Coupled Equations Research Articles

The evolution of grain boundary structure mediated by disclinations in magnesium alloys under superplastic deformation

Superplastic deformation in metals and alloys, characterized by ultrahigh ductility (exceeding 300%) without cracking at elevated temperatures, is a critical process for manufacturing complex-shaped components. While a few grain-boundary (GB)-mediated deformation mechanisms have been identified as essential contributors to superplasticity in fine-grained polycrystals (grain size is typically less than 10 μm), it is still a challenge to maintain a steady fine-grained microstructure and sustainable plastic flow at high temperatures. Partially due to the lack of a quantitative description of dislocation-GB reactions, it has not been well recognized how grain coarsening can be suppressed by the external loading during superplastic deformation. In this work, we address this challenge by formulating a disclination-dislocation coupling equation within the Lie-algebra framework, providing a quantitative understanding of the interactions between disclinations, dislocations, and GBs. Using quasi-in-situ electron backscattered diffraction (EBSD) analysis in Mg alloys, we systematically investigate the multiscale interactions of the defects and their impact on grain structure evolution. Three key mechanisms that suppress conventional grain coarsening have been identified, i.e., disclination-assisted GB accommodation, disclination-GB pinning, and disclination-induced sub-GB crossing, all of which are captured by the proposed equation. This study contributes to the broader field of plasticity by linking macroscopic deformation behavior with microscopic mechanisms, offering new insights into the theory of superplastic deformation in metals and alloys.

Numerical Treatment of the Coupled Fredholm Integro-Differential Equations by Compact Finite Difference Method

Numerical Treatment of the Coupled Fredholm Integro-Differential Equations by Compact Finite Difference Method

Numerical analysis of vortex-induced vibration of deepwater drilling riser based on Van der Pol wake oscillator model

Coupled equations of the dynamic model and the Van der Pol wake oscillator model are solved by the central finite difference method for the deepwater drilling riser. The vortex-induced vibration (VIV) response and fatigue life of the riser are calculated and the model validation is performed by comparing with the published experimental and simulation results. The effects of the top tension, current flow velocity, flow velocity profile, internal flow velocity as well as the installation of buoyancy modules on the VIV are discussed. Dynamic response and fatigue life of the riser under In-Line (IL) and Cross-Flow (CF) VIV are preliminarily analyzed. The results show that the VIV of the riser has multiple modes and exhibits a mixed behavior of standing waves and traveling waves. With the increase of top tension and flow velocity profile coefficient, the order of the VIV dominant mode and the peak values of the root mean square (RMS) of VIV displacement decrease, and the fatigue life of the riser is extended. With the increase of current flow velocity, the order of the VIV dominant mode increases and the riser fatigue life decreases. The effect of internal flow velocity on the riser VIV is neglectable. The installation of buoyancy modules can improve the riser stress state and extend the fatigue life. Compared with the CF VIV model, the calculated minimum fatigue life of the riser is extended under the IL and CF coupled VIV model due to the decrease of bending moment and the changing position of fatigue weak point.

Optimal Control of a Domestic Frost-Free Refrigerator Inlet Freezer from the Perspective of Reducing Energy Consumption

This study is dedicated to the development of a mathematical model based on shape optimal control of the inlet domestic frost-free refrigerator in the context of energy consumption reduction. A three-dimensional thermofluid model is established for the numerical studies. As the physical system (with shelves and fruits) can be seen as a porous medium, the coupled state equations of the transport mechanism (for velocity and temperature fields) are governed by Navier–Stokes–Forchheimer/Fourier equations. Then, based on the finite element method, the numerical scheme computation is made with FreeFem++ software (Version 4.6). Numerical results are shown for three different cases: empty without shelves, empty with shelves, and loaded with foods. For each case, the velocity and temperature fields’ results are discussed for the optimal configurations. The characterization of these physical parameters would help engineers in domestic frost-free refrigerator design.

Formation of Optical Fractals by Chaotic Solitons in Coupled Nonlinear Helmholtz Equations

Formation of Optical Fractals by Chaotic Solitons in Coupled Nonlinear Helmholtz Equations

New and Modified Homotopy Perturbation Methods for Addressing Burger’s Non-linear Equation

Objectives: The aim of this study is to find a novel solution procedure for solving a fluid dynamical problem, especially to solve two-dimensional coupled Burger’s non-linear equation. Methods: New and Modified homotopy perturbation techniques are used to solve the two-dimensional coupled Burger’s equation. The methods intend to make homotopy perturbation method a more robust and trustworthy tool for fluid dynamics researchers by addressing convergence concerns, improving solution accuracy and allowing it to handle a broader range of problems. Findings: The solution for two-dimensional coupled Burger’s non-linear equation is obtained by constructing homotopies. Numerical results thus obtained are compared with the exact solution. The obtained results and the exact solutions to the stated problem are closely related. Novelty: The New and Modified Homotopy Perturbation Method proposed in this paper are effective mathematical tools for getting an exact solution to the coupled Burgers' equations. It is also a potential approach for solving various linear and nonlinear partial differential equations. Keywords: Perturbation, Homotopy, New homotopy perturbation, Modified homotopy perturbation, Two­dimension, Linear, Coupled burgers' equation

Analytical Solutions of the Fractional Hirota–Satsuma Coupled KdV Equation along with Analysis of Bifurcation, Sensitivity and Chaotic Behaviors

Analytical Solutions of the Fractional Hirota–Satsuma Coupled KdV Equation along with Analysis of Bifurcation, Sensitivity and Chaotic Behaviors

Levinson–Smith Dissipative Equations and Geometry of GENERIC Formalism and Contact Hamiltonian Mechanics

We apply Jacobi’s Last Multiplier theory to construct the non-standard Lagrangian and Hamiltonian structures for the Levinson–Smith equations satisfying the Chiellini integrability condition. Then after a brief exposition of the contact geometry, we explore its connection with the non-standard Hamiltonian structures. We present the formulation of the Levinson–Smith equation in terms of General Equation for the Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) method and also study the gradient-type flow. We give a geometric formulation of GENERIC and apply this to general Levinson–Smith equations.

Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy Jumps

We establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have q players and are driven by a general-dimensional vector Lévy process. By establishing a vector-form Ito-Ventzell formula and a 4-tuple vector-field solution to the unified SPDE, we obtain a Pareto optimal Nash equilibrium policy process or a saddle point policy process to the SDG in a non-zero-sum or zero-sum sense. The unified SPDE is in both a general-dimensional vector form and forward–backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position parameters over a domain. Since the unified SPDE is of general nonlinearity and a general high order, we extend our recent study from the existing Brownian motion (BM)-driven backward case to a general Lévy-driven forward–backward coupled case. In doing so, we construct a new topological space to support the proof of the existence and uniqueness of an adapted solution of the unified SPDE, which is in a 4-tuple strong sense. The construction of the topological space is through constructing a set of topological spaces associated with a set of exponents {γ1,γ2,…} under a set of general localized conditions, which is significantly different from the construction of the single exponent case. Furthermore, due to the coupling from the forward SPDE and the involvement of the discontinuous Lévy jumps, our study is also significantly different from the BM-driven backward case. The coupling between forward and backward SPDEs essentially corresponds to the interaction between noise encoding and noise decoding in the current hot diffusion transformer model for generative AI.

Semi-analytical layerwise solutions for FG MEE complex shell structures

This paper presents a unique analytical layerwise solution for functionally graded magneto-electric-elastic shell with complex geometry. The middle surface of the shell is modeled by a parametric equation and the Lamé parameter and radii of curvature is modeled by Differential Geometry. The mechanical displacements, electrical and magnetic potentials are written in term of a simple cosine layerwise based on a unified formulation. The highly coupled differential equations are discretized by the Chebyshev-Gauss-Lobatto grid points and solved numerically via the Differential Quadrature Method (DQM). Lagrange interpolation polynomials are employed as the basis functions. The highly coupled differential equations are solved for shells subjected to different loads and boundary conditions. Since extremely few results on this topic is available in the literature, benchmarks complex shell problems and their solutions are introduced in this paper for the very first time.

Global quaternion generalized minimal residual method for generalized coupled Sylvester quaternion matrix equations with application to colour image encryption and decryption

ABSTRACT Sylvester matrix equations play important roles in control and system theory. We consider a class of the generalized coupled Sylvester quaternion matrix equations, which includes a lot of special matrix equations such as Stein matrix equation, Lyapunov matrix equation and Sylvester matrix equation on the quaternion skew field. First, the quaternion Krylov subspace and its F-orthogonal basis are constructed via a new quaternion matrix product and the global quaternion Arnoldi process. Then a global quaternion generalized minimal residual (Gl-QGMRES) method is proposed for solving the generalized coupled Sylvester quaternion matrix equations. The convergence of the proposed method is theoretically analysed. Finally, the effectiveness and feasibility of the proposal are verified by some numerical experiments. As an application of Gl-QGMRES method, we establish a kind of colour image encryption and decryption by embedding watermark images.

A new thermo-optical system with a fractional Caputo operator for a rotating spherical semiconductor medium immersed in a magnetic field

PurposeUnderstanding the mechanical and thermal behavior of materials is the goal of the branch of study known as fractional thermoelasticity, which blends fractional calculus with thermoelasticity. It accounts for the fact that heat transfer and deformation are non-local processes that depend on long-term memory. The sphere is free of external stresses and rotates around one of its radial axes at a constant rate. The coupled system equations are solved using the Laplace transform. The outcomes showed that the viscoelastic deformation and thermal stresses increased with the value of the fractional order coefficients.Design/methodology/approachThe results obtained are considered good because they indicate that the approach or model under examination shows robust performance and produces accurate or reliable results that are consistent with the corresponding literature.FindingsThis study introduces a proposed viscoelastic photoelastic heat transfer model based on the Moore-Gibson-Thompson framework, accompanied by the incorporation of a new fractional derivative operator. In deriving this model, the recently proposed Caputo proportional fractional derivative was considered. This work also sheds light on how thermoelastic materials transfer light energy and how plasmas interact with viscoelasticity. The derived model was used to consider the behavior of a solid semiconductor sphere immersed in a magnetic field and subjected to a sudden change in temperature.Originality/valueThis study introduces a proposed viscoelastic photoelastic heat transfer model based on the Moore-Gibson-Thompson framework, accompanied by the incorporation of a new fractional derivative operator. In deriving this model, the recently proposed Caputo proportional fractional derivative was considered. This work also sheds light on how thermoelastic materials transfer light energy and how plasmas interact with viscoelasticity. The derived model was used to consider the behavior of a solid semiconductor sphere immersed in a magnetic field and subjected to a sudden change in temperature.

Steric effect induces enhancement of electroconvective flow near electrochemical interfaces.

Electroconvection, occurring near electrochemical interfaces, propels the movement of ions and water, leading to intricate phenomena rooted in the fine interplay between fluid, voltage, and ion. Here, neglecting ionic interactions, by incorporating the steric term into the Poisson-Nernst-Planck-Stokes coupling equation, direct numerical simulations of electroconvective vortex near nanoslot-bulk interfaces are conducted. For the steric effect, the steric number is introduced to discuss the factors and laws affecting the vortex. We illustrate the substantial enhancement of electroconvective vortex due to the steric effect of ions within the nanoslot. Upon increasing the steric number, the cation concentration in the nanoslot is enhanced, resulting in the expansion of the electric double layer (EDL). The EDLs on the walls inside the nanoslot come into contact with each other, causing the EDLs to overlap, consequently increasing the total charge within the EDLs inside the nanoslot. This EDL overlap enhances the charge density of the extended space charge layer, leading to the enhancement of the electroconvective vortex. Further, our scaling analysis, corroborated by direct numerical simulation and existing data, establishes the scaling of slip velocity, jointly regulated by the steric number and voltage difference. By modulating the membrane transport characteristics, the steric effect reduces flow structure size and flux fluctuations, which offers new perspectives for manipulating ion transport and flow instability.

Non-similar solution of Casson fluid flow over a curved stretching surface with viscous dissipation; Artificial neural network analysis

PurposeThe main focus is to provide a non-similar solution for the magnetohydrodynamic (MHD) flow of Casson fluid over a curved stretching surface through the novel technique of the artificial intelligence (AI)-based Lavenberg–Marquardt scheme of an artificial neural network (ANN). The effects of joule heating, viscous dissipation and non-linear thermal radiation are discussed in relation to the thermal behavior of Casson fluid.Design/methodology/approachThe non-linear coupled boundary layer equations are transformed into a non-linear dimensionless Partial Differential Equation (PDE) by using a non-similar transformation. The local non-similar technique is utilized to truncate the non-similar dimensionless system up to 2nd order, which is treated as coupled ordinary differential equations (ODEs). The coupled system of ODEs is solved numerically via bvp4c. The data sets are constructed numerically and then implemented by the ANN.FindingsThe results indicate that the non-linear radiation parameter increases the fluid temperature. The Casson parameter reduces the fluid velocity as well as the temperature. The mean squared error (MSE), regression plot, error histogram, error analysis of skin friction, and local Nusselt number are presented. Furthermore, the regression values of skin friction and local Nusselt number are obtained as 0.99993 and 0.99997, respectively. The ANN predicted values of skin friction and the local Nusselt number show stability and convergence with high accuracy.Originality/valueAI-based ANNs have not been applied to non-similar solutions of curved stretching surfaces with Casson fluid model, with viscous dissipation. Moreover, the authors of this study employed Levenberg–Marquardt supervised learning to investigate the non-similar solution of the MHD Casson fluid model over a curved stretching surface with non-linear thermal radiation and joule heating. The governing boundary layer equations are transformed into a non-linear, dimensionless PDE by using a non-similar transformation. The local non-similar technique is utilized to truncate the non-similar dimensionless system up to 2nd order, which is treated as coupled ODEs. The coupled system of ODEs is solved numerically via bvp4c. The data sets are constructed numerically and then implemented by the ANN.

Stochastic Maximum Principle for Fully Coupled Forward-Backward Stochastic Differential Equations Driven by Subdiffusion

Stochastic Maximum Principle for Fully Coupled Forward-Backward Stochastic Differential Equations Driven by Subdiffusion

Numerical Solution for Conformable Fractional PDEs by Using a New Double Conformable Integral Transform-Decomposition Method

The conformable Sumudu-Elzaki Transform Decomposition Method CSETDM is used in this research to handle the approximate numerical solutions for the regular and singular one-dimensional conformable Fractional Coupled Burger's Equation CFCBsE and the Nonlinear Singular Conformable Pseudohyperbolic equation NSCPE. This method is generalized in the sense of conformable derivatives. Essential results and theorems concerning this method are discussed. Some numerical examples are given to exhibit the proposed technique's viability, applicability, and effortlessness.

Numerical Modeling and Analysis of Pendant Installation Method Dynamics Using Absolute Nodal Coordinate Formulation

Accurately simulating the deployment process of coupled systems in deep-sea environments remains a significant challenge. This study employs the Absolute Nodal Coordinate Formulation (ANCF) to dynamically model and analyze multi-body systems based on the Pendant Installation Method (PIM). Utilizing the principle of energy conversion, this study calculates the stiffness, generalized elastic forces, mass matrices, and Morison equation, formulating a motion equation for the dynamic coupling of nonlinear time-domain forces in cables during pendulum deployment, which is numerically solved using the implicit generalized-α method. By comparing the simulation results of this model with those from the catenary theory model, the advanced modeling capabilities of this model are validated. Lastly, the sensitivity of the multi-body system under various boundary conditions is analyzed. The results indicate that deployment operations are more effective in environments with strong ocean currents. Furthermore, upon comparing the impacts of structural mass and deployment depth on the system, it was found that deployment depth has a more significant effect. Consequently, the findings of this study provide a scientific basis for formulating subsequent optimization strategies.

Quantitative analysis of heat and mass transfer in MoS2-Al2O3/EG hybrid flow between parallel surfaces with suction/injection by numerical modeling of HPM method

This description focuses on how the magnetic field affects mass and heat transfer in a hybrid nanofluid (Hnf) between two parallel, rotating plates. By dispersing aluminum oxide (Al2O3) and molybdenum disulfide (MoS2) nanoparticles (NPs) in ethylene glycol (EG), a hybrid nanofluid (Hnf) is created. This research aims to analyze the heat and mass transfer characteristics in the flow of a hybrid nanofluid (MoS2-Al2O3/EG) between two rotating parallel plates under the influence of a magnetic field. Furthermore, the statistical technique of response surface methodology (RSM) has been employed to optimize the parameters of velocity, temperature, and concentration of the nanofluid within the flow region bounded by the rotating plates. Dimensionless differential equations have been calculated and checked using the Homotopy perturbation method. This study introduces a novel approach by utilizing the RSM method to identify optimal points for velocity and temperature parameters of nanofluids between two stretching plates for the first time. Additionally, the article innovatively applies the exact HPM method to validate dimensionless linear and non-linear coupling equations. As the Reynolds number and the suction/injection coefficient of nanofluids flowing between two plates under tension increase, the results indicate a decrease in the velocity field. This decrease in velocity field can be attributed to the reduction in fluid diffusion as viscous forces diminish with varying Reynolds numbers. The ideal temperature distribution for nanofluids flowing between two parallel plates occurs when they are uniformly dispersed at the midpoint between them. As the distance from the initial point of nanofluid entry to the end of the plates increases, along with the vertical distance from the bottom plate, the temperature gradient diminishes, reducing the thickness of the thermal boundary layer. The velocity gradient and the rate of heat flux transfer between the nanofluid and plate rise by 34 % when the volume percentage is raised from 1 % to 5 %.

Shortcut Approximation to the Coupled Equations of Heat and Electric Currents

A self-consistent approximation to the coupled equations of heat and electric current densities is developed to facilitate the integration of the relaxation time approximation of the Boltzmann transport equation in a realistic design, analysis, and optimization of energy conversion devices. By introducing two auxiliary energies in the empirical Boltzmann transport equations, the coupled equations can be simply expressed in these auxiliary energies. These auxiliary energies are then determined by optimizing both the heat current through charge neutrality equation without the presence of electric current and the carrier concentrations with the presence of electric current. The heat and the electric current densities as well as the transport properties can then be calculated. The calculated transport properties agreed reasonably well with existing experimental and computational data.

An investigation of heat transfer and optimization of entropy in bio-convective flow of Eyring-Powell nanomaterial with gyrotactic microorganisms

Entropy generation in nanofluid flows is a critical parameter that influences the performance, efficiency, and sustainability of thermal systems. Understanding and optimizing entropy creation can lead to noteworthy advancements in numerous engineering applications, from renewable energy systems to industrial processes and biomedical equipments. The purpose of this article is to examine the entropy produced in the bioconvection flow of Eyring-Powell nanomaterial with gyrotactic microorganisms. The mathematical equations representing the flow are modeled considering the flow towards a porous surface of a cylinder. Together with the effects of the governing parameters, the mass and heat transfer components of the problem are discussed. The thermal effects of different natures are used in a new way. Overall, entropy creation is designed with regard to the second law of thermodynamics. Activation energy, chemical reaction, thermal radiation, and internal friction force effects are accounted for the development of the mathematical model. Coupled non-linear dimensional equations have been altered into ordinary differential equations (ODEs) and then treated using the MATLAB Finite Difference Method (FDM). Consequence of diverse sundry parameters on entropy production, thermal field, mass concentration profile, Bejan quantity, and motile density of microorganisms is deliberated via plots. Through tables, engineering quantities are analyzed. According to the findings, the velocity profile rises as the curvature and Eyring-Powell fluid parameters rise, while it falls when the magnetic parameter is enhanced. Additionally, it is noted that as the curvature variable enhances, the rate of heat transfer and skin friction coefficient decays.