Nonhomogeneous Boundary Conditions Research Articles

Petrov-Galerkin method for small deflections in fourth-order beam equations in civil engineering

Abstract This study explores the Petrov–Galerkin method’s application in solving a linear fourth-order ordinary beam equation of the form u ″ ″ + q u = f u^{\prime\prime} ^{\prime\prime} +qu=f . The equation entails two distinct boundary conditions: pinned–pinned conditions on u u and u ′ u^{\prime} , and clamped–clamped conditions on u u and u ″ {u}^{^{\prime\prime} } . To satisfy these boundary conditions, we have built two sets of basis functions. The explicit forms of all spectral matrices were reported. The nonhomogeneous boundary conditions were easily handled using perfect transformations, ensuring the numerical solution’s accuracy. Detailed analysis of the method’s convergence was studied. Some numerical examples were presented, accompanied by comparisons with other existing methods in the literature.

Mathematical and numerical analysis of reduced order interface conditions and augmented finite elements for mixed dimensional problems

In this paper, we are interested in the mathematical properties of methods based on a fictitious domain approach combined with reduced-order interface coupling conditions, which have been recently introduced to simulate 3D-1D fluid-structure or structure-structure coupled problems. To give insights on the approximation properties of these methods, we investigate them in a simplified setting by considering the Poisson problem in a two-dimensional domain with non-homogeneous Dirichlet boundary conditions on small inclusions. The approximated reduced problem is obtained using a fictitious domain approach combined with a projection on a Fourier finite-dimensional space of the Lagrange multiplier associated to the Dirichlet boundary constraints, obtaining in this way a Poisson problem with defective interface conditions. After analyzing the existence of a solution of the reduced problem, we prove its convergence towards the original full problem, when the size of the holes tends towards zero, with a rate which depends on the number of modes of the finite-dimensional space. In particular, our estimates highlight the fact that to obtain a good convergence on the Lagrange multiplier, one needs to consider more modes than the first Fourier mode (constant mode). This is a key issue when one wants to deal with real coupled problems, such as fluid-structure problems for instance. Next, the numerical discretization of the reduced problem using the finite element method is analyzed in the case where the computational mesh does not fit the small inclusion interface. As is standard for these types of problem, the convergence of the solution is not optimal due to the lack of regularity of the solution. Moreover, convergence exhibits a well-known locking effect when the mesh size and the inclusion size are of the same order of magnitude. This locking effect is more apparent when increasing the number of modes and affects the Lagrange multiplier convergence rate more heavily. To resolve these issues, we propose and analyze a stabilized method and an enriched method for which additional basis functions are added without changing the approximation space of the Lagrange multiplier. Finally, the properties of numerical strategies are illustrated by numerical experiments.

Hierarchical Stackelberg Control for a Two-Stroke Linear System in Population Dynamics

We study a unique hierarchical control problem for a two-stoke linear system, adjoint to an age and space dependent population dynamics problem. Using Stackelberg's method, we introduce two levels of control: a boundary control to achieve optimal flow regulation and a distributed control for null controllability. Our approach employs Carleman inequalities to address non-homogeneous Dirichlet boundary conditions, leading to new insights in controlling invasive species populations. These results highlight the applicability of hierarchical controls in ecological systems, providing a robust framework for future studies in control theory and population dynamics.

Semi-Analytical Design of PDE Endpoint Controller for Flexible Manipulator With Non-Homogenous Boundary Conditions

Semi-Analytical Design of PDE Endpoint Controller for Flexible Manipulator With Non-Homogenous Boundary Conditions

A bulk-surface mechanobiochemical modelling approach for single cell migration in two-space dimensions

In this work, we present a mechanobiochemical model for two-dimensional cell migration which couples mechanical properties of the cell cytosol with biochemical processes taking place near or on the cell plasma membrane. The modelling approach is based on a recently developed mathematical formalism of evolving bulk-surface partial differential equations of reaction–diffusion type. We solve these equations using finite element methods within a moving-mesh framework derived from the weak formulation of the evolving bulk-surface PDEs. In the present work, the cell cytosol interior (bulk) dynamics are coupled to the cell membrane (surface) dynamics through non-homogeneous Dirichlet boundary conditions. The modelling approach exhibits both directed cell migration in response to chemical cues as well as spontaneous migration in the absence of such cues. As a by-product, the approach shows fundamental characteristics associated with single cell migration such as: (i) cytosolic and membrane polarisation, (ii) actin dependent protrusions, and (iii) continuous shape deformation of the cell during migration.Cell migration is an ubiquitous process in life that is mainly triggered by the dynamics of the actin cytoskeleton and therefore is driven by both mechanical and biochemical processes. It is a multistep process essential for mammalian organisms and is closely linked to a vast diversity of processes; from embryonic development to cancer invasion. Experimental, theoretical and computational studies have been key to elucidate the mechanisms underlying cell migration. On one hand, rapid advances in experimental techniques allow for detailed experimental measurements of cell migration pathways, while, on the other, computational approaches allow for the modelling, analysis and understanding of such observations. The bulk-surface mechanobiochemical modelling approach presented in this work, set premises to study single cell migration through complex non-isotropic environments in two- and three-space dimensions.

Residual-based reduced order models for parameterized Navier–Stokes equations with nonhomogeneous boundary condition

In this paper, we propose a residual-based reduced-order model (ROM) framework that utilizes available data to increase the ROM accuracy and stability. The available snapshots are utilized to obtain the original ROM systems and their projection coefficients by performing proper orthogonal decomposition. Then a time-parameter varying closure term is added to the original ROM systems to obtain the new ROM systems, and the values of the closure term at discrete time-parameter points with respect to snapshots are evaluated by computing the residual of the original ROM systems with projection coefficients. In an online testing stage, the values of the closure term at unknown time-parameter points are approximated by linear interpolation, and the new ROM systems are utilized to approximate solutions for unseen parameter values. In addition, the closure term in the new ROM systems is instead discarded when we predict systems' evolution outside the time interval with respect to snapshots. Numerical results show that the proposed method not only improves the prediction accuracy of original ROM but also extends the applicability of it.

On analysis of a system of non-homogenous boundary value problems using hausdorff derivative with exponential kernel

AbstractIn recent years, the fractals (Hausdorff) derivatives with fractional order under various types kernel have gained attention from researchers. The aforesaid area has many applications in the description of intricate and irregular geometry of various processes. Numerous studies utilizing the fractional derivatives (HFDs) for initial value problems have been carried out. But the boundary value problems using the said concepts have been very rarely studied. Thus, a coupled system with non-homogenous boundary conditions (BCs) is examined in this study by using fractals fractional derivative in Caputo Fabrizio sense. To establish the required conditions for the existence and uniqueness of solution to the considered problem, we apply the Banach and Krasnoselskii’s fixed point theorems. Furthermore, some results related to Hyers-Ulam (H-U) stability have also deduced. We have included two pertinent examples to verify our results.

Generalized Rainbow Differential Privacy

We study a new framework for designing differentially private (DP) mechanisms via randomized graph colorings, called rainbow differential privacy. In this framework, datasets are nodes in a graph, and two neighboring datasets are connected by an edge. Each dataset in the graph has a preferential ordering for the possible outputs of the mechanism, and these orderings are called rainbows. Different rainbows partition the graph of connected datasets into different regions. We show that if a DP mechanism at the boundary of such regions is fixed and it behaves identically for all same-rainbow boundary datasets, then a unique optimal $(\epsilon,\delta)$-DP mechanism exists (as long as the boundary condition is valid) and can be expressed in closed-form. Our proof technique is based on an interesting relationship between dominance ordering and DP, which applies to any finite number of colors and for $(\epsilon,\delta)$-DP, improving upon previous results that only apply to at most three colors and for $\epsilon$-DP. We justify the homogeneous boundary condition assumption by giving an example with non-homogeneous boundary condition, for which there exists no optimal DP mechanism.

A Necessary Optimality Condition on the Control of a Charged Particle

We consider an optimal control problem with the boundary functional for a Schrödinger equation describing the motion of a charged particle. By using the existence of an optimal solution, we search the necessary optimality conditions for the examined control problem. First, we constitute an adjoint problem by a Lagrange multiplier that is related to constraints of theory on symmetries and conservation laws. The adjoint problem obtained is a boundary value problem with a nonhomogeneous boundary condition. We prove the existence and uniqueness of the solution of the adjoint problem. Then, we demonstrate the differentiability of the objective functional in the sense of Frechet and get a formula for its gradient. Finally, we give a necessary optimality condition in the form of a variational inequality.

A Boundary Control Problem for Stochastic 2D-Navier–Stokes Equations

AbstractWe study a stochastic velocity tracking problem for the 2D-Navier–Stokes equations perturbed by a multiplicative Gaussian noise. From a physical point of view, the control acts through a boundary injection/suction device with uncertainty, modeled by stochastic non-homogeneous Navier-slip boundary conditions. We show the existence and uniqueness of the solution to the state equation, and prove the existence of an optimal solution to the control problem.

Finite Difference Method for Solving Parabolic-Type Integro-Differential Equations in Multidimensional Domain with Nonhomogeneous First-Order Boundary Conditions

Introduction. We investigate a multidimensional (in terms of spatial variables) parabolic-type integro-differential equation with nonhomogeneous first-order boundary conditions. The locally one-dimensional finite difference scheme developed herein can be applied to solve various applied problems leading to multidimensional parabolic-type integro-differential equations. Examples include mathematical modelling of cloud processes, addressing the issue of active intervention in convective clouds to prevent hail and artificially enhance precipitation, as well as describing the droplet mass distribution function due to microphysical processes such as condensation, coagulation, fragmentation, and freezing of droplets in convective clouds.Materials and Methods. In this study, an approximate solution to the initial-boundary value problem is constructed using the locally one-dimensional scheme of A.A. Samarsky with a specified order of approximation О(h2 +τ). The primary research method employed is the method of energy inequalities.Results. A priori estimates have been obtained in the discrete interpretation, from which uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem follow, with a convergence rate equal to the order of approximation of the difference scheme.Discussion and Conclusions. The research findings can be utilized for further development of boundary value problem theory for parabolic equations with variable coefficients. Additionally, they may find applications in the fields of difference scheme theory, computational mathematics, and numerical modelling.

Solving Boundary Value Problems by Sinc Method and Geometric Sinc Method

This paper introduces an efficient numerical method for approximating solutions to geometric boundary value problems. We propose the multiplicative sinc–Galerkin method, tailored specifically for solving multiplicative differential equations. The method utilizes the geometric Whittaker cardinal function to approximate functions and their geometric derivatives. By reducing the geometric differential equation to a system of algebraic equations, we achieve computational efficiency. The method not only proves to be computationally efficient but also showcases a valuable symmetric property, aligning with inherent patterns in geometric structures. This symmetry enhances the method’s compatibility with the often-present symmetries in geometric boundary value problems, offering both computational advantages and a deeper understanding of geometric calculus. To demonstrate the reliability and efficiency of the proposed method, we present several examples with both homogeneous and non-homogeneous boundary conditions. These examples serve to validate the method’s performance in practice.

An efficient unconditional energy stable scheme for the simulation of droplet formation

We have developed an efficient and unconditionally energy-stable method for simulating droplet formation dynamics. Our approach involves a novel time-marching scheme based on the scalar auxiliary variable technique, specifically designed for solving the Cahn-Hilliard-Navier-Stokes phase field model with variable density and viscosity. We have successfully applied this method to simulate droplet formation in scenarios where a Newtonian fluid is injected through a vertical tube into another immiscible Newtonian fluid. To tackle the challenges posed by nonhomogeneous Dirichlet boundary conditions at the tube entrance, we have introduced additional nonlocal auxiliary variables and associated ordinary differential equations. These additions effectively eliminate the influence of boundary terms. Moreover, we have incorporated stabilization terms into the scheme to enhance its numerical effectiveness. Notably, our resulting scheme is fully decoupled, requiring the solution of only linear systems at each time step. We have also demonstrated the energy decaying property of the scheme, with suitable modifications. To assess the accuracy and stability of our algorithm, we have conducted extensive numerical simulations. Additionally, we have examined the dynamics of droplet formation and explored the impact of dimensionless parameters on the process. Overall, our work presents a refined method for simulating droplet formation dynamics, offering improved efficiency, energy stability, and accuracy.

Mixed convection flow in a square lid-driven cavity subject to inclined magnetic field with highly accurate wavelet-homotopy solutions

Numerical studies of flow in square enclosures attain great interest due to their wide applications in engineering, such as in the meteorology, navigation, mechanology and mining, etc. In this paper, the solutions are obtained to the square lid driven cavity describing classical mixed convection in the presence of an inclined magnetic field, chemical radiation and heat absorption/generation. The problem is mathematically reduced into Navier-Stokes equations. The Coiflet wavelet homotopy approach, a newly suggested computational technique, is then used to solve these nonlinear partial differential equations with their non-homogeneous boundary conditions. The numerical solutions for streamlines, isotherms, iso-concentrations contours, Nusselt number are carefully determined. The solutions so obtained are analyzed through various plots to demonstrate the effects of several physical parameters such as Reynolds numbers (Re), Richardson number (Ri), Grashof number (Gr), Hartman number (Ha), thermal radiation parameter (NR), chemical reaction coefficient (K), heat generation/absorption coefficient (Qh), the angle of inclination of magnetic field (γ), slipping parameter (λ) and Schmidt number (Sc) and their physical insights are also reported. Rigid comparison of with previous studies shows excellent computational effectiveness of newly proposed method. Additionally, according to the results, our recently proposed approach has superior characteristics and strong nonlinear processing capabilities, which render it more appropriate than the Coiflet wavelet analysis method and the traditional homotopy analysis method for tackling complex nonlinear problems.

A multi-domain spectral collocation method for the Fokker–Planck equation in an infinite channel

In this paper, we propose a multi-domain spectral collocation method for partial differential equations on two-dimensional unbounded domains. Some approximation results on the composite generalized Laguerre-Legendre interpolation and quasi-orthogonal projecting are established, respectively. These results play a significant role in related spectral collocation method. As an application, a multi-domain spectral collocation scheme is provided for the Fokker–Planck equation with absorption or non-homogeneous boundary conditions. The convergence of the proposed algorithm is performed. An efficient implementation is presented. Numerical experiments demonstrate the effectiveness and high accuracy of the algorithm.

Analytical approach to solving linear diffusion–advection–reaction equations with local and nonlocal boundary conditions

Initial–boundary value problems for a linear diffusion–advection–reaction equation are considered, with general nonhomogeneous linear boundary conditions and general linear nonlocal boundary conditions. Analytical solutions are obtained using an embedding method. The solutions are expressed in terms of time‐varying functions that satisfy coupled linear Volterra integral equations of the first kind. A boundary element method is applied to numerically solve the integral equations. Three examples are given to demonstrate the accuracy of the numerical solutions when compared with the analytical solutions. The embedding method is applicable to problems with bounded and unbounded spatial domains.

Critical Point Approaches for a Class of Differential Equations with Sturm-Liouville Type Nonhomogeneous Boundary Conditions

A class of p-Laplacian equations with Sturm-Liouville type nonhomogeneous boundary value problem with nonlinear derivative depending on two control parameters is investigated. Existence and multiplicity of solutions are discussed by means of variational methods and critical point theory. Two examples supporting our theoretical results are also presented.

Boundary regularity results for minimisers of convex functionals with (p, q)-growth

Abstract We prove improved differentiability results for relaxed minimisers of vectorial convex functionals with ( p , q ) \left(p,q) -growth, satisfying a Hölder-growth condition in x x . We consider both Dirichlet and Neumann boundary data. In addition, we obtain a characterisation of regular boundary points for such minimisers. In particular, in case of homogeneous boundary conditions, this allows us to deduce partial boundary regularity of relaxed minimisers on smooth domains for radial integrands. We also obtain some partial boundary regularity results for non-homogeneous Neumann boundary conditions.

A three-step defect-correction stabilized algorithm for incompressible flows with non-homogeneous Dirichlet boundary conditions

A three-step defect-correction stabilized algorithm for incompressible flows with non-homogeneous Dirichlet boundary conditions

A hybrid Lagrangian–Eulerian particle finite element method for free‐surface and fluid–structure interaction problems

AbstractThe dynamics of fluid flows with free surfaces and interacting with highly deformable structures is a complex problem, attracting considerable attention. The Particle Finite Element Method (PFEM) is one of the various numerical methods recently proposed in the literature to simulate this type of problems. It is a mesh‐based Lagrangian approach, particularly suited for problems with fast changes in the domain topology, since the fluid boundaries and the Fluid–Structure Interaction (FSI) interface are naturally tracked by the position of the mesh nodes. However, when nonhomogeneous boundary conditions are imposed on velocities or when there are regions where the topology varies moderately, for example, in confined portions of the fluid domain characterized by fixed boundaries, an Eulerian formulation turns out to be more convenient. To exploit the advantages of both formulations, an adaptive hybrid Lagrangian–Eulerian approach is presented in this work. According to the proposed method, nodes on the fluid free‐surface and on the FSI interface are treated as Lagrangian, while the remaining nodes can be either Eulerian or Lagrangian. Furthermore, to increase the efficiency of the method, an algorithm to automatically detect runtime the transition zone between the two kinematic descriptions is devised. To validate the proposed approach, several numerical examples are developed and their results are compared to those available in the literature.