Coupled Boundary Value Problems Research Articles

Mixed Galerkin- Implicit Differences Methods for Solving Coupled Parabolic Boundary Value Problems with Variable Coefficients

Mixed Galerkin- Implicit Differences Methods for Solving Coupled Parabolic Boundary Value Problems with Variable Coefficients

A decomposition method for two and three dimensional fluid-solid interaction scattering problem

In this paper, we propose a decomposition formula for the fluid-solid interaction scattering problem. By the decomposition of the displacement, the corresponding problem is transformed into a coupled boundary value problem. For the two dimensional case, the coupled boundary value problem is about two scalar Helmholtz equations, and the other one is a scalar Helmholtz equation and a vector Helmholtz equation for the three dimensional. Then the coupled boundary value problem is solved by the method of fundamental solution. At last we give some examples to show the effectiveness of the present method in two and three dimensional case. A benchmark example is available for checking the feasibility and accuracy of the presented method. In the numerical examples, a part of exact solution for the boundary value problem is including for checking the accuracy.

Existence, uniqueness, and Ulam stability of solutions of fractional conformable Langevin system on the ethane graph

The present study examines the Langevin coupled boundary value problem with the fractional conformable derivative on the ethane graph. The main objective of this research is to establish the existence and uniqueness of solutions with the help of the fixed point theorem and to analyze the Ulam stability of the corresponding problem. Several examples are provided to demonstrate the obtained results.

Solving coupled non-linear higher order BVPs using improved shooting method

Abstract The purpose of this work is to propose a new efficient initial guess algorithm rather than the conventional Newton method to meet the adjoint terminal conditions, rapidly. Thus, the focus of this paper is to use better algorithms for obtaining refined initial guesses with shooting method to solve the coupled boundary value problems. The boundary value problem is formulated as a system of equations i.e. initial value problems with one unknown initial condition. The efficiency and accuracy of the Shooting method is enhanced by improving the initial guess and then solving the problem iteratively.

Coupled Nonlocal Boundary Value Problems for Fractional Integro-differential Langevin System via Variable Coefficient

In this paper, we aim to study a new coupled system of nonlinear fractional integro-differential Langevin equations with coupled multipoint boundary conditions. The existence and uniqueness of solution are investigated by using the Banach’s and Krasnoselskii’s fixed point theorems. The Ulam-Hyers stability of the mentioned equation is provided by applying the classical technique of functional analysis. Two examples are presented to verify our analysis.

A highly accurate indirect boundary integral equation solution for three dimensional elastic scattering problem

In this paper, we propose a numerical formula for three dimensional elastic wave scattering problem. Different from the classical boundary integral equation method, the three dimensional elastic scattering problem is transformed into a coupled boundary value problem including a scalar Helmholtz equation and a vector Helmholtz equation by the decomposition of the displacement. The coupled boundary value problem can be solved by the boundary integral equations based on the Helmholtz equation. A denseness result is given to support this algorithm. A comparative study with the classical method of fundamental solutions getting from the boundary integral equation method is given to check the accuracy with different types of boundary conditions. From the numerical results, the present method can give more accurate results than the classical method of fundamental solutions method.

On the qualitative evaluation of the variable-order coupled boundary value problems with a fractional delay

The logical progression from the constant order differential equations is the field of variable-order differential equations. Such equations can frequently give a more succinct description of problems in the real world. In light of this, we therefore take into account a class of coupled boundary value problems under variable-order differentiation. By utilizing the fixed-point techniques of Banach and Schauder, we investigate the existence and uniqueness of solutions to the proposed problem. Also, sufficient results are documented for the necessary needs. Furthermore, some stability results based on the ideas of Ulam, Hyers, and Rassias are elaborated upon. Ultimately, appropriate examples and in-depth analysis are presented to support our results.

Darcy–Forchheimer dynamics of hybrid nanofluid due to a porous Riga surface capitalizing Cattaneo–Christov theory

Due to superior heat transport offered by hybrid nanofluids compared to traditional nanofluids, an investigation on the dynamics of a hybrid nanofluid consisting of (Fe3O4+Al2O3)/water over a porous Riga stretching surface using the double-diffusion theory of Cattaneo–Christov is investigated. The hybrid mixture (Fe3O4+Al2O3) is placed in a Darcy–Forchheimer-porous-medium, and the significance of Brownian and thermo diffusions are also taken into account. The thermophysical attributes of solid particles and base fluid are used to model the problem along with the basic transport equations of fluid dynamics. Similarity transformations are used to make the transport equations dimensionless. The final version of the coupled boundary value problem is then numerically solved via RKFST (Runge–Kutta–Fehlberg method based on shooting technique). The post-processing of the solution involves computing the velocity field, skin-friction, Nusselt number, Sherwood number, temperature field, streamlines, isotherms, and concentration field against various estimations of the involved parameters. The streamlines and isotherms are observed to be more effective for the hybrid nanofluid than the nanofluid, whereas the temperature is reduced for the hybrid nanofluid compared to the nanofluid. Temperature is reduced with the development of thermal-relaxation multiplier, while concentration is also declined with the higher estimation of concentration-relaxation multiplier.

Fragmentation of inflated elastic brittle rings: Emergence of the quasi-equidistant spacing of cracks

The order of appearance and the position of meridional cracks in brittle domes is a delicate question of mechanics. This paper investigates a dimension-reduced model, a pressurized brittle ring constrained to the plane, to show that a simple, deterministic approach based on the Griffith theory of fracture predicts quasi-equidistant, i.e., close to an equal spacing of the emerging cracks. It is also demonstrated that the order of emergence is significantly affected by the variation of the in-plane elastic support as well the bending rigidity of the ring. In the model, the energy minimization is first recast into a sequence of coupled boundary value problems. The relationship between the mechanical model and geometric properties of the emerging pattern is studied numerically. The pattern is shown to be driven by elaborate co-dimension one bifurcations and by the requirement of global energy minimization.

NUMERICAL study of MAGNETO convective Buongiorno nanofluid flow in a rectangular enclosure under oblique magnetic field with heat generation/absorption and complex wall conditions

A mathematical model is presented to analyze the magnetohydrodynamic (MHD) convective flow in a rectangular enclosure. Earlier studies on the Tiwai-Das volume fraction nanofluid model did not consider the Buongiorno nanofluid model. This is the focus of the present analysis which examines the laminar mixed convection magnetohydrodynamic flow of a nanofluid in a differentially heated rectangular enclosure with complex boundary conditions under an inclined magnetic field. Buongiorno's two-component nanofluid model is employed, which incorporates the effects of Brownian motion and thermophoretic diffusion of nanoparticles. Magnetic nanofluids have considerable potential for enhancing transport processes in energy systems such as hybrid fuel cells. The study is essential with heat generation/absorption effects. Additionally, the work is highlighted by the general case of an oblique (inclined) magnetic field. The conservation equations for mass, primary and secondary momentum, energy, and nanoparticle concentration with wall boundary conditions are dimensionless using appropriate scaling transformations. A finite-difference computational scheme known as the Harlow-Welch Marker and Cell (MAC) method is employed to solve the dimensionless nonlinear coupled boundary value problem. A mesh independence study is included. Graphical plots are presented for the impact of key control parameters on streamline contours, isotherm contours, iso-concentration (nanoparticle mass) contours, and local Nusselt number. With heat sink (absorption), the Nusselt number is enhanced in magnitude whereas it is suppressed with heat generation since there is a heat reduction transmitted to the boundary. The configurations of streamlines, isotherms, and iso-concentrations are mostly invariant to magnetic field direction changes. The obtained results show interesting behaviors of the flow and thermal fields, which mainly involve the effect of Brownian motion and thermophoresis parameters, as well as unsteady regimes, depending on specific values of the Schmidt number, Richardson number, and Prandtl numbers. Increasing the Schmidt number induces a contraction in the central cooler zone in the enclosure and also reduces the iso-concentration magnitudes in the central region across the enclosure. The core region of the enclosure heats up as the thermophoresis and Brownian motion parameters rise, pushing the previously cooler top and bottom wall zones further away from the center. There is also a decrease in iso-concentration magnitudes in particular at the upper and lower boundaries at higher values of Brownian motion and thermophoresis parameters. At decreasing buoyancy ratios, the left vortex cell first decelerates while the right vortex cell accelerates. However, when the buoyancy ratio increases, the left vortex cell streamlines magnitudes increase with a contraction in vortex size, while the right cell develops. A Very minor alteration is observed in the isotherm and iso-concentration contours with an increasing buoyancy ratio. When the Richardson number increases, the vortex cell structures shift from a strong circulation cell on the left to a weaker cell on the right, resulting in reverse distribution. With the rising Richardson number, significant cooling is also caused in the core zone, as well as a drop in iso-concentrations, with the original dual low-concentration upper and lower zones merging into a single center zone. The original symmetric left and right vortex cells are gradually twisted diagonally towards the right wall as the magnetic field increases, yet the stronger right cell and the weaker left cell are maintained.

On a Coupled Differential System Involving (k,ψ)-Hilfer Derivative and (k,ψ)-Riemann–Liouville Integral Operators

We investigate a nonlinear, nonlocal, and fully coupled boundary value problem containing mixed (k,ψ^)-Hilfer fractional derivative and (k,ψ^)-Riemann–Liouville fractional integral operators. Existence and uniqueness results for the given problem are proved with the aid of standard fixed point theorems. Examples illustrating the main results are presented. The paper concludes with some interesting findings.

Curing spurious magneto-mechanical coupling in soft non-magnetic materials.

The present work is concerned with the issue of spurious coupling effects that are pervasive in fully coupled magneto-mechanical finite element simulations involving very soft non-magnetic or air-like media. We first address the characterization of the spurious magneto-mechanical effects and their intuitive interpretation based on energy considerations. Then, as main contribution, we propose two new cures for the issue under consideration that completely prune the undesired spurious magneto-mechanical coupling in non-magnetic media. The proposed methods are compared with established methods in the context of magnetic bodies embedded in (i) air or vacuum and (ii) very soft elastic non-magnetic media. The comparison shows that the proposed approaches are accurate and effective. They, furthermore, allow for a consistent linearization of the coupled boundary value problems, which is crucial for the simulation of compliant structures. For reproducibility and accessibility of the proposed methods, we provide our implementations with Netgen/NGSolve as well as all codes necessary for the reproduction of our results as supplementary material.

Finite-Element Modeling of the Hysteresis Behavior of Tetragonal and Rhombohedral Polydomain Ferroelectroelastic Structures.

The influence of the domain structure's initial topology and its evolution on the hysteresis curves of tetragonal and rhombohedral polydomain structures of ferroelectroelastic materials is studied. Based on the analysis of electrical and mechanical compatibility conditions, all possible variants of representative volume elements of tetragonal and rhombohedral second-rank-domain laminate structures were obtained and used in simulations. Considerable local inhomogeneity of stress and electric fields within the representative volume, as well as domain interaction, necessitates the use of numerical methods. Hysteresis curves for laminated domain patterns of the second rank were obtained using finite-element homogenization. The vector-potential finite-element formulation as the most effective method was used for solving nonlinear coupled boundary value problems of ferroelectroelasticity. A significant anisotropy of the hysteresis properties of domain structures was established both within individual phases and when comparing the tetragonal and rhombohedral phases. The proposed approach describes the effects of domain hardening and unloading nonlinearity.

A boundary integral equation method for the fluid-solid interaction problem

<abstract><p>In this paper, a boundary integral equation method is proposed for the fluid-solid interaction scattering problem, and a high-precision numerical method is developed. More specifically, by introducing the Helmholtz decomposition, the corresponding problem is transformed into a coupled boundary value problem for the Helmholtz equation. Based on the integral equation method, the coupled value problem is reduced to a system of three coupled hypersingular integral equations. Semi-discrete and fully-discrete collocation methods are proposed for the singular integral equations. The presented method is based on trigonometric interpolation and discretized singular operators applied to differentiated interpolation. The convergence of the method is verified by a numerical experiment.</p></abstract>

NUMERICAL STUDY OF FREE CONVECTION Ag-WATER NANOFLUID FLOW IN A SQUARE ENCLOSURE WITH VISCOUS DISSIPATION AND HEAT GENERATION/ABSORPTION EFFECTS IN A POROUS MEDIUM WITH COMPLEX WALL CONDITIONS

Nanofluids hold great promise in improving transport processes in energy systems including hybrid fuel cells. In this present work, a mathematical model is developed for laminar free convection flow of Ag-water nano-additives in an enclosure in a porous medium with complex boundary conditions. Additionally, heat generation/absorption and viscous dissipation effects are included. Via appropriate scaling transformations, the conservation equations for mass, primary and secondary momentum, energy, and nanoiparticle vorticity with wall boundary conditions are rendered dimensionless. A finite-difference computational scheme known as the marker and cell (MAC) method, developed by Harlow and Welch, is occupied to solve the dimensionless, nonlinear coupled boundary value problem. A mesh independence study is included. The impact of parameters such as Eckert number (Ec), Darcy number (Da), Grashof number (Gr), Prandtl number (Pr), Reynolds number (Re), and Richardson number (Ri) are observed with physical framework. Graphical plots are presented for the impact of key control parameters on streamline contours, isotherm contours, and local Nusselt number. By heat sink (absorption), the Nusselt number is increased, whereas by heat generation it is reduced since there is a decrease in heat transferred to the boundary. The presence of viscous dissipation effects moves the streamlines toward the blue core and allows the temperature to increase in the neighborhood of the hot wall of the envelope. An increase in Richardson number induces a flip in vortex cell structures from an initially strong circulation cell on the left and weaker cell on the right, to the opposite distribution. Significant cooling is also induced in the core zone with an increasing Richardson number, and a decrease in vorticity is observed.

On numerical study of constrained coupled shape optimization problems based on a new shape derivative method

AbstractIn this article, we deal with a numerical method for the approximation of a class of coupled shape optimization problems, which consist in minimizing an appropriate general volume cost functional subjected to coupled boundary value problems by means of a Neumann boundary transmission condition. We show the existence of the shape derivative of the cost functional and express it by means of support functions, using a new formula of shape derivative on a family of convex domains. This allows us to avoid the disadvantages related to the classical shape derivative method using vectors field. Then the numerical discretization is performed using the dual reciprocity boundary element method in order to avert the remeshing task required for the finite element method. Finally, we give some numerical results, based on the gradient method, showing the efficiency of the proposed approach.

Influence of variable thermal conductivity and dissipation on magnetic Carreau fluid flow along a micro-cantilever sensor in a squeezing regime

Mathematical modeling of squeezing flows finds numerous applications in biological, mechanical, and medical engineering. Sensors feature such flows and can be effectively utilized to control vibrations and regulate lubrication. Magnetic fluids are critical to modern sensor systems. Micro-cantilevers are utilized in biomedical applications as biological, physical, or chemical sensors and operate via the detection of variations in the vibrational frequency or cantilever bending. Hence, a theoretical study is conducted to explore the dissipative flow and thermal characteristics in non-Newtonian boundary layer flow along a micro-cantilever sensor surface suspended in a squeezing regime between parallel plates. To achieve a more refined simulation, the effects of variable thermal conductivity and Joule magnetic dissipation are incorporated. The governing conservation equations for unsteady magnetic Carreau squeezing flow and heat transfer are rendered dimensionless and self-similar via appropriate scaling transformations. Emerging nonlinear coupled boundary value problem is then solved by using RK-4 method. The magnifying Weissenberg parameter decays the flow field. Skin friction is elevated with increasing Carreau power-law index and Weissenberg number. The novelty of the present study is the inclusion of dissipation and thermal conductivity variation effects which extends previous investigations and provides a more accurate appraisal of thermal characteristics in sensor squeezing flows.

A Numerical Algorithm for a Coupled Hyperbolic Boundary Value Problem

This paper develops an algorithm to find the quadruple [Formula: see text] solution to a double Goursat–Cauchy boundary value problem in a triangular domain. The functions [Formula: see text] and [Formula: see text] are coupled through the function [Formula: see text]. Numerical examples are provided to illustrate the algorithm. This work finds applicability in inverse spectral problems.

A spectral boundary integral method for the elastic obstacle scattering problem in three dimensions

In this paper, we consider the scattering of a plane wave by a rigid obstacle embedded in a homogeneous and isotropic elastic medium in three dimensions. Based on the Helmholtz decomposition, the elastic scattering problem is reduced to a coupled boundary value problem for the Helmholtz and Maxwell equations. A novel system of boundary integral equations is formulated and a spectral boundary integral method is developed for the coupled boundary value problem. Numerical experiments are presented to demonstrate the superior performance of the proposed method.

Numerical outlook of a viscoelastic nanofluid in an inclined channel via Keller box method

Background and objectivesThis work is a theoretical investigation of a viscoelastic fluid of which heat transfer properties are explored in the presence of nanoparticles. Further the viscous dissipation has been mathematically modeled in the energy equation. MethodologyThe physical model is handled using fluid dynamics fundamental equations. Buongiorno nanofluid model is utilized to study the nanofluid properties. The resulting coupled boundary value problem is nondimensionalized and solved by Keller box method. SignificanceBackflow in the inclined channels can be witnessed at any Reynolds number and at any opening angle of the channel. In the present study, the same has been investigated when nanoparticles are added to the viscoelastic fluid flowing through the inclined channel. Numerical findingsThe viscous dissipation elevated the temperature profile and Schmidt number upsurged the nanoparticles concentration in the converging channel for stretching case and vice versa behaviour is observed for the diverging channel.