Traveling wave solutions in predator–prey models with competition

This paper studies the minimal wave speed of traveling wave solutions in predator–prey models, in which there are several groups of predators that compete among different groups. We investigate the existence and nonexistence of traveling wave solutions modeling the invasion of predators and coexistence of these species. When the positive solution of the corresponding kinetic system converges to the unique positive steady state, a threshold that is the minimal wave speed of traveling wave solutions is obtained. To finish the proof, we construct contracting rectangles and upper–lower solutions and apply the asymptotic spreading theory of scalar equations. Moreover, multiple propagation thresholds in the corresponding initial value problem are presented by numerical examples, and one threshold may be the minimal wave speed of traveling wave solutions.

Periodic Solutions of Second-Order Differential Equations in Hilbert Spaces

We prove the existence of periodic solutions of some infinite-dimensional systems by the use of the lower/upper solutions method. Both the well-ordered and non-well-ordered cases are treated, thus generalizing to systems some well-established results for scalar equations.

General Finite-Element Framework of the Virtual Fields Method in Nonlinear Elasticity

This paper presents a method to derive the virtual fields for identifying constitutive model parameters using the Virtual Fields Method (VFM). The VFM is an approach to identify unknown constitutive parameters using deformation fields measured across a given volume of interest. The general principle for solving identification problems with the VFM is first to derive parametric stress field, where the stress components at any point depend on the unknown constitutive parameters, across the volume of interest from the measured deformation fields. Applying the principle of virtual work to the parametric stress fields, one can write scalar equations of the unknown parameters and solve the obtained system of equations to deduce the values of unknown parameters. However, no rules have been proposed to select the virtual fields in identification problems related to nonlinear elasticity and there are multiple strategies possible that can yield different results. In this work, we propose a systematic, robust and automatic approach to reconstruct the systems of scalar equations with the VFM. This approach is well suited to finite-element implementation and can be applied to any problem provided that full-field deformation data are available across a volume of interest. We also successfully demonstrate the feasibility of the novel approach by multiple numerical examples. Potential applications of the proposed approach are numerous in biomedical engineering where imaging techniques are commonly used to observe soft tissues and where alterations of material properties are markers of diseased states.

General Finite-Element Framework of the Virtual Fields Method in Nonlinear Elasticity

This paper presents a method to derive the virtual fields for identifying constitutive model parameters using the Virtual Fields Method (VFM). The VFM is an approach to identify unknown constitutive parameters using deformation fields measured across a given volume of interest. The general principle for solving identification problems with the VFM is first to derive parametric stress field, where the stress components at any point depend on the unknown constitutive parameters, across the volume of interest from the measured deformation fields. Applying the principle of virtual work to the parametric stress fields, one can write scalar equations of the unknown parameters and solve the obtained system of equations to deduce the values of unknown parameters. However, no rules have been proposed to select the virtual fields in identification problems related to nonlinear elasticity and there are multiple strategies possible that can yield different results. In this work, we propose a systematic, robust and automatic approach to reconstruct the systems of scalar equations with the VFM. This approach is well suited to finite-element implementation and can be applied to any problem provided that full-field deformation data are available across a volume of interest. We also successfully demonstrate the feasibility of the novel approach by multiple numerical examples. Potential applications of the proposed approach are numerous in biomedical engineering where imaging techniques are commonly used to observe soft tissues and where alterations of material properties are markers of diseased states.

Some Novel Sixth-Order Iteration Schemes for Computing Zeros of Nonlinear Scalar Equations and Their Applications in Engineering

In this paper, we propose two novel iteration schemes for computing zeros of nonlinear equations in one dimension. We develop these iteration schemes with the help of Taylor’s series expansion, generalized Newton-Raphson’s method, and interpolation technique. The convergence analysis of the proposed iteration schemes is discussed. It is established that the newly developed iteration schemes have sixth order of convergence. Several numerical examples have been solved to illustrate the applicability and validity of the suggested schemes. These problems also include some real-life applications associated with the chemical and civil engineering such as adiabatic flame temperature equation, conversion of nitrogen-hydrogen feed to ammonia, the van der Wall’s equation, and the open channel flow problem whose numerical results prove the better efficiency of these methods as compared to other well-known existing iterative methods of the same kind.

Dichotomy. Dichotomy? Dichotomy! A modified method of dichotomy for solution of nonlinear scalar equations and some results of its investigation

Introduction: when creating modern automatic control systems for various processes and objects operating in real time, very often one has to face the problem of solving various kinds of nonlinear scalar equations. In the first part of this work entitled “Dichotomy. Dichotomy? Dichotomy!: fundamentals, terminology problems and inspection analysis of the dichotomy method”, a modified version of the dichotomy method was proposed, which has all the main advantages of the modified method. This method has a number of advantages in comparison with other methods for solving nonlinear equations, but at present it has not found wide practical use. The main reason for its low popularity is a low rate of convergence of the sequence of approximate solutions, and a large amount of computation required to obtain sufficiently accurate solutions. Purpose of the study: to propose a modified version of the dichotomy method, which allows one to obtain more rapidly converging sequences of approximate solutions to nonlinear scalar equations and requires significantly less computations required to obtain solutions with the desired accuracy, to illustrate, a higher convergence rate of the sequence of approximate solutions calculated using the modified dichotomy method by solving a number of specific nonlinear equations and, thereby, to substantiate the advantage of the new method for its use in creating various automatic control and regulation systems. Results: a modification of the method for dividing a segment in half is proposed, which has all the main advantages of the modified method. The results of solving 4 nonlinear equations are presented illustrating a higher rate of convergence of solutions calculated using the proposed modification. Practical significance: the research results can be used in the development of modern automatic control systems for various technological processes and objects.

$ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains

<p style='text-indent:20px;'>In this paper, we establish the <inline-formula><tex-math id="M1">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates for solutions of second order elliptic problems with drift terms in bounded Lipschitz domains by using a real variable method. For scalar equations, we prove that the <inline-formula><tex-math id="M2">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates hold for <inline-formula><tex-math id="M3">\begin{document}$ \frac{3}{2}-\varepsilon&lt;p&lt;3+\varepsilon $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M4">\begin{document}$ d\geq3 $\end{document}</tex-math></inline-formula>, and the range for <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula> is sharp. For elliptic systems, we prove that the <inline-formula><tex-math id="M6">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates hold for <inline-formula><tex-math id="M7">\begin{document}$ \frac{2d}{d+1}-\varepsilon&lt;p&lt;\frac{2d}{d-1}+\varepsilon $\end{document}</tex-math></inline-formula> under the assumption that the Lipschitz constant of the domain is small.</p>