System Of Coupled Equations Research Articles

Sturm–Liouville problem for a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems

The problem of constructing eigenfunctions of a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems is considered. The corresponding Sturm–Liouville problem is formulated using Fourier’s separation of variables applied to a coupled system of thermoelasticity equations, assuming that the heat transfer rate is finite. It is shown that the eigenfunctions of the one-dimensional thermoelastic operator are expressed in terms of well-known trigonometric, cylinder, and spherical functions. However, coupled thermoelasticity problems are solved analytically only under certain boundary conditions, whose form is determined by the properties of the eigenfunctions.

First-principle modeling of parallel-flow regenerative kilns and their optimization with genetic algorithm and gradient-based method

We present a one-dimensional first-principle model for parallel-flow regenerative kilns that accounts for the most important effects. These include the kinetics and thermal effects of the limestone decomposition as well as the heat transfer between the gaseous and solid phases. The model consists of two coupled equation systems for the upper and lower part of the kiln. The results of the model are validated qualitatively and are used to train an artificial neural network that predicts the conversion and the temperature in the crossover channel. The artificial neural network performs very well with values of the root mean squared error that are two to three orders of magnitudes lower than the range covered within the data. Finally, we use a genetic algorithm to optimize the feed mass flows such that the conversion and the fuel efficiency are improved in a Pareto-optimal manner. The results are compared to those of a gradient-based optimization method, which shows the usefulness and validity of the approach with the genetic algorithm.

Calderón–Zygmund theory for strongly coupled linear system of nonlocal equations with Hölder-regular coefficient

Abstract We extend the Calderón–Zygmund theory for nonlocal equations to strongly coupled system of linear nonlocal equations ℒ A s ⁢ u = f {\mathcal{L}^{s}_{A}u=f} , where the operator ℒ A s {\mathcal{L}^{s}_{A}} is formally given by ℒ A s ⁢ u = ∫ ℝ n A ⁢ ( x , y ) | x - y | n + 2 ⁢ s ⁢ ( x - y ) ⊗ ( x - y ) | x - y | 2 ⁢ ( u ⁢ ( x ) - u ⁢ ( y ) ) ⁢ 𝑑 y . \mathcal{L}^{s}_{A}u=\int_{\mathbb{R}^{n}}\frac{A(x,y)}{|x-y|^{n+2s}}\frac{(x-% y)\otimes(x-y)}{|x-y|^{2}}(u(x)-u(y))\,dy. For 0 < s < 1 {0<s<1} and A : ℝ n × ℝ n → ℝ {A:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}} taken to be symmetric and serving as a variable coefficient for the operator, the system under consideration is the fractional version of the classical Navier–Lamé linearized elasticity system. The study of the coupled system of nonlocal equations is motivated by its appearance in nonlocal mechanics, primarily in peridynamics. Our regularity result states that if A ⁢ ( ⋅ , y ) {A(\,\cdot\,,y)} is uniformly Holder continuous and inf x ∈ ℝ n ⁡ A ⁢ ( x , x ) > 0 {\inf_{x\in\mathbb{R}^{n}}A(x,x)>0} , then for f ∈ L loc p {f\in L^{p}_{\rm loc}} , for p ≥ 2 {p\geq 2} , the solution vector u ∈ H loc 2 ⁢ s - δ , p {u\in H^{2s-\delta,p}_{\rm loc}} for some δ ∈ ( 0 , s ) {\delta\in(0,s)} .

The Robin Problems in the Coupled System of Wave Equations on a Half-Line

This article investigates the local well-posedness of a coupled system of wave equations on a half-line, with a particular emphasis on Robin boundary conditions within Sobolev spaces. We provide estimates for the solutions to linear initial-boundary-value problems related to the coupled system of wave equations, utilizing the Unified Transform Method in conjunction with the Hadamard norm while considering the influence of external forces. Furthermore, we demonstrate that replacing the external force with a nonlinear term alters the iteration map defined by the unified transform solutions, making it a contraction map in a suitable solution space. By employing the contraction mapping theorem, we establish the existence of a unique solution. Finally, we show that the data-to-solution map is locally Lipschitz continuous, thus confirming the local well-posedness of the coupled system of wave equations under consideration.

وجود حل محلي لنظام الجذب الكيميائي والتنافر مع الانتشار غير الخطي والمصدر اللوجستي

In this paper, we are concerned with a model arising from biology, which is a coupled system of chemotaxis equations and viscous compressible fluid equations through transport and external forcing. The local existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for an attraction-repulsion chemotaxis model system over three space dimensions, we obtain local existence and uniqueness of convergence on classical solutions near constant states. We prove local existence of unique solutions in three dimensions by using energy estimates.

A Numerical Scheme of a Fractional Coupled System of Volterra Integro-Differential Equations with the Caputo Fabrizio Fractional Derivative

In this paper, we have developed a new computational scheme for solving coupled systems of fractional order Volterra-type integro-differential equation (FVIDE). We construct new operational matrices, which serve as building blocks for converting the FVIDE into a Sylvester-type algebraic structure that is more easily solvable. The fractional derivatives and their inverses (integrals) are considered in the Caputo-Fabrizio sense. This computational scheme is based on Legendre polynomials. The assessment of the proposed method’s convergence involves utilizing appropriate error norms to measure the level of convergence. The algorithm is designed in such a way that it can be simulated using any computational package; we have used Matlab for simulating the proposed scheme. Graphical visualization and tabulation of data are performed to confirm convergence and to analyze errors.

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.

Nontrivial solutions for multiparameter k-Hessian systems

This paper is devoted to studying the coupled system of multiparameter k-Hessian equations where B = {x ∈ ℝ n : |x| < 1}, λ1, λ2 are positive parameters, ρi (i = 1, 2) is singular near the boundary ∂B, f i(i = 1, 2) satisfies a combined superlinear condition at ∞. Employing the fixed point theorem of Krasnosel’skii type in Banach space, the existence and multiplicity of nontrivial radial solutions are established. In particular, the dependence of the solutions on the parameters λ1, λ2 is also discussed. Finally, we study the nonexistence results of nontrivial radial solutions for the case λ1, λ2 ≫ 1.

Study of Caputo fractional derivative and Riemann–Liouville integral with different orders and its application in multi‐term differential equations

In this article, we initially provided the relationship between the RL fractional integral and the Caputo fractional derivative of different orders. Additionally, it is clear from the literature that studies into boundary value problems involving multi‐term operators have been conducted recently, and the aforementioned idea is used in the formulation of several novel models. We offer a unique coupled system of fractional delay differential equations with proper respect for the role that multi‐term operators play in the research of fractional differential equations, taking into account the newly established solution for fractional integral and derivative. We also made the assumptions that connected integral boundary conditions would be added on top of ‐fractional differential derivatives. The requirements for the existence and uniqueness of solutions are also developed using fixed‐point theorems. While analyzing various sorts of Ulam's stability results, the qualitative elements of the underlying model will also be examined. In the paper's final section, an example is given for purposes of demonstration.

Particle trajectories in the KP-II equation

In the current work, we consider particle trajectories beneath traveling soliton solutions described by the Kadomtsev–Petiviashvili-II (KP-II) equation, which is a model for small-amplitude water waves in shallow water. The KP-II equation has a large number of exact solutions describing crossing line solitons. Here, we describe the particle drift induced by the single line soliton and various two- and three-soliton solutions on a two-dimensional horizontal domain.First, the derivation of the KP-II equation is used to exhibit expressions for the three components of the fluid velocity vector induced by a surface wave profile given by the KP-II equation. These velocity components can be used to specify a coupled system of ordinary differential equations (ODEs) which describe the motion of a fluid particle. Given an initial position inside the fluid for a specific particle, as well as continuously providing time-dependent values for the surface deflection, the ODE system can be solved numerically to find the particle path induced by the passage of a wave. A special numerical integration grid tracking the particle is used to handle integral terms occurring in the fluid velocity expressions.

Existence results for a coupled system of fractional stochastic differential equations involving Hilfer derivative

Abstract In this paper, we consider a coupled system of fractional stochastic differential equations involving the Hilfer derivative of order 1 2 &lt; α &lt; 1 {\frac{1}{2}&lt;\alpha&lt;1} . Under some assumptions, we prove the existence of mild solutions for our system based on Perov’s and Schaefer’s fixed point theorems. An example illustrating our result is provided.

Bilayer graphene in periodic and quasiperiodic magnetic superlattices

Starting from the effective Hamiltonian arising from the tight-binding model, we study the behaviour of low-lying excitations for bilayer graphene placed in periodic external magnetic fields by using irreducible second-order supersymmetry transformations. The coupled system of equations describing these excitations is reduced to a pair of periodic Schrödinger Hamiltonians intertwined by a second-order differential operator. The direct implementation of more general second-order supersymmetry transformations allows to create non-singular Schrödinger potentials with periodicity defects and bound states embedded in the forbidden bands, which turn out to be associated with quasiperiodic magnetic superlattices. Applications in quantum metamaterials stem from the ability to engineer and control such bound states which could lead to a fast development of the subject in the near future.

Special function models of indecomposable sl(2) representations: the Laguerre case

In this paper, we point out connections between certain types of indecomposable representations of sl(2) and generalizations of well-known orthogonal polynomials. Those representations take the form of infinite dimensional chains of weight or generalised weight spaces, for which the Cartan generator acts in a diagonal way or via Jordan blocks. The other generators of the Lie algebras sl(2) act as raising and lowering operators but are now allowed to relate the different chains as well. In addition, we construct generating functions, we calculate the action of the Casimir invariant and present relations to systems of non-homogeneous second-order coupled differential equations. We present different properties as higher-order linear differential equations for building blocks taking the form of one variable polynomials. We also present insight into the zeroes and recurrence relations.

Exploration of linear and exponential heat source/sink with the significance of thermophoretic particle deposition on ZnO-SAE50 nano lubricant flow past a curved surface

The present research focuses on exploring the impact of linear and exponential heat source/sink on double-diffusive MHD convective flow heat and mass transport of ZnO – SAE50 nano lubricants in an exponentially stretching curved sheet along with the effect of thermophoretic particle deposition. The heat transmission is considered to be caused by both radiation and convection. The mathematical modelling of the considered problem results in a coupled system of partial differential equations, using suitable similarity transformation variables these equations are reduced to a coupled non-linear system of ordinary differential equations and are solved numerically by adopting the finite difference method to obtain the solution for the field variables. The quantity and spatial distribution of linear heat sources/sinks affect the physical dynamics and optimum status of the entire system and the application of exponential heat sources/sinks can improve heat dispersion and energy efficiency in industrial operations such as nuclear reactors and steel production. The thermophoretic particle deposition parameter controls particle concentration by allowing particles to accumulate on surfaces or in specific areas of the fluid. A rise in Hartmann number causes a mechanical damping force to act in the opposite direction of the fluid motion, lowering the speed of the fluid in the curved sheet. The heat and mass transmission properties are analyzed through the Levenberg – Marquardt backpropagating algorithm and the correlation coefficient has ideal values of unity for training, validation, testing, and overall. In light of this, the anticipated behaviour of the LMNN-BPA indicated that the heat transfer profile numerical data and the network output are in good consensus.

Multiscale model reduction for the time fractional thermoporoelasticity problem in fractured and heterogeneous media

In this paper, we consider the time fractional thermoporoelasticity problem in fractured and heterogeneous media. The mathematical model with a time memory formalism is described by a coupled system of equations for pressure, temperature and displacements. We use an implicit finite difference approximation for temporal discretization. We present a fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) for two-dimensional model problems. Further, we use the Generalized Multiscale Finite Element Method (GMsFEM) for coarse grid approximation. The primary concept behind the proposed method is to streamline the complexity inherent in the thermoporoelasticity problem. Given that our model equation incorporates multiple fractional powers, leading to multiple unknowns with memory effects, we aim to address this intricacy by optimizing the problem’s dimensionality. As a result, the solution is sought on a coarse grid, a strategic choice that not only simplifies the computational cost but also contributes to significant time savings. We present numerical results for the two-dimensional model problems in heterogeneous fractured porous media. We derive relative errors between the reference fine grid solution and the multiscale solution for different numbers of multiscale basis functions. The results confirm that the proposed method is able to achieve good accuracy with a few degrees of freedoms on the coarse grid.

Fractional Sequential Coupled Systems of Hilfer and Caputo Integro-Differential Equations with Non-Separated Boundary Conditions

Fractional Sequential Coupled Systems of Hilfer and Caputo Integro-Differential Equations with Non-Separated Boundary Conditions

Parameter identification and uncertainty propagation of hydrogel coupled diffusion-deformation using POD-based reduced-order modeling

AbstractThis study explores reduced-order modeling for analyzing time-dependent diffusion-deformation of hydrogels. The full-order model describing hydrogel transient behavior consists of a coupled system of partial differential equations in which the chemical potential and displacements are coupled. This system is formulated in a monolithic fashion and solved using the finite element method. We employ proper orthogonal decomposition as a model order reduction approach. The reduced-order model performance is tested through a benchmark problem on hydrogel swelling and a case study simulating co-axial printing. Then, we embed the reduced-order model into an optimization loop to efficiently identify the coupled problem’s material parameters using full-field data. Finally, a study is conducted on the uncertainty propagation of the material parameter.

Parametrically excited shape distortion of a submillimeter bubble

The existence of finite amplitude shape distortion caused by parametrically excited surface instabilities for a gas bubble in water driven by a temporally periodic, spatially uniform pressure field in an axisymmetric geometry is investigated. Employing a nonlinear coupled system of equations which includes shape mode interactions to third order, the resultant spherical oscillations, translation, and shape distortion of the bubble are modelled, placing no restriction on the size of the spherical oscillations. The model accounts for viscous and thermal damping with compressibility effects. The existence of synchronous and higher order parametrically induced sustained, finite amplitude, periodic shape deformation is demonstrated. The excitement of an odd shape mode via the synchronous mechanism is shown to give rise to linear bubble self-propulsion. For larger driving amplitudes, it is shown that more than one shape mode can be parametrically excited at the same driving frequency but by different resonance mechanisms, leading to more involved shape deformation and the increased possibility of bubble self-propulsion.

Solving the System of Two-Dimensional Coupled Burgers' Equations byModified Variational Iteration Method with Genetic Algorithm

Solving the System of Two-Dimensional Coupled Burgers' Equations byModified Variational Iteration Method with Genetic Algorithm

Coupled systems of conformable fractional differential equations

This paper deals with some existence of solutions for some classes of coupled systems of conformable fractional differential equations with initial and boundary conditions in Banach and Fréchet spaces. Our results are based on some fixed point theorems. Some illustrative examples are presented in the last section.