Extended scaling invariance of one‐dimensional models of liquid dynamics in a horizontal capillary

In this paper, we consider the adaptive numerical solution of one‐dimensional models of liquid dynamics in a horizontal capillary. The bulk liquid is assumed to be initially at rest and is put into motion by capillarity: the smaller is the capillary radius, the steeper becomes the initial transitory of the meniscus location derivative, and as a consequence, the numerical solution to a prescribed accuracy becomes harder to achieve. Therefore, in order to solve a capillary problem effectively, it would be advisable to apply an adaptive numerical method. Here, we show how an extended scaling invariance that can be used to define a family of solutions from a computed one. In the viscous case, the similarity transformation involves solutions of liquids with different density, surface tension, viscosity, and capillary radii, whereas in the inviscid case, we can generate a family of solutions for the same liquid and capillaries with different radii. With our study, we are able to prove that the monitor function, used in the adaptive algorithm, is invariant with respect to the considered scaling group. It follows, from this particular results, that all the solutions within the generated family verify the adaptive criteria used for the computed one. Moreover, all the solutions have the same order of accuracy even if the maximum value of the step size varies under the action of the scaling group. Copyright © 2012 John Wiley & Sons, Ltd.     

Simplified immersed interface methods for elliptic interface problems with straight interfaces

In this article, we propose simplified immersed interface methods for elliptic partial/ordinary differential equations with discontinuous coefficients across interfaces that are few isolated points in 1D, and straight lines in 2D. For one‐dimensional problems or two‐dimensional problems with circular interfaces, we propose a conservative second‐order finite difference scheme whose coefficient matrix is symmetric and definite. For two‐dimensional problems with straight interfaces, we first propose a conservative first‐order finite difference scheme, then use the Richardson extrapolation technique to get a second‐order method. In both cases, the finite difference coefficients are almost the same as those for regular problems. Error analysis is given along with numerical example. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 188–203, 2012     

Convergence analysis of a vertex‐centered finite volume scheme for a copper heap leaching model

In this paper a two‐dimensional solute transport model is considered to simulate the leaching of copper ore tailing using sulfuric acid as the leaching agent. The mathematical model consists in a system of differential equations: two diffusion–convection‐reaction equations with Neumann boundary conditions, and one ordinary differential equation. The numerical scheme consists in a combination of finite volume and finite element methods. A Godunov scheme is used for the convection term and an P1‐FEM for the diffusion term. The convergence analysis is based on standard compactness results in L². Some numerical examples illustrate the effectiveness of the scheme. Copyright © 2009 John Wiley & Sons, Ltd.     

Stability and bifurcation in epidemic models describing the transmission of toxoplasmosis in human and cat populations

A five‐dimensional ordinary differential equation model describing the transmission of Toxoplamosis gondii disease between human and cat populations is studied in this paper. Self‐diffusion modeling the spatial dynamics of the T. gondii disease is incorporated in the ordinary differential equation model. The normalized version of both models where the unknown functions are the proportions of the susceptible, infected, and controlled individuals in the total population are analyzed. The main results presented herein are that the ODE model undergoes a trans‐critical bifurcation, the system has no periodic orbits inside the positive octant, and the endemic equilibrium is globally asymptotically stable when we restrict the model to inside of the first octant. Furthermore, a local linear stability analysis for the spatially homogeneous equilibrium points of the reaction diffusion model is carried out, and the global stability of both the disease‐free and endemic equilibria are established for the reaction–diffusion system when restricted to inside of the first octant. Finally, numerical simulations are provided to support our theoretical results and to predict some scenarios about the spread of the disease. Copyright © 2017 John Wiley & Sons, Ltd.     

A second‐order finite difference approximation for a mathematical model of erythropoiesis

We present a second‐order finite difference scheme for approximating solutions of a mathematical model of erythropoiesis, which consists of two nonlinear partial differential equations and one nonlinear ordinary differential equation. We show that the scheme achieves second‐order accuracy for smooth solutions. We compare this scheme to a previously developed first‐order method and show that the first order method requires significantly more computational time to provide solutions with similar accuracy. We also compare this numerical scheme with other well‐known second‐order methods and show that it has better capability in approximating discontinuous solutions. Finally, we present an application to recovery after blood loss. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Singly diagonally implicit runge‐kutta method for time‐dependent reaction‐diffusion equation

In this article, an efficient fourth‐order accurate numerical method based on Padé approximation in space and singly diagonally implicit Runge‐Kutta method in time is proposed to solve the time‐dependent one‐dimensional reaction‐diffusion equation. In this scheme, we first approximate the spatial derivative using the second‐order central finite difference then improve it to fourth‐order by applying Padé approximation. A three stage fourth‐order singly diagonally implicit Runge‐Kutta method is then used to solve the resulting system of ordinary differential equations. It is also shown that the scheme is unconditionally stable, and is suitable for stiff problems. Several numerical examples are solved by the scheme and the efficiency and accuracy of the new scheme are compared with two widely used high‐order compact finite difference methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1423–1441, 2011     

Example of a suspension bridge ODE model exhibiting chaotic dynamics: A topological approach

Using an elementary phase-plane analysis combined with some recent results on topological horseshoes and fixed points for planar maps, we prove the existence of infinitely many periodic solutions as well as the presence of chaotic dynamics for a simple second order nonlinear ordinary differential equation arising in the study of Lazer-McKenna suspension bridges model.     

Solitary Waves and N‐Particle Algorithms for a Class of Euler–Poincaré Equations

We study a class of partial differential equations (PDEs) in the family of the so‐called Euler–Poincaré differential systems, with the aim of developing a foundation for numerical algorithms of their solutions. This requires particular attention to the mathematical properties of this system when the associated class of elliptic operators possesses nonsmooth kernels. By casting the system in its Lagrangian (or characteristics) form, we first formulate a particle system algorithm in free space with homogeneous Dirichlet boundary conditions for the evolving fields. We next examine the deformation of the system when nonhomogeneous “constant stream” boundary conditions are assumed. We show how this simple change at the boundary deeply affects the nature of the evolution, from hyperbolic‐like to dispersive with a nontrivial dispersion relation, and examine the potentially regularizing properties of singular kernels offered by this deformation. From the particle algorithm viewpoint, kernel singularities affect the existence and uniqueness of solutions to the corresponding ordinary differential equations systems. We illustrate this with the case when the operator kernel assumes a conical shape over the spatial variables, and examine in detail two‐particle dynamics under the resulting lack of Lipschitz continuity. Curiously, we find that for the conically shaped kernels the motion of the related two‐dimensional waves can become completely integrable under appropriate initial data. This reduction projects the two‐dimensional system to the one‐dimensional completely integrable Shallow‐Water equation [1], while retaining the full dependence on two spatial dimensions for the single channel solutions. Finally, by comparing with an operator‐splitting pseudospectral method we illustrate the performance of the particle algorithms with respect to their Eulerian counterpart for this class of nonsmooth kernels.     

Stability of the space identification problem for the elliptic‐telegraph differential equation

The present paper is devoted to study the space identification problem for the elliptic‐telegraph differential equation in Hilbert spaces with the self‐adjoint positive definite operator. The main theorem on the stability of the space identification problem for the elliptic‐telegraph differential equation is proved. In applications, theorems on the stability of three source identification problems for one dimensional with nonlocal conditions and multidimensional elliptic‐telegraph differential equations are established.     

On synchronization of a forced delay dynamical system via the Galerkin approximation

A forced scalar delay dynamical system is analyzed from the perspective of bifurcation and synchronization. In general first order differential equations do not exhibit chaos, but introduction of a delay feedback makes the system infinite dimensional and shows chaoticity. In order to study the dynamics of such a system, Galerkin projection technique is used to obtain a finite dimensional set of ordinary differential equations from the delay differential equation. We compare the results of simulation with those obtained from direct numerical simulation of the delay equation to ascertain the accuracy of the truncation process in the Galerkin approximation. We have considered two cases, one with five and the other with eight shape functions. Next we study two types of synchronization by considering coupling of the above derived equations with a forced dynamical system without delay. Our analysis shows that it is possible to have synchronization between two such systems. It has been shown that the chaotic system with delay feedback can drive the system without delay to achieve synchronization and the opposite case is also equally valid. This is confirmed by the evaluation of the conditional Lyapunov exponents of the systems.     

MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems

We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.     

Spots and stripes in nonlinear Schrödinger‐type equations with nearly one‐dimensional potentials

We consider the existence of spots and stripes for a class of nonlinear Schrödinger‐type equations in the presence of nearly one‐dimensional localized potentials. Under suitable assumptions on the potential, we construct various types of waves that are localized in the direction of the potential and have single‐ or multihump, or periodic profile in the perpendicular direction. The analysis relies upon a spatial dynamics formulation of the existence problem, together with a center manifold reduction. This reduction procedure allows these waves to be realized as unipulse or multipulse homoclinic orbits, or periodic orbits in a reduced system of ordinary differential equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal control problem of the uncertain second‐order circuit based on first hitting criteria

First hitting criteria of a system are to initially achieve some performance indeces of the target state set. This paper primarily investigates the optimal control problem of the uncertain second‐order circuit based on first hitting criteria. First, considering time efficiency and different from the ordinary expected utility criterion over an infinite time horizon, two first hitting criteria which are reliability index and reliable time criteria are innovatively proposed. Second, assuming the circuit output voltage as an uncertain variable when the historical data is lacking, we better model the real circuit system with the uncertain second‐order differential equation which is essentially the uncertain fractional‐order differential equation. Then, based on the first hitting time theorem of the uncertain fractional‐order differential equation, the distribution function of the first hitting time under the second‐order circuit system is proposed and the uncertain second‐order circuit optimal control model (reliability index and reliable time‐based model) is transformed into corresponding crisp optimal problem. Lastly, analytic expressions of the optimal control for the reliability index model are obtained. Meanwhile, sufficient condition and guidance for parameters for the optimal solution of the reliable time‐based model are derived, and corresponding numerical examples are also given to demonstrate the fluctuation of our optimal solution for different parameters.     

Stationary solutions to phase field crystal equations

This paper is devoted to the analytical and numerical study of stationary solutions of the one‐dimensional Phase Field Crystal Equation. This new model recently introduced by K. Elder and M. Grant describes phase transformations at the atomic level on large time scales. By using bifurcation methods, we investigate the quantitative and qualitative properties of these solutions: multiplicity, stability, and periodicity. Quite unusual bifurcation diagrams are obtained by numerical simulations. Copyright © 2010 John Wiley & Sons, Ltd.     

Analytic approximate solution of three‐dimensional Navier‐Stokes equations of flow between two stretchable disks

The similarity transform for the steady three‐dimensional Navier‐Stokes equations of flow between two stretchable disks gives a system of nonlinear ordinary differential equations which is analytically solved by applying a newly developed method, namely, the homotopy analysis method. The analytic solutions of the system of nonlinear ordinary differential equations are constructed in the series form. The convergence of the obtained series solutions is analyzed. The validity of our solutions is verified by the numerical results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Pressure‐based adaption indicator for compressible euler equations

We consider the Euler equations of gas dynamics and develop a new adaption indicator, which is based on the weak local residual measured for the nonconservative pressure variable. We demonstrate that the proposed indicator is capable of automatically detecting discontinuities and distinguishing between the shock and contact waves when they are isolated from each other. We then use the developed indicator to design a scheme adaption algorithm, according to which nonlinear limiters are used only in the vicinity of shocks. The new adaption algorithm is realized using a second‐order limited and a high‐order nonlimited central‐upwind scheme. We demonstrate robustness and high resolution of the designed method on a number of one‐ and two‐dimensional numerical examples. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1844–1874, 2015     

A relaxation scheme for the approximation of the pressureless Euler equations

In the present work, we consider the numerical approximation of pressureless gas dynamics in one and two spatial dimensions. Two particular phenomena are of special interest for us, namely δ‐shocks and vacuum states. A relaxation scheme is developed which reliably captures these phenomena. In one space dimension, we prove the validity of several stability criteria, i.e., we show that a maximum principle as well as the TVD property for the discrete velocity component and the validity of discrete entropy inequalities hold. Several numerical tests considering not only the developed first‐order scheme but also a classical MUSCL‐type second‐order extension confirm the reliability and robustness of the relaxation approach. The article extends previous results on the topic: the stability conditions for relaxation methods for the pressureless case are refined, useful properties for the time stepping procedure are established, and two‐dimensional numerical results are presented. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Optimal combination of spatial basis functions for the model reduction of nonlinear distributed parameter systems

Spectral methods are among the most extensively used techniques for model reduction of distributed parameter systems in various fields, including fluid dynamics, quantum mechanics, heat conduction, and weather prediction. However, the model dimension is not minimized for a given desired accuracy because of general spatial basis functions. New spatial basis functions are obtained by linear combination of general spatial basis functions in spectral method, whereas the basis function transformation matrix is derived from straightforward optimization techniques. After the expansion and truncation of spatial basis functions, the present spatial basis functions can provide a lower dimensional and more precise ordinary differential equation system to approximate the dynamics of the systems. The numerical example shows the feasibility and effectiveness of the optimal combination of spectral basis functions for model reduction of nonlinear distributed parameter systems.     

Uniform stabilization of a nonlinear structural acoustic model with a Timoshenko beam interface

This paper is concerned with a nonlinear model which describes the interaction of sound and elastic waves in a two‐dimensional acoustic chamber in which one flat ‘wall’, the interface, is flexible. The composite dynamics of the structural acoustic model is described by the linearized equations for a gas defined on the interior of the chamber and the nonlinear Timoshenko beam equations on the interface. Uniform stability of the energy associated with the interactive system of partial differential equations is achieved by incorporating a nonlinear feedback boundary damping scheme in the equations for the gas and the beam. Copyright © 2006 John Wiley & Sons, Ltd.