System Of Nonlinear Partial Differential Equations Research Articles

Outlier-resistant non-fragile observer-based fuzzy boundary control for interconnected nonlinear PDE systems

This paper investigates an observer-based boundary controller design for interconnected nonlinear partial differential equation (PDE) systems. First, the Takagi–Sugeno (T–S) fuzzy model is adopted to accurately describe the target systems. Then, boundary measurements are employed to reduce the number of sensors. Next, considering the phenomenon of abnormal interference that may lead to measurement outliers and observer parameters’ uncertainties, an outlier-resistant non-fragile observer expressed by a saturation function is designed to guarantee the desired control objectives. Moreover, the boundary control approach is employed to trade-off the cost of system design and system performance. Furthermore, utilizing the membership function-dependent Lyapunov functions and free-weight matrixes, sufficient conditions ensuring the closed-loop systems’ exponential stability are obtained while decreasing the conservativeness of the system stability analysis. Finally, the proposed method’s feasibility and effectiveness are validated by an example.

Nonlinear thermal radiation analysis for bioconvection peristaltic transport of fourth-grade nanofluid

Nanoparticles mixed with base liquids result in the formation of nanofluids, known for their exceptional thermal conductivity and their ability to enhance the performance of electrical and mechanical devices. Therefore, this study investigates the peristaltic activity of fourth-grade nanomaterial by taking the consideration of magnetic aspects. Analysis is carried out in the presence of motile gyrotactic microorganisms, Ohmic heating, and viscous dissipation. Thermal transport incorporated effects of nonlinear thermal radiation along with Brownian motion and thermophoresis. Thermal and mass convective conditions are imposed on elastic symmetric channels. The assumptions of low Reynolds number and long wavelength approximations have led to the simplification of the nonlinear partial differential equations system into ordinary differential equations. The governing system is solved numerically by NDSolve technique via Mathematica. Graphs are plotted and examined for velocity, thermal field, density of motile gyrotactic microorganisms, and heat transfer rate.

Modeling viscous dissipation in MHD boundary layer flow: A two-stage in time and compact in space finite difference approach

It is suggested that an explicit-implicit third-order in-time numerical scheme be used to solve time-dependent partial differential equations (PDEs). A fourth-order compact scheme in space is adopted for discretizing space variables. The work creatively blends a third-order explicit-implicit numerical scheme in time with a fourth-order compact scheme in space to solve time-dependent PDEs. Notably, the proposed numerical scheme demonstrates stability for scalar PDEs and exhibits convergence for both linear and nonlinear PDE systems. In addition, a mathematical model is given for the heat transfer of magnetohydrodynamic (MHD) flow over flat and oscillatory sheets, and the effect of viscous dissipation is also considered. For various governing parameters, features of the flow and heat transfer characteristics are discussed and examined. Some results are verified using an exact solution. We use an explicit-implicit finite difference method guaranteed to be stable and convergent under all conditions to solve the coupled nonlinear partial differential equations that describe this problem. Physical aspects of the problem are discussed in light of the system’s embedded parameters, and numerical and graphical results are presented. To sum up, progress in our knowledge of such systems relies heavily on creating an accurate and efficient numerical scheme for modeling MHD boundary layer flows while considering the influence of viscous dissipation. This study aims to improve the current state of MHD flow simulations and provide a useful resource for engineers and scientists engaged in developing and improving MHD-based technologies.

PDE-based control synthesis for a planar cable-driven continuum arm

Classical wave and beam models cannot describe the arc-like planar configuration of a continuum arm. Based on the Cosserat rod theory, a strain vector and the rotation matrix are introduced into this work. Two groups of nonlinear partial differential equations (PDEs) are obtained by analyzing kinetic and potential energy densities and virtual power done by the cable tension, which are explicit both in the spatial coordinates and in the physical parameters. Although involving algebraic–trigonometric descriptions, the obtained dynamical model is the most concise formulation but difficult to use for control design. We develop a feedforward plus proportional–derivative feedback boundary control scheme for the nonlinear PDE system to regulate the planar shape of the continuum arm and then linearize the dynamical equations about the equilibrium to assess the control performance. Some interesting results are the closed-loop stability and the well-posedness analysis of the linearized model, which have not previously been noted. Finally, numerical results validate the dynamical equations and the performance of the controller.

Polynomial Fuzzy Observer-Based Feedback Control for Nonlinear Hyperbolic PDEs Systems.

This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler's homogeneous theorem, Lyapunov-Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.

Dynamics of Tangent Hyperbolic Fluid Past a Semi-infinite Plate with the Significance of Joule Heating, Thermal Radiation and Soret-Dufour Mechanisms

The present investigation concentrates on the unsteady flow of tangent hyperbolic liquid past a vertical plate under the influence of Lorentz force, Joule heating, and viscous dissipation. The mathematical modelling leads to nonlinear coupled partial differential equations (PDEs). Suitable non-dimensional quantities are applied to the governing PDEs to obtain dimensionless systems of equations. The transformed boundary layer PDEs are solved with the aid of the spectral relaxation method (SRM). The SRM employs the Gauss-Seidel techniques to linearize and decouple the system of nonlinear PDEs. The applied magnetic field acts as an opposition to the flow by producing the Lorentz force. The Weissenberg parameter, alongside the magnetic parameter, is observed to decline the velocity profile. An increment in thermal radiation parameter is observed to enhance the thickness of the hydrodynamic and thermal boundary layer. Therefore, the thermal condition and convective flow are improved with heat generation and thermal radiation in the flow phenomenon. This investigation is unique because it investigates the combined influence of Soret-Dufour and MHD, viscous dissipation, and Joule heating. This study plays a significant role in astrophysics, heat exchanger devices, MHD power generation, and geothermal energy extraction. When this study is compared to studies that have already been done, it agrees with those studies.

Thermal Analysis on Kerosene Oil-Based Two Groups of Ternary Hybrid Nanoparticles (CNT-Gr-Fe3O4 and MgO-Cu-Au) Mix Flow over a Bidirectional Stretching Sheet: A Comparative Approach

The present work examines the impressions of radiation impact on the three-dimensional non-Newtonian MHD Casson flow of ternary hybrid nanofluids over a symmetrical stretching sheet with magnetic impression and heat generation/absorption. The unique boost in thermal efficiency and development of the rate of heat transport as valid to the dynamics of energy and coolant in automobiles is what has led to an increase in knowledge of hybrid nanofluid. For the study, two groups of ternary nanoparticles (CNT-Gr-Fe3O4 and MgO-Cu-Au) are combined with the base fluid kerosene oil. A nonlinear partial differential equation system is created while keeping in mind some reasonable presumptions. Using the similarities transformation, PDE’s are changed into nonlinear ODE’s. Also, it is then mathematically simplified with the bvp4c technique. The consequences of an exclusive group of unique impacts on motion characteristics, skin friction coefficient, thermal field impressions, heat transport rate, concentration distribution, and mass transfer rate are described clearly. The motion in the x and y directions decays with increasing the Casson fluid parameter 0.04 ≤ β ≤ 0.06 and magnetic impact 6 ≤ M ≤ 10 for ternary groups I and II. An energy upsurge profile appears for radiation impression (1 ≤ Nr ≤ 20) and heat source/sink (0.1 ≤ Q ≤ 1.5). When compared to rising Dufour number (0.1 ≤ Du ≤ 0.9) and heat source/sink values (0.1 ≤ Q ≤ 1.5), the Nusselt number decreases. The volume fraction 0.1 ≤ ϕ ≤ 0.3 of ternary nanoparticles rises the velocity (in both directions) and thermal distributions. Also, the Nusselt number enhances for both ternary groups when increasing thermal radiation (1 ≤ Nr ≤ 20).

Theoretical Investigation of the Nonlinear General Rate Model with the Bi-Langmuir Adsorption Isotherm Using Core-Shell Adsorbents.

Core-shell particles enable the separation of intricate mixtures in a highly efficient and rapid manner. The porous shell particles increased the separation efficiency with expedited flow rates due to an abatement in the pore volume accessible for longitudinal diffusion and a decrease in diffusion path length. This study focuses on the numerical approximation of a nonlinear isothermal general rate model applied to stationary bed columns that are replete with inert core adsorbents featuring double adsorption sites. The transport of solute in heterogeneous porous media can be modeled by a nonlinear convection acquiescent partial differential equation system together with a specific nonlinear algebraic relation: the bi-Langmuir adsorption isotherm. Therefore, it is important to develop accurate and reliable numerical techniques that can perform accurate numerical simulations of these models. We extended and implemented a second-order, semidiscrete, high-resolution finite volume method to simulate the governing equations of the model. Single solute flow and multi component mixture flows are assessed through a series of numerical experiments to theoretically illustrate the repercussions of intraparticle diffusion, film mass resistance, axial dispersion, and the size of the inert core radius upon simulated elution curves. Standard performance criteria are assessed to determine the optimal core radius fraction range for optimizing the separation performance.

Numerical analysis of radiative hybrid nanomaterials flow across a permeable curved surface with inertial and Joule heating characteristics

The water-based Cu and CoFe2O4 hybrid nano liquid flow across a permeable curved sheet under the consequences of inertial and Lorentz forces has been reported in this analysis. The Joule heating and Darcy Forchheimer effects on fluid flow have been also examined. In the presence of copper (Cu) and cobalt iron oxide (CoFe2O4) nanoparticles, the hybrid nano liquid is synthesized. Radiation and heat source features are additionally incorporated to perform thermodynamics analysis in detail. The second law of thermodynamics is employed in order to estimate the overall generation of entropy. The nonlinear system of PDEs (partial differential equations) is transformed into a dimensionally-free set of ODEs (ordinary differential equations) by employing a similarity framework. The Mathematica built in package ND Solve method is applied to compute the resulting set of nonlinear differential equations numerically. Along with the velocity, and temperature profiles, skin friction and Nusselt number are also computed. Figures and tables illustrate the effects of flow factors on important profiles. Evidently, the outcomes reveal that hybrid nanofluid (Cu + CoFe2O4+H2O) is more progressive than nanofluid (Cu + H2O) and base fluid (H2O) in thermal phenomena. Furthermore, the velocity profile is improved with the greater values of curvature parameter, while the inverse trend is observed against the magnetic parameters. Also, the velocity and energy distribution of hybrid nano-liquid flow boosts with the inclusion of Cu and CoFe2O4 nanoparticles into the base fluid. Velocity distribution diminishes with the increment of volume friction. For high values of inertial factor, skin friction improve while velocity and Nusselt number declines.

Numerical investigations of activation energy on the peristaltic transport of Carreau nanofluid through a curved asymmetric channel

Nanometric particles with base liquids cause the production of nanofluids, which are distinguished by their outstanding thermally conductive fluid properties and the expansion of electrical and mechanical devices. Based on these considerations, we devised a study to investigate the effect of activation energy on the peristaltic motion of Carreau nanofluid inside a curved asymmetric channel under the influence of a magnetic field. The governing equations for the curved channel of non-Newtonian fluid flow are formulated. The nonlinear partial differential equations system has been reduced to ordinary differential equations by the assumptions of low Reynolds number and long wavelength approximations. The resulting nonlinear coupled differential equations are numerically solved directly using NDSolve (numerical differential equation solver) coding of computational mathematical software Mathematica, and velocity, temperature, concentration, and streamlines are plotted. With graphical demonstrations, the influence of essential parameters on velocity, temperature, concentration, and streamlines is explained in detail. The dimensionless temperature distribution grows as the activation energy parameter grows. In reality, the number of energetic particles (with energies equal to or greater than activation energy) increases, resulting in improved temperature distribution.

Observer-Based Boundary Fuzzy Control Design of Nonlinear Parabolic PDE Systems Using Mobile Sensors

This paper studies the boundary fuzzy control problem for nonlinear parabolic partial differential equation (PDE) systems under spatially noncollocated mobile sensors. In a real setup, sensors and actuators can never be placed at the same location, and the noncollocated setting may be beneficial in some application scenarios. The control design is very difficult due to the noncollocated mobile observation, which can be solved by an observer-based technique. At first, a Takagi-Sugeno fuzzy PDE model is devoted to accurately representing the nonlinear parabolic PDE system. Next, we present a state estimation scheme including fuzzy Luenberger-type PDE state observer plus mobile sensor guidance. Then, an observer-based boundary fuzzy controller is posed to render the resulting closedloop system exponentially stable, and the exponential decay rate is increased by the designed mobile sensor guidance laws. At last, two examples verify the proposed method.

An IMEX-based approach for the pricing of equity warrants under fractional Brownian motion models

In this paper, the pricing of equity warrants under a class of fractional Brownian motion models is investigated numerically. By establishing a new nonlinear partial differential equation (PDE) system governing the price in terms of the observable stock price, we solve the pricing system effectively by a robust implicit-explicit numerical method. This is fundamentally different from the documented methods, which first solve the price with respect to the firm value analytically, by assuming that the volatility of the firm is constant, and then compute the price with respect to the stock price and estimate the firm volatility numerically. It is shown that the proposed method is stable in the maximum-norm sense. Furthermore, a sharp theoretical error estimate for the current method is provided, which is also verified numerically. Numerical examples suggest that the current method is efficient and can produce results that are, overall, closer to real market prices than other existing approaches. A great advantage of the current method is that it can be extended easily to price equity warrants under other complicated models. doi: 10.1017/S1446181123000159

Conformable mathematical modeling of the COVID‐19 transmission dynamics: A more general study

Many challenges are still faced in bridging the gap between mathematical modeling and biological sciences. Measuring population immunity to assess the epidemiology of health and disease is a challenging task and is currently an active area of research. However, to meet these challenges, mathematical modeling is an effective technique in shaping the population dynamics that can help disease control. In this paper, we introduce a susceptible–infected–recovered (SIR) model and a susceptible–exposed–infected–recovered–deceased (SEIRD) model based on conformable space‐time partial differential equations (PDEs) for the coronavirus disease 2019 (COVID‐19) pandemic. As efficient analytical tools, we present new modifications based on the fractional exponential rational function method (ERFM) and an analytical technique based on the Adomian decomposition method for obtaining the solutions for the proposed models. These analytical approaches are more efficacious for obtaining analytical solutions for nonlinear systems of PDEs with conformable derivatives. The interesting result of this paper is that it yields new exact and approximate solutions to the proposed COVID‐19 pandemic models with conformable space‐time partial derivatives.

Dynamics and steady state of squirmer motion in liquid crystal

AbstractWe analyse a nonlinear partial differential equation system describing the motion of a microswimmer in a nematic liquid crystal environment. For the microswimmer’s motility, the squirmer model is used in which self-propulsion enters the model through the slip velocity on the microswimmer’s surface. The liquid crystal is described using the well-established Beris–Edwards formulation. In previous computational studies, it was shown that the squirmer, regardless of its initial configuration, eventually orients itself either parallel or perpendicular to the preferred orientation dictated by the liquid crystal. Furthermore, the corresponding solution of the coupled nonlinear system converges to a steady state. In this work, we rigorously establish the existence of steady state and also the finite-time existence for the time-dependent problem in a periodic domain. Finally, we will use a two-scale asymptotic expansion to derive a homogenised model for the collective swimming of squirmers as they reach their steady-state orientation and speed.

Numerical study of MHD radiative Williamson fluid over an inclined moving plate covering of water-based hybrid nanomaterial

Modern developments in nanotechnology have provided a fantastic foundation for creating a better ultra-high-performing coolant known as nanofluids for many applications in manufacturing and engineering. Numerous scholars have been drawn in by the hybrid nanomaterials’ capacity to improve heat transmission more to examine the working fluid. This study uses the mathematical solution to explore the effect of viscous dissipative on the radiative-convective Williamson hybrid nanofluid flow through an angled moving plate with a magnetic impression. Focusing on certain reasonable assumptions, a nonlinear partial differential equation system is developed and then numerically solved using the bvp4c method. It is thoroughly explained how a particular collection of distinctive factors affects the motion features, shear stress, temperature field profiles and heat transfer. The motion declined with increasing Williamson fluid for both stretching and shrinking surfaces. An upsurge profile for energy is showing for radiation impression. The shear rate falls with the inclined plate and increases with the buoyancy impression.

Analysis of a viscoelastic fluid flow with Cattaneo–Christov heat flux and Soret–Dufour effects

The present study concentrates on the motion of a third-grade fluid with magneto-hydrodynamics (MHD) over a stretching surface. Additionally, the Cattaneo–Christov model is employed to derive information about the heat flux, which is subsequently utilized to ascertain the heat transmission properties. The inquiry involves an assessment of the impact of thermal relaxation time on the boundary layer. The considered fluid is a type of non-Newtonian fluid that deal with both shear thinning and shear thickening behaviors. The study also involved calculating all normal stress components that are not accounted for by either the power law model or the second grade model. The motion of fluid is attributed by linear stretching, and the momentum flow is observed with the help of Soret–Dufour effects. The Cattaneo and Christov model is particularly beneficial for describing transmission of heat in materials with high thermal conductivity, where the classical Fourier’s law may not be accurate. The examination of the transport properties of velocity, heat, and mass involves the consideration of the temperature effect on variable viscosity and thermal conductivity, which are linearly associated with each other. A set of partial differential equations (PDEs) that are nonlinear in nature has been derived, and some boundary conditions have been identified that yield satisfactory results. Using similarity transformation a system of nonlinear PDE’s are modify into dimensionless system of ordinary differential equations (ODE’s). The homotopy analysis method (HAM) is applied in a convenient manner to obtain solutions for transform equations, while the impact of relevant flow parameters is visually demonstrated. Adequate graphical and tabular results are achieved using a semi-analytical approach. Moreover, the Cattaneo–Christov equation can provide a better understanding of the thermal behavior. It allows for a more comprehensive analysis of the thermal response, ensuring that the heat transfer models used are appropriate for such conditions.

Dynamic Fuzzy Boundary Output Feedback Control for Nonlinear Delayed Parabolic Partial Differential Equation Systems Under Noncollocated Boundary Measurement

For nonlinear space-varying parabolic partial differential equation systems (PPDESs) with random time-varying delay, this paper introduces a dynamic fuzzy boundary output feedback (DFBOF) control under non-collocated boundary measurement (NCBM). Initially, the nonlinear delayed PPDESs are represented by Takagi-Sugeno (T–S) fuzzy models and random time-varying delay is considered by taking the influence of uncertain factors, which belongs to two intervals in a probabilistic way. Since the system state is not fully available and NCBM makes the boundary control design very difficult, a fuzzy observer under NCBM is presented to surmount the design difficulty. Subsequently, an observer-based fuzzy boundary controller is proposed and spatial linear matrix inequality (SLMI) based sufficient conditions to ensure mean square exponential stability are obtained for closed-loop delayed PPDESs by utilizing the Lyapunov direct method and Wirtinger inequality. Then, to solve the SLMIs, the feasibility conditions of DFBOF controller design for nonlinear delayed PPDES are expressed in LMIs. Lastly, two examples are offered to demonstrate the validity of the presented dynamic fuzzy boundary control approach.

Sub-optimal controller design for time-delay nonlinear partial differential equation systems: an extended state-dependent differential Riccati equation approach

ABSTRACT In this article, a sub-optimal controller based on Extended State-Dependent Differential Riccati Equation (ESDDRE) is suggested for a class of time-delay nonlinear Partial Differential Equation (PDE) systems. At first, an extended pseudo linearisation presentation for parameterisation form using State-Dependent Coefficients (SDC) is proposed. In this presentation, all the time-delay parts are placed in system matrices as well as in input vectors. By defining a cost function and a Hamiltonian related to the PDE systems, the sub-optimal control law regarded on the ESDDRE is obtained. The stability of the closed-loop system based on the ESDDRE control approach is ensured by using a proper Lyapunov function and also Poincaré inequality. Numerical simulation results for three time-delay nonlinear PDE systems illustrate the appropriate performance of the proposed control approach.

Boundary Layer Flow of Williamson Hybrid Ferrofluid over A Permeable Stretching Sheet with Thermal Radiation Effects

This research investigated the convective boundary layer flow and heat transfer of Williamson hybrid ferrofluid over a permeable stretching sheet with thermal radiation effects. Human blood is employed as a based fluid while magnetite (Fe3O4) and copper (Cu) are taken as the hybrid ferroparticle. The study started with transforming the non-linear partial differential equation system that governed the model to a more convenience non-linear dimensionless ordinary differential equations using the similarity transformation. The transformed equations obtained then are solved numerically using the Runge-Kutta-Fehlberg (RKF45) method in Maple software. The characteristics and effects of stretching parameter, permeability parameter, thermal radiation parameter as well as the ferroparticle volume fraction in the Williamson hybrid ferrofluid towards the temperature profiles, velocity profiles as well as the Nusselt number and the skin friction coefficient are analysed and discussed. The result of this research for various pertinent parameter varies differently. It can be concluded that the increase in magnetic parameter, the Williamson parameter, the stretching parameter, and the permeability rate parameter increase the skin friction and reduced the velocity profile. Furthermore, the increase in stretching parameter, thermal radiation parameter and the permeability rate results to the increase in the Nusselt number.

Application of Iterative Three-Dimensional Laplace Transform Method for 2-Dimensional Nonlinear Klein-Gordon Equation

In the present study, the exact analytical solutions of the 2-dimensional nonlinear Klein-Gordon equation (NLKGE) is investigated using the 3-dimensional Laplace transform method in conjunction with the Daftardar-Gejji and Jafari Method (Iterative method). Through this method, the linear part of the problem is solved by using the 3-dimensional Laplace transform method, while the noise terms from the nonlinear part of the equation disappear through a successive iteration process of the Daftardar-Gejji and Jafari Method (DJM), where a single iteration gives the exact solution to the problem. Three test modeling problems from mathematical physics nonlinear Klein-Gordon equations are taken to confirm the performance and efficiency of the presented technique. For each illustrative example, the convergence of the novel iterative approach is shown. The findings also suggest that the proposed method could be applied to other types of nonlinear partial differential equation systems. HIGHLIGHTS Partial differential equations play a crucial role in the formulation of fundamental laws of nature and the mathematical analysis in different fields of studies The Klein-Gordon equation (KGE) is a nonlinear hyperbolic class of partial differential equations that occur in relativistic quantum mechanics and field theory, both of which are crucial to high-energy physicists The 3-dimensional Laplace transform method provides rapid convergence of the exact solution without any restrictive assumptions about the solution for linear differential equations The Laplace Transform method is often combined with other methods like the Daftardar-Gejji and Jafari (the new Iterative) method to solve complicated nonlinear differential equations The new iterative technique yields a series that can be summed up to get an analytical formula or used to build an appropriate approximation with a faster convergent series solution GRAPHICAL ABSTRACT