Showing papers in "Applications of Mathematics in 2021"

Uniqueness of Weak Solutions to a Keller-Segel-Navier-Stokes Model with a Logistic Source

Abstract: We prove a uniqueness result of weak solutions to the nD (n ⩾ 3) Cauchy problem of a Keller-Segel-Navier-Stokes system with a logistic term.

Exact Solution of the Time Fractional Variant Boussinesq-Burgers Equations

Abstract: In the present article, we consider a nonlinear time fractional system of variant Boussinesq-Burgers equations. Using Lie group analysis, we derive the infinitesimal groups of transformations containing some arbitrary constants. Next, we obtain the system of optimal algebras for the symmetry group of transformations. Afterward, we consider one of the optimal algebras and construct similarity variables, which reduces the given system of fractional partial differential equations (FPDEs) to fractional ordinary differential equations (FODEs). Further, under the invariance condition we construct the exact solution and the physical significance of the solution is investigated graphically. Finally, we study the conservation law of the system of equations.

Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method

Abstract: A method of characterizing all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in Gergelits, Mardal, Nielsen, and Strakos (2019). Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elasticity operator with tensor data, discretized with the conforming finite element method, and preconditioned by the inverse of a matrix of the same operator with different data. Our results hold for mixed Dirichlet and Robin or periodic boundary conditions applied to the original and preconditioning problems. The bounds are two-sided, guaranteed, easily accessible, and depend solely on the material data.

On the stabilization of laminated beams with delay

Abstract: Of concern in this paper is the laminated beam system with frictional damping and an internal constant delay term in the transverse displacement. Under suitable assumptions on the weight of the delay, we establish that the system’s energy decays exponentially in the case of equal wave speeds of propagation, and polynomially in the case of non-equal wave speeds.

Variational Gaussian Process for Optimal Sensor Placement

Abstract: Sensor placement is an optimisation problem that has recently gained great relevance. In order to achieve accurate online updates of a predictive model, sensors are used to provide observations. When sensor location is optimally selected, the predictive model can greatly reduce its internal errors. A greedy-selection algorithm is used for locating these optimal spatial locations from a numerical embedded space. A novel architecture for solving this big data problem is proposed, relying on a variational Gaussian process. The generalisation of the model is further improved via the preconditioning of its inputs: Masked Autoregressive Flows are implemented to learn nonlinear, invertible transformations of the conditionally modelled spatial features. Finally, a global optimisation strategy extending the Mutual Information-based optimisation and fine-tuning of the selected optimal location is proposed. The methodology is parallelised to speed up the computational time, making these tools very fast despite the high complexity associated with both spatial modelling and placement tasks. The model is applied to a real three-dimensional test case considering a room within the Clarence Centre building located in Elephant and Castle, London, UK.

On a Deformed Version of the Two-Disk Dynamo System

Abstract: We give some deformations of the Rikitake two-disk dynamo system. Particularly, we consider an integrable deformation of an integrable version of the Rikitake system. The deformed system is a three-dimensional Hamilton-Poisson system. We present two Lie-Poisson structures and also symplectic realizations. Furthermore, we give a prequantization result of one of the Poisson manifold. We study the stability of the equilibrium states and we prove the existence of periodic orbits. We analyze some properties of the energy-Casimir mapping $${\cal E}{\cal C}$$ associated to our system. In many cases the dynamical behavior of such systems is related with some geometric properties of the image of the energy-Casimir mapping. These connections were observed in the cases when the image of EC is a convex proper subset of ℝ2. In order to point out new connections, we choose deformation functions such that $${\mathop{\rm Im} olimits} \left({{\cal E}{\cal C}} \right) = {\mathbb{R}^2}$$ . Using the images of the equilibrium states through the energy-Casimir mapping we give parametric equations of some special orbits, namely heteroclinic orbits, split-heteroclinic orbits, and split-homoclinic orbits. Finally, we implement the mid-point rule to perform some numerical integrations of the considered system.

A New Model to Describe the Response of a Class of Seemingly Viscoplastic Materials

Abstract: A new model is proposed to mimic the response of a class of seemingly viscoplastic materials. Using the proposed model, the steady, fully developed flow of the fluid is studied in a cylindrical pipe. The semi-inverse approach is applied to obtain an analytical solution for the velocity profile. The model is used to fit the shear-stress data of several supposedly viscoplastic materials reported in the literature. A numerical procedure is developed to solve the governing ODE and the procedure is validated by comparison with the analytical solution.

A New Nonmonotone Adaptive Trust Region Algorithm

Abstract: We propose a new and efficient nonmonotone adaptive trust region algorithm to solve unconstrained optimization problems. This algorithm incorporates two novelties: it benefits from a radius dependent shrinkage parameter for adjusting the trust region radius that avoids undesirable directions and exploits a new strategy to prevent sudden increments of objective function values in nonmonotone trust region techniques. Global convergence of this algorithm is investigated under some mild conditions. Numerical experiments demonstrate the efficiency and robustness of the proposed algorithm in solving a collection of unconstrained optimization problems from the CUTEst package.

Continuous Dependence on Parameters and Boundedness of Solutions to a Hysteresis System

Abstract: We analyze an ordinary differential system with a hysteresis-relay nonlinearity in two cases when the system is autonomous or nonautonomous. Sufficient conditions for both the continuous dependence on the system parameters and the boundedness of the solutions to the system are obtained. We give a supporting example for the autonomous system.

An XFEM/DG Approach for Fluid-Structure Interaction Problems with Contact

Abstract: In this work, we address the problem of fluid-structure interaction (FSI) with moving structures that may come into contact. We propose a penalization contact algorithm implemented in an unfitted numerical framework designed to treat large displacements. In the proposed method, the fluid mesh is fixed and the structure meshes are superimposed to it without any constraint on the conformity. Thanks to the Extended Finite Element Method (XFEM), we can treat discontinuities of the fluid solution on the mesh elements intersecting the structure. The coupling conditions at the fluid-structure interface are enforced via a discontinuous Galerkin mortaring technique, which is a penalization method that ensures the consistency of the scheme with the underlining problem. Concerning the contact problem, we consider a frictionless contact model in a master/slave approach. By considering the coupled FSI-contact problem, we perform some numerical tests to assess the sensitivity of the proposed method with respect to the discretization and contact parameters and we show some examples in the case of contact between a flexible body and a rigid wall and between two deformable structures.

A Spatially Sixth-Order Hybrid L1-CCD Method for Solving Time Fractional Schrödinger Equations

Abstract: We consider highly accurate schemes for nonlinear time fractional Schrodinger equations (NTFSEs). While an L1 strategy is employed for approximating the Caputo fractional derivative in the temporal direction, compact CCD finite difference approaches are incorporated in the space. A highly effective hybrid L1-CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order 2 − γ in time, where 0 < γ < 1 is the order of the Caputo fractional derivative involved. It is proved rigorously that the hybrid numerical method accomplished is unconditionally stable in the Fourier sense. Numerical experiments are carried out with typical testing problems to validate the effectiveness of the new algorithms.

Solution of option pricing equations using orthogonal polynomial expansion

Abstract: We study both analytic and numerical solutions of option pricing equations using systems of orthogonal polynomials Using a Galerkin-based method, we solve the parabolic partial differential equation for the Black-Scholes model using Hermite polynomials and for the Heston model using Hermite and Laguerre polynomials We compare the obtained solutions to existing semi-closed pricing formulas Special attention is paid to the solution of the Heston model at the boundary with vanishing volatility

Non-Local Damage Modelling of Quasi-Brittle Composites

Abstract: Most building materials can be characterized as quasi-brittle composites with a cementitious matrix, reinforced by some stiffening particles or elements. Their massive exploitation motivates the development of numerical modelling and simulation of behaviour of such material class under mechanical, thermal, etc. loads, including the evaluation of the risk of initiation and development of micro- and macro-fracture. This paper demonstrates the possibility of certain deterministic prediction, applying the dynamical approach using the Kelvin viscoelastic model and cohesive interface properties. The existence and convergence results rely on the semilinear computational scheme coming from the method of discretization in time, using several types of Rothe sequences, coupled with the extended finite element method (XFEM) for practical calculations. Numerical examples refer to cementitious samples reinforced by short steel fibres, with increasing number of applications as constructive parts in civil engineering.

A Convergence Result and Numerical Study for a Nonlinear Piezoelectric Material in a Frictional Contact Process with a Conductive Foundation

Abstract: We consider two static problems which describe the contact between a piezoelectric body and an obstacle, the so-called foundation. The constitutive relation of the material is assumed to be electro-elastic and involves the nonlinear elastic constitutive Hencky’s law. In the first problem, the contact is assumed to be frictionless, and the foundation is nonconductive, while in the second it is supposed to be frictional, and the foundation is electrically conductive. The contact is modeled with the normal compliance condition with finite penetration, the regularized Coulomb law, and the regularized electrical conductivity condition. The existence and uniqueness results are provided using the theory of variational inequalities and Schauder’s fixed-point theorem. We also prove that the solution of the latter problem converges towards that of the former as the friction and electrical conductivity coefficients converge towards zero. The numerical solutions of the problems are achieved by using a successive iteration technique; their convergence is also established. The numerical treatment of the contact condition is realized using an Augmented Lagrangian type formulation that leads us to use Uzawa type algorithms. Numerical experiments are performed to show that the numerical results are consistent with the theoretical analysis.

Asymptotic Lower Bounds for Eigenvalues of the Steklov Eigenvalue Problem with Variable Coefficients

Abstract: In this paper, using a new correction to the Crouzeix-Raviart finite element eigenvalue approximations, we obtain asymptotic lower bounds of eigenvalues for the Steklov eigenvalue problem with variable coefficients on d-dimensional domains (d = 2, 3). In addition, we prove that the corrected eigenvalues converge to the exact ones from below. The new result removes the conditions of eigenfunction being singular and eigenvalue being large enough, which are usually required in the existing arguments about asymptotic lower bounds. Further, we prove that the corrected eigenvalues still maintain the same convergence order as uncorrected eigenvalues. Finally, numerical experiments validate our theoretical results.

Herbivore Harvesting and Alternative Steady States in Coral Reefs

Abstract: Coral reefs can undergo relatively rapid changes in the dominant biota, a phenomenon referred to as phase shift. Degradation of coral reefs is often associated with changes in community structure towards a macroalgae-dominated reef ecosystem due to the reduction in herbivory caused by overfishing. We investigate the coral-macroalgal phase shift due to the effects of harvesting of herbivorous reef fish by means of a continuous time model in the food chain. Conditions for local asymptotic stability of steady states are derived. We have shown that under certain conditions the system is uniformly persistent in presence of all the organisms. Moreover, it is shown that the system undergoes a Hopf bifurcation when the carrying capacity of macroalgae crosses certain critical value. Computer simulations have been carried out to illustrate different analytical results.

An Improved Regularity Criteria for the MHD System Based on Two Components of the Solution

Abstract: As observed by Yamazaki, the third component b3 of the magnetic field can be estimated by the corresponding component u3 of the velocity field in Lλ (2 ⩽ λ ⩽ 6) norm. This leads him to establish regularity criterion involving u3, j3 or u3, ω3. Noticing that λ can be greater than 6 in this paper, we can improve previous results.

Construction of convergent adaptive weighted essentially non-oscillatory schemes for Hamilton-Jacobi equations on triangular meshes

Abstract: We propose a method of constructing convergent high order schemes for Hamilton-Jacobi equations on triangular meshes, which is based on combining a high order scheme with a first order monotone scheme. According to this methodology, we construct adaptive schemes of weighted essentially non-oscillatory type on triangular meshes for non-convex Hamilton-Jacobi equations in which the first order monotone approximations are occasionally applied near singular points of the solution (discontinuities of the derivative) instead of weighted essentially non-oscillatory approximations. Through detailed numerical experiments, the convergence and effectiveness of the proposed adaptive schemes are demonstrated.

On inversions of van der Grinten projections

Abstract: Approximately 150 map projections are known, but the inverse forms have been published for only two-thirds of them. This paper focuses on finding the inverse forms of van der Grinten projections I-IV, both by non-linear partial differential equations and by the straightforward inverse of their projection equations. Taking into account the particular cases, new derivations of coordinate functions are also presented. Both the direct and inverse equations have the analytic form, are easy to implement and are applicable to the coordinate transformations.

Solvability of the Power Flow Problem in DC Overhead Wire Circuit Modeling

Abstract: Proper traffic simulation of electric vehicles, which draw energy from overhead wires, requires adequate modeling of traction infrastructure. Such vehicles include trains, trams or trolleybuses. Since the requested power demands depend on a traffic situation, the overhead wire DC electrical circuit is associated with a non-linear power flow problem. Although the Newton-Raphson method is well-known and widely accepted for seeking its solution, the existence of such a solution is not guaranteed. Particularly in situations where the vehicle power demands are too high (during acceleration), the solution of the studied problem may not exist. To deal with such cases, we introduce a numerical method which seeks maximal suppliable power demands for which the solution exists. This corresponds to introducing a scaling parameter to reduce the demanded power. The interpretation of the scaling parameter is the amount of energy which is absent in the system, and which needs to be provided by external sources such as on-board batteries. We propose an efficient two-stage algorithm to find the optimal scaling parameter and the resulting potentials in the overhead wire network. We perform a comparison with a naive approach and present a real-world simulation in the part of the Pilsen city in the Czech Republic. These simulations are performed in the traffic micro-simulator SUMO, a popular open-source traffic simulation platform.

The Effect of a Magnetic Field on the Onset Of Bénard Convection in Variable Viscosity Couple-Stress Fluids Using Classical Lorenz Model

Abstract: The Rayleigh-Benard convection for a couple-stress fluid with a thermorheological effect in the presence of an applied magnetic field is studied using both linear and non-linear stability analysis. This problem discusses the three important mechanisms that control the onset of convection; namely, suspended particles, an applied magnetic field, and variable viscosity. It is found that the thermorheological parameter, the couple-stress parameter, and the Chandrasekhar number influence the onset of convection. The effect of an increase in the thermorheological parameter leads to destabilization in the system, while the Chandrasekhar number and the couple-stress parameter have the opposite effect. The generalized Lorenz’s model of the problem is essentially the classical Lorenz model but with coefficients involving the impact of three mechanisms as discussed earlier. The classical Lorenz model is a fifth-order autonomous system and found to be analytically intractable. Therefore, the Lorenz system is solved numerically using the Runge-Kutta method in order to quantify heat transfer. An effect of increasing the thermorheological parameter is found to enhance heat transfer, while the couple-stress parameter and the Chandrasekhar number diminishes the same.

The Generalized Finite Volume Sushi Scheme for the Discretization of the Peaceman Model

Abstract: We demonstrate some a priori estimates of a scheme using stabilization and hybrid interfaces applying to partial differential equations describing miscible displacement in porous media. This system is made of two coupled equations: an anisotropic diffusion equation on the pressure and a convection-diffusion-dispersion equation on the concentration of invading fluid. The anisotropic diffusion operators in both equations require special care while discretizing by a finite volume method SUSHI. Later, we present some numerical experiments.

Dirichlet Boundary Value Problem for an Impulsive Forced Pendulum Equation with Viscous and Dry Frictions

Abstract: Sufficient conditions are given for the solvability of an impulsive Dirichlet boundary value problem to forced nonlinear differential equations involving the combination of viscous and dry frictions. Apart from the solvability, also the explicit estimates of solutions and their derivatives are obtained. As an application, an illustrative example is given, and the corresponding numerical solution is obtained by applying Matlab software.

A frictional contact problem with adhesion for viscoelastic materials with long memory

Abstract: We consider a quasistatic contact problem between a viscoelastic material with long-term memory and a foundation. The contact is modelled with a normal compliance condition, a version of Coulomb’s law of dry friction and a bonding field which describes the adhesion effect. We derive a variational formulation of the mechanical problem and, under a smallness assumption, we establish an existence theorem of a weak solution including a regularity result. The proof is based on the time-discretization method, the Banach fixed point theorem and arguments of lower semicontinuity, compactness and monotonicity.

A new energy conservative scheme for regularized long wave equation

Abstract: An energy conservative scheme is proposed for the regularized long wave (RLW) equation. The integral method with variational limit is used to discretize the spatial derivative and the finite difference method is used to discretize the time derivative. The energy conservation of the scheme and existence of the numerical solution are proved. The convergence of the order O(h2 + τ2) and unconditional stability are also derived. Numerical examples are carried out to verify the correctness of the theoretical analysis.

Logarithmic Stabilization of the Kirchhoff Plate Transmission System with Locally Distributed Kelvin-Voigt Damping

Abstract: We are concerned with a transmission problem for the Kirchhoff plate equation where one small part of the domain is made of a viscoelastic material with the Kelvin-Voigt constitutive relation. We obtain the logarithmic stabilization result (explicit energy decay rate), as well as the wellposedness, for the transmission system. The method is based on a new Carleman estimate to obtain information on the resolvent for high frequency. The main ingredient of the proof is some careful analysis for the Kirchhoff transmission plate equation.

Verified Numerical Computations for Large-Scale Linear Systems

Abstract: This paper concerns accuracy-guaranteed numerical computations for linear systems. Due to the rapid progress of supercomputers, the treatable problem size is getting larger. The larger the problem size, the more rounding errors in floating-point arithmetic can accumulate in general, and the more inaccurate numerical solutions are obtained. Therefore, it is important to verify the accuracy of numerical solutions. Verified numerical computations are used to produce error bounds on numerical solutions. We report the implementation of a verification method for large-scale linear systems and some numerical results using the RIKEN K computer and the Fujitsu PRIMEHPC FX100, which show the high performance of the verified numerical computations.

Global existence and Lp decay estimate of solution for Cahn-Hilliard equation with inertial term

Abstract: The Cauchy problem of the Cahn-Hilliard equation with inertial term in multi space dimension is considered Based on detailed analysis of Green’s function, using fixed-point theorem, we get the global existence in time of classical solution with large initial data Furthermore, we get Lp decay rate of the solution

Kinetic BGK model for a crowd: Crowd characterized by a state of equilibrium

Abstract: This article focuses on dynamic description of the collective pedestrian motion based on the kinetic model of Bhatnagar-Gross-Krook. The proposed mathematical model is based on a tendency of pedestrians to reach a state of equilibrium within a certain time of relaxation. An approximation of the Maxwellian function representing this equilibrium state is determined. A result of the existence and uniqueness of the discrete velocity model is demonstrated. Thus, the convergence of the solution to that of the continuous BGK equation is proven. Numerical simulations are presented to validate the proposed mathematical model.

A blow-up criterion for the strong solutions to the nonhomogeneous Navier-Stokes-Korteweg equations in dimension three

Abstract: This paper proves a Serrin’s type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density ϱ and velocity field u satisfy $$\Vert abla\varrho\Vert_{L^{\infty}(0,T;W^{1,q})}+\Vert u\Vert_{L^{s}(0,T;L_{\omega}^{r})}<\infty$$ for some q > 3 and any (r, s) satisfying 2/s + 3/r ⩽ 1, 3 < r ⩽ ∞, then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over [0, T]. Here L denotes the weak Lr space.