1):

    f5(x,t)=3/106γβ+26βω(1100γ2p2ω2β2p1ψ)(p1p2ψ)1106β3/2ω2(1100γ2p2ω2β2p1ψ)2γ1(p1p2ψ)2. (3.10)

    Solution Group 4. For Set 2 (ζ<0) provided that ω=120, we obtain the following set of solutions for Eq (3.1), where in this case

    ζ=1400γ2ω2β2,ψ=5/66βωxαγαωtαα,
    f6(x,t)=s2(p1p2ζtanh(ζψ))2(1400γ2p2ω2β2p1ζtanh(ζψ))2+s1(p1p2ζtanh(ζψ))(1400γ2p2ω2β2p1ζtanh(ζψ))1+s0+26βω(1400γ2p2ω2β2p1ζtanh(ζψ))(p1p2ζtanh(ζψ))1+s2ω2β(1400γ2p2ω2β2p1ζtanh(ζψ))2(p1p2ζtanh(ζψ))2 (3.11)

    or

    f7(x,t)=s2(p1p2ζcoth(ζψ))2(1400γ2p2ω2β2p1ζcoth(ζψ))2+s1(p1p2ζcoth(ζψ))(1400γ2p2ω2β2p1ζcoth(ζψ))1+s0+26βω(1400γ2p2ω2β2p1ζcoth(ζψ))(p1p2ζcoth(ζψ))1+s2ω2β(1400γ2p2ω2β2p1ζcoth(ζψ))2(p1p2ζcoth(ζψ))2. (3.12)

    Solution Group 5. For Set 2 (ζ>0) provided that ω=120, we obtain the following set of solutions for Eq (3.1):

    f8(x,t)=s2(p1+p2ζtan(ζψ))2(1400γ2p2ω2β2+p1ζtan(ζψ))2+s1(p1+p2ζtan(ζψ))(1400γ2p2ω2β2+p1ζtan(ζψ))1+s0+26βω(1400γ2p2ω2β2+p1ζtan(ζψ))(p1+p2ζtan(ζψ))1+s2ω2β(1400γ2p2ω2β2+p1ζtan(ζψ))2(p1+p2ζtan(ζψ))2 (3.13)

    or

    f9(x,t)=s2(p1p2ζcot(ζψ))2(1400γ2p2ω2β2p1ζcot(ζψ))2+s1(p1p2ζcot(ζψ))(1400γ2p2ω2β2p1ζcot(ζψ))1+s0+26βω(1400γ2p2ω2β2p1ζcot(ζψ))(p1p2ζcot(ζψ))1+s2ω2β(1400γ2p2ω2β2p1ζcot(ζψ))2(p1p2ζcot(ζψ))2. (3.14)

    Solution Group 6. For Set 2 (ζ=0), we obtain the following set of solutions for Eq (3.1):

    f10(x,t)=s2(p1p2ψ)2(1400γ2p2ω2β2p1ψ)2+s1(p1p2ψ)(1400γ2p2ω2β2p1ψ)1+s0+26βω(1400γ2p2ω2β2p1ψ)(p1p2ψ)1+s2ω2β(1400γ2p2ω2β2p1ψ)2(p1p2ψ)2. (3.15)

    In this work, we have used the Riccati-Bernoulli sub-ODE approach in conjunction with the Bäcklund transformation to obtain explicit solitonic solutions of the fractional Oskolkov equation. Although finding these solutions was possible due to the use of computational tools such as Maple, the emphasis is much more than just writing down these solutions. The solutions offer essential information on the nonlinear dynamics determined by the fractional differential operator (α), especially for the description of solitonic phenomena in multiscale systems. Thus, we build a strong theoretical background by the detailed analysis of the fractional order and the proper consideration of the results for the fluid dynamics and the wave propagation. This framework also aided in improving knowledge about the fractional Oskolkov equation as well as providing assessment of the concepts in fractional calculus and real-world applications of fractional phenomena. The novelty of this work is to connect these solutions to the physical systems and, thus, to the field of applied mathematics and theory of nonlinear waves. In the next section, we discuss the spatial visualization of wave solutions attained through the fractional Oskolkov equation. The identified solution types, trigonometric, hyperbolic, and rational, are shown in the following figures in 3D and 2D views Figures 14. This equation turns out to be extremely useful in arriving at the dimensions and geometry of thin-walled pressure vessels like tanks and reactors and hence forms a part of the standard tools used by chemical and mechanical engineers. In particular, it contributes to such vessels' design, which are destined to operate at elevated pressure and temperature, typical for severe service conditions [47]. Moreover, to predict the mechanical properties, stresses, and strains of those pressure vessels, one makes use of the Oskolkov equation. Table 1 has been created to presents side by side comparison between solutions achieved in this study and those obtained using modified Kudryashov method in other study. This table presents the difference and advantage of the current approach and shows how our work deviates from the previous method used in similar issue.

    Table 1.  Comparison of the Riccati-Bernaoulli sub-ODE along with Bäcklund transformation with the modified Kudryashov method [47].
    Present method Modified Kudryashov method
    Case I: ζ<0, f(x,t)=3/106γβ+26βω(1100γ2p2ω2β2p1ζtanh(ζψ))(p1p2ζtanh(ζψ)) M(x,t)=12α56βNN+(NJ)m±1kn(m)6β{±2(NN+(NJ)m±1k(m)6βξ)}
    ψ=5/66βωxαγαωtαα 106β3/2ω2(1100γ2p2ω2β2p1ζtanh(ζψ))2γ(p1p2ζtanh(ζψ))2 for ξ=kx6kαΓ(σ+1)5δ6βτδ
    Case II: ζ>0, f(x,t)=3/106γβ+26βω(1100γ2p2ω2β2+p1ζtan(ζψ))(p1+p2ζtan(ζψ)) M(x,t)=12α56β{±1N2{N+(NJ)m±1kln(m)6βξ}2}
    ψ=5/66βωxαγαωtαα 106β3/2ω2(1100γ2p2ω2β2+p1ζtan(ζψ))2γ(p1+p2ζtan(ζψ))2 for ξ=kx6kαΓ(σ+1)5δ6βτδ
    Case III: ζ=0, f5(x,t)=3/106γβ+26βω(1100γ2p2ω2β2p1ψ)(p1p2ψ) M(x,t)=12iα56β{±12NN+(NJ)m±ikln(m)6βξ}{(NN+(NJ)m±ikln(m)6βξ)2}
    ψ=5/66βωxαγαωtαα 106β3/2ω2(1100γ2p2ω2β2p1ψ)2γ(p1p2ψ)2 for ξ=kx6ikαΓ(σ+1)5δ6βτδ

     | Show Table
    DownLoad: CSV
    Figure 1.  Different variation analysis of the derivative parameter (α) of the solution f1(x,t).
    Figure 2.  Different variation analysis of the derivative parameter (α) of the solution f4(x,t).
    Figure 3.  Different variation analysis of the derivative parameter (α) of the solution f6(x,t).
    Figure 4.  Different variation analysis of the derivative parameter (α) of the solution f10(x,t).

    The solution for an anti-kink wave is depicted in Figure 1. Thus, one may claim that as the value of the fractional order parameter (α) increases, the wave propagation features change dramatically. More particularly, the amplitude of the solution increases manifold in the medium in which the wave propagates.

    The graphical solutions presented in Figure 1 can be discussed distinguishing some characteristics that can be significant considering wave propagation, especially in areas of fluid dynamics. Such large amplitude solutions indicate large coupling within the medium which is again important for quantifying the wave phenomena in different systems. This understanding proves very useful in areas of engineering, particularly in the estimation and design of thin-walled pressure vessels. In chemical and mechanical engineering, these designs have to be optimized with significant awareness of wave behavior at different pressures and temperatures. For instance, the solutions proffered out of the time-space fractional Oskolkov equation are instrumental in estimating stress, strain, and mechanical characteristics of materials toward enhancing the design effectiveness and reliability of engineered structures.

    Figure 2 plots the fuzziness diagnostic of the soliton solution kink type provided in this paper with the deterioration of the fractional-order derivative parameter (α). The deviation, as a result, with an application of the fractional order parameter with reduced values aligns to Figure 1 below. In the physical sense, this implies that smaller fractional orders for the reduced order model can have a beneficial effect on the dynamic characteristics of the system. In particular, the increase in the amplitudes for α=0.99 and α=0.98 compared to α=1 hints at the possibility that the interior force within the nuclear matter could be boosted by minimal fractions. Such enhancement can, therefore, result in deeper kink formations, implying improved angular density, and hence, sharper and profound state transition during the fission and fusion process inherent in nuclear reactions.

    In detail, Figure 3 illustrates the dependence of the amplitude of a kink-type solitary solution on the fractional operator parameter (α). When varying the fractional parameter, a noticeable decline of the amplitude is observed in the bottom zones, whereas the amplitude in the top zones does not get altered. Such selectiveness indicates that (α) affects the solution energy distribution in a differential manner. At a physical level, it could be the manifestation, within this model, of the selective impact of the fractional parameter on the energy of the system. The trough in the lower energy state fluctuates more due to the memory effect and the strength of the nuclei matter. On the other hand, the crests which are a location of the higher energy states are left unscathed hence there is a sense of nuclear transitions in the fission and fusion at these locations to be stable and uninterfered with. This understanding could have crucial implications for the analysis of energy interactions across behaviors, especially in structural and dynamic nuclear physical systems and engineering where wave properties impact the structure stability and energy changes.

    It is clearly seen from Figure 4 that the effect of an increase in the fractional order parameter (α) is uniform and is manifested through the decrease in the amplitude of the kink-type solitary solution at all points. From this plot, one can infer that with the decrease in memory effects and interaction strengths in nuclear matters, the total strength of the kink-type solution decreases. From a physical perspective, it shows that lesser fractional weights negatively affect the solution by providing a weaker bend. Also, notable behavior is essential for enhancing thin-wall pressure vessels utilized in high-pressure and high-temperature susceptible applications. It is useful in predicting mechanical properties such as stress and strain by applying the Oskolkov equation, thereby guaranteeing high performance and reliability, especially in operations involving fluids and nuclear- related processes.

    Here we considered the spatial nature and dynamic properties of the solutions obtained from the fractional Oskolkov equation, and especially the solitons of the kink type. It is shown that by applying systematic Bäcklund transformation and the sub-ODE of the given Riccati-Bernoulli equation, the types of different solutions derived and discussed include trigonometric, hyperbolic, and rational forms.

    As shown in our results depicted in Figures 14, a considerable difference in the amplitude and propagation of solutions is distinguished when the value of the fractional order parameter is changed. Reduction of the fractional order also improves the dynamic characteristics of the system that increases its strength within the nuclear matter and, therefore, sharpens the kink formations. These ideas have great relevance to the generation and modeling of thin-walled pressure vessels in chemical and mechanical engineering. Owing to the Oskolkov equation, it is possible to precisely forecast and regulate mechanical characteristics, including stress and strain, to increase the efficiency within high-pressure and high-temperature conditions. The given approach based on the Bäcklund transformation with the use of the Riccati-Bernoulli sub-ODE method is a solid platform for further detailed examination of emerging wave solutions in the context of fractional systems.

    The study is beneficial for developing the quantitative descriptions on the realistic wave systems with potential applications in hydrodynamics, plasma physics, and nonlinear optics. Applying conformable fractional derivatives, the work provides an enriched insight into the behavior of dynamic waves and the significance of the use of fractional-order models for a better description of the nonlocal phenomena and memory of the physical processes. It is noteworthy that the aforementioned analysis and the two proposed methods, namely the Riccati-Bernoulli sub-ODE method and Bäcklund transformation, provide efficient ways of obtaining the exact solutions for the time-space fractional Oskolkov equation. For instance, the solutions that have been found are limited by the range of fractional orders, and how these solutions might behave when the orders transcend beyond these stated limits has not been explained. Also, it must be noted that the model is still simplified and deals with the concepts of an idealized environment, and the effects of turbulence, the higher-order effects, or the interactions between multiple solitons have not been fully considered. Also it was found that using two and three dimensionals plots can help us understand the solitonic behavior of the analytically computed solutions but the experimental confirmation of these theoretical findings has been left for another future work. These limitations present the scope for coming up with additional research to enhance this approach. Future work could include looking at the theoretical and experimental results of this work and making comparisons with other nonlinear models and an in-depth analysis of boundary layers. This will further strengthen the usage as well as the relialibility of the models presented here.

    Conceptualization, M.M.A; Data curation, H.Y; Formal analysis, A.M.M; Resources, M.M.A; Investigation, H.Y; Project administration, A.M.M; Validation, M.M.A.; Software, H.Y; Validation, A.M.M; Visualization, M.M.A.; Validation, H.Y.; Visualization, M.M.A.; Resources, A.M.M.; Project administration, M.M.A.; Writingreview & editing, A.M.M; Funding, H.Y. All authors have read and agreed to the published version of the manuscript.

    This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (KFU242264).

    This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (KFU242264).

    The authors declare that they have no conflicts of interest.



    [1] Y. Gu, C. Budati, Energy-aware workflow scheduling and optimization in clouds using bat algorithm, Future Gener. Comput. Syst., 113 (2020), 106–112. https://doi.org/10.1016/j.future.2020.06.031 doi: 10.1016/j.future.2020.06.031
    [2] L. Mishra, S. Varma, Performance evaluation of real-time stream processing systems for Internet of Things applications, Future Gener. Comput. Syst., 113 (2020), 207–217. https://doi.org/10.1016/j.future.2020.07.012 doi: 10.1016/j.future.2020.07.012
    [3] X. Chen, Y. Xu, A. Liu, Cross layer design for optimizing transmission reliability, energy efficiency, and lifetime in body sensor networks, Sensors, 17 (2017), 900. https://doi.org/10.3390/s17040900 doi: 10.3390/s17040900
    [4] S. L. Chen, M. C. Tuan, H. Y. Lee, T. L. Lin, VLSI implementation of a cost-efficient micro control unit with an asymmetric encryption for wireless body sensor networks, IEEE Access, 5 (2017), 4077–4086. https://doi.org/10.1109/ACCESS.2017.2679123 doi: 10.1109/ACCESS.2017.2679123
    [5] A. H. Sodhro, M. S. Obaidat, Q. H. Abbasi, P. Pace, S. Pirbhulal, G. Fortino, et al., Quality of service optimization in an IoT-driven intelligent transportation system, IEEE Wireless Commun., 26 (2019), 10–17. https://doi.org/10.1109/MWC.001.1900085 doi: 10.1109/MWC.001.1900085
    [6] K. G. Mkongwa, Q. Liu, C. Zhang, Link reliability and performance optimization in wireless body area networks, IEEE Access, 7 (2019), 155392–155404. https://doi.org/10.1109/ACCESS.2019.2944573 doi: 10.1109/ACCESS.2019.2944573
    [7] A. H. Sodhro, S. Pirbhulal, V. H. C. De Albuquerque, Artificial intelligence-driven mechanism for edge computing-based industrial applications, IEEE Trans. Ind. Inf., 15 (2019), 4235–4243. https://doi.org/10.1109/TII.2019.2902878 doi: 10.1109/TII.2019.2902878
    [8] H. Li, Q. Zheng, W. Yan, R. Tao, X. Qi, Z. Wen, Image super-resolution reconstruction for secure data transmission in Internet of Things environment, Math. Biosci. Eng., 18 (2021), 6652–6671. https://doi.org/10.3934/mbe.2021330 doi: 10.3934/mbe.2021330
    [9] K. Babber, R. Randhawa, A cross-layer optimization framework for energy efficiency in wireless sensor networks, Wireless Sensor Network, 9 (2017), 189. https://doi.org/10.4236/wsn.2017.96011 doi: 10.4236/wsn.2017.96011
    [10] A. H. Sodhro, S. Pirbhulal, M. Qaraqe, S. Lohano, G. H. Sodhro, N. Ur R. Junejo, et al., Power control algorithms for media transmission in remote healthcare systems, IEEE Access, 6 (2018), 42384–42393. https://doi.org/10.1109/ACCESS.2018.2859205 doi: 10.1109/ACCESS.2018.2859205
    [11] Y. Zhou, Z. Sheng, C. Mahapatra, V. CM Leung, P. Servati, Topology design and cross-layer optimization for wireless body sensor networks, Ad Hoc Networks, 59 (2017), 48–62. https://doi.org/10.1016/j.adhoc.2017.01.005 doi: 10.1016/j.adhoc.2017.01.005
    [12] A. R. Bhangwar, A. Ahmed, U. A. Khan, T. Saba, K. Almustafa, K. Haseeb, et al., WETRP: Weight based energy & temperature aware routing protocol for wireless body sensor networks, IEEE Access, 7 (2019), 87987–87995. https://doi.org/10.1109/ACCESS.2019.2925741 doi: 10.1109/ACCESS.2019.2925741
    [13] I. Saidu, S. Subramaniam, A. Jaafar, Z. A. Zukarnain, An efficient battery lifetime aware power saving (EBLAPS) mechanism in IEEE 802.16 e networks, Wireless Pers. Commun., 80 (2015), 29–49. https://doi.org/10.1007/s11277-014-1993-7 doi: 10.1007/s11277-014-1993-7
    [14] R. Gravina, P. Alinia, H. Ghasemzadeh, G. Fortino, Multi-sensor fusion in body sensor networks: State-of-the-art and research challenges, Inf. Fusion, 35 (2017), 68–80. https://doi.org/10.1016/j.inffus.2016.09.005 doi: 10.1016/j.inffus.2016.09.005
    [15] A. Sangwan, P. P. Bhattacharya, An optimization to routing approach under WBAN architectural constraints, In Intelligent Systems Technologies and Applications, Springer, Cham, (2016), 75–89.
    [16] H. Karvonen, J. Iinatti, M. Hämäläinen, A cross-layer energy efficiency optimization model for WBAN using IR-UWB transceivers, Telecommun. Syst., 58 (2015), 165–177. https://doi.org/10.1007/s11235-014-9900-9 doi: 10.1007/s11235-014-9900-9
    [17] R. Talat, M. S. Obaidat, M. Muzammal, A. H. Sodhro, Z. Luo, S. Pirbhulal, A decentralised approach to privacy preserving trajectory mining, Future Gener. Comput. Syst., 102 (2020), 382–392. https://doi.org/10.1016/j.future.2019.07.068 doi: 10.1016/j.future.2019.07.068
    [18] A. H. Sodhro, S. Pirbhulal, A. Sangaiah, Convergence of IoT and product lifecycle management in medical health care, Future Gener. Comput. Syst., 86 (2018), 380–391. https://doi.org/10.1016/j.future.2018.03.052 doi: 10.1016/j.future.2018.03.052
    [19] J. Zhao, G. Li, Study on real-time wearable sport health device based on body sensor networks, Comput. Commun., 154 (2020), 40–47. https://doi.org/10.1016/j.comcom.2020.02.045 doi: 10.1016/j.comcom.2020.02.045
    [20] Y. Lin, X. Jin, J. Chen, A. H. Sodhro, Z. Pan, An analytic computation-driven algorithm for Decentralized Multicore Systems, Future Gener. Comput. Syst., 96 (2019), 101–110. https://doi.org/10.1016/j.future.2019.01.031 doi: 10.1016/j.future.2019.01.031
    [21] A. H. Sodhro, N. Zahid, L. Wang, S. Pirbhulal, Y. Ouzrout, A. Sekhari, et al., Towards ML-based Energy-Efficient Mechanism for 6G Enabled Industrial Network in Box Systems, IEEE Trans. Ind. Inf., 17 (2020). https://doi.org/10.1109/TII.2020.3026663 doi: 10.1109/TII.2020.3026663
    [22] W. Jiang, X. Ye, R. Chen, F. Su, M. Lin, Y. Ma, et al., Wearable on-device deep learning system for hand gesture recognition based on FPGA accelerator, Math. Biosci. Eng., 18 (2021), 132–153. https://doi.org/10.3934/mbe.2021007 doi: 10.3934/mbe.2021007
    [23] W. Aziz, L. Hussain, I. R. Khan, J. S. Alowibdi, M. H. Alkinani, Machine learning based classification of normal, slow and fast walking by extracting multimodal features from stride interval time series, Math. Biosci. Eng., 18 (2021), 495–517. https://doi.org/10.3934/mbe.2021027 doi: 10.3934/mbe.2021027
    [24] A. Lakhan, M. A. Dootio, A. H. Sodhro, S. Pirbhulal, T. M. Groenli, M. S. Khokhar, et al., Cost-efficient service selection and execution and blockchain-enabled serverless network for internet of medical things, Math. Biosci. Eng., 18(2021), 7344–7362. https://doi.org/10.3934/mbe.2021363 doi: 10.3934/mbe.2021363
    [25] F. Li, G. Zhou, J. Lei, Reliable data transmission in wireless sensor networks with data decomposition and ensemble recovery, Math. Biosci. Eng., 16 (2019), 4526–4545. https://doi.org/10.3934/mbe.2019226 doi: 10.3934/mbe.2019226
    [26] A. Lakhan, J. Li, T. M. Groenli, A. H. Sodhro, N. A. Zardari, A. S. Imran, et al., Dynamic application partitioning and task-scheduling secure schemes for biosensor healthcare workload in mobile edge cloud, Electronics, 10 (2021), 2797. https://doi.org/10.3390/electronics10222797 doi: 10.3390/electronics10222797
    [27] A. Lakhan, M. A. Dootio, T. M. Groenli, A. H. Sodhro, M. S. Khokhar, Multi-layer latency aware workload assignment of e-transport iot applications in mobile sensors cloudlet cloud networks, Electronics, 10 (2021), 1719. https://doi.org/10.3390/electronics10141719 doi: 10.3390/electronics10141719
    [28] L. Cui, C. Xu, S. Yang, J. Z. Huang, J. Li, X. Wang, et al., Joint optimization of energy consumption and latency in mobile edge computing for Internet of Things, IEEE Internet Things J., 6 (2018), 4791–4803. https://doi.org/10.1109/JIOT.2018.2869226 doi: 10.1109/JIOT.2018.2869226
    [29] C. Iwendi, J. H. Anajemba, C. Biamba, D. Ngabo, Security of things intrusion detection system for smart healthcare, Electronics, 10 (2021), 1375. https://doi.org/10.3390/electronics10121375 doi: 10.3390/electronics10121375
    [30] N. Zahid, A. H. Sodhro, R. F. Zafar, B. Zahid, S. A. Khan, F. Akhter, Regression-based transmission power control for green healthcare, in 2019 2nd international conference on computing, mathematics and engineering technologies (iCoMET), IEEE, 2019. https://doi.org/10.1109/ICOMET.2019.8673532
    [31] A. H. Sodhro, N. Zahid, AI-enabled framework for fog computing driven e-healthcare applications, Sensors, 21 (2021), 8039. https://doi.org/10.3390/s21238039 doi: 10.3390/s21238039
    [32] S. T. Abbas, H. J. Mohammed, J. S. Ahmed, A. S. Rashid, B. Alhayani, A. Alkhayyat, The optimization efficient energy cooperative communication image transmission over WSN, Appl. Nanosci., (2021), 1–13. https://doi.org/10.1007/s13204-021-02100-2 doi: 10.1007/s13204-021-02100-2
    [33] A. H. Sodhro, Y. Li, M. A. Shah, Energy-efficient adaptive transmission power control for wireless body area networks, IET Commun., 10 (2016), 81–90. https://doi.org/10.1049/iet-com.2015.0368 doi: 10.1049/iet-com.2015.0368
    [34] A. Alkhayyat, S. F. Jawad, S. B. Sadkhan, Cooperative communication based: Efficient power allocation for wireless body area networks, in 2019 1st AL-Noor International Conference for Science and Technology (NICST), IEEE, (2019), 106–111. https://doi.org/10.1109/NICST49484.2019.9043843
    [35] A. H. Sodhro, M. S. Al-Rakhami, L. Wang, H. Magsi, N. Zahid, S. Pirbhulal, et al., Decentralized energy efficient model for data transmission in IoT-based healthcare system, in 2021 IEEE 93rd Vehicular Technology Conference (VTC2021-Spring), IEEE, (2021), 1–5. https://doi.org/10.1109/VTC2021-Spring51267.2021.9448886
    [36] J. F. A.Rida, A. Alkhayyat, Remote Health Care based on mobile wireless communication Networks, J. Appl. Sci. Eng., 24 (2021), 799–805. https://doi.org/10.6180/jase.202110_24(5).0016 doi: 10.6180/jase.202110_24(5).0016
    [37] A. H. Sodhro, S. Pirbhulal, G. H. Sodhro, A. Gurtov, M. Muzammal, Z. Luo, A joint transmission power control and duty-cycle approach for smart healthcare system, IEEE Sensors J., 19 (2018), 8479–8486. https://doi.org/10.1109/JSEN.2018.2881611 doi: 10.1109/JSEN.2018.2881611
    [38] A. H. Sodhro, L. Chen, A. Sekhari, Y. Ouzrout, W. Wu, Energy efficiency comparison between data rate control and transmission power control algorithms for wireless body sensor networks, Int. J. Distrib. Sensor Networks, 14 (2018), 1550147717750030. https://doi.org/10.1177/1550147717750030
    [39] L. Hanlen, V. Chaganti, B. Gilbert, D. Rodda, T. Lamahewa, D. Smith, Open-source testbed for body area networks: 200 sample/sec, 12 hrs continuous measurement, in 2010 IEEE 21st International Symposium on Personal, Indoor and Mobile Radio Communications Workshops, (2010), 66–71. https://doi.org/10.1109/PIMRCW.2010.5670518
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4373) PDF downloads(268) Cited by(30)

Figures and Tables

Figures(9)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog