Volume 1 (2016): Issue 2 (July 2016)

Reaction-diffusion equations have enjoyed a considerable amount of scientific interest. The reason for the large amount of work put into studying these equations is not only their practical relevance, but also interesting phenomena that can arise from such equations. Fisher equation is commonly used in biology for population dynamics models and in bacterial growth problems as well as development and growth of solid tumours. The physical aspects of this equation are not fully understood without getting deeper into the concept of conservation laws. In [4], Anco and Bluman gave a general treatment of a direct conservation law method for partial differential equations expressed in a standard Cauchy-Kovaleskaya form. In this work we study the well known density dependent diffusion-reaction equation. We derive conservation laws by using the direct method of the multipliers.

We analyzed the first set of complete CCD light curves of the W UMa type eclipsing binary IK Boo in the BVRI bands by using the PHOEBE code and deduced its first photometric parameters with, mass ratio q = 0.648 and orbital inclination i = 63^o. We have applied a spotted model due to the light curves asymmetry. The system shows a distinct O’Connell effect. The best solution fit to the light curves suggested the influence of star spot(s) on both components. Such presence of star spot(s) is common among the RS CVn and W UMa chromospheric active late type stars.

We also present an analysis of mid–eclipse time measurements of IK Boo. The analysis indicates a period decrease rate dP/dt = −1.68 × 10⁻⁷d/yr, which can be interpreted in terms of mass transfer of rate 3.1 × 10⁻⁷M_⊙/yr, from the more massive to the less massive component.

The bird strike damage on aircrafts is a widely studied matter [1] with a high economic impact on stakeholders finances. Some authors estimate it in about USD1.2 Billion for nowadays commercial worldwide activity [2], and more than USD937 million in direct and other monetary losses per year just for the United States, as an example of civil aviation industry [3]. The present techniques to face this problem have been previously analyzed in order to decrease the wild life hazards at the airport facilities [4] however nowadays there is a new point of view to prevent this risk at airports that requires an interesting approach in relationship with industrial process improvement examples, such approach lies on preserving the natural life at the airport facilities by developing raptor micro-habitats than change into exclusion areas when the risk of being hunted is recognized by the existing wildlife.

Therefore, the main goal of this paper is to share several experiences developed at the Spanish dual airport (military & civilian) of San Javier (Spain), as a case of study in where the mathematics and nonlinear sciences provides the foundations of the ontological knowledge for falconry performance as a Wildlife Control Technique for airport facilities.

In this paper, we find the values of three important domination parameters namely, connected domination number, total domination number and total edge domination number of molecular graph of octane isomers. Further, we show that these parameters are highly correlated with physical properties of octane isomers. Finally we carry out QSPR (Quantitative Structure-Property Relationship) analysis using several physicochemical properties of octane isomers.

In this paper we focus on integrated Reconnaissance/Strike LAV, in order to reveal the evolution regularity when group LAVs combats cooperatively. The evolution of cooperative behavior of group LAVs, which is described with finite state machine, can be regarded as a conversion process of a LAV in different task states, using the rate equation for probability analysis. Then based on the missions of integration of reconnaissance, attack, and damage effectiveness evaluation, we build the model of finite state machine based on behavior state transition. Solved with Runge-Kutta method. We can analyze how the key technology quota of LAV impact on the operational effectiveness of Group LAVs.the fractional order control approach.

The paper presents Elastohydrodynamic lubrication line contact problem with bio-based oil as lubricants for an isothermal case. The fast convergence method for the solution of Elastohydrodynamic lubrication line contact problem with seed oil as lubricant is analyzed using Multigrid, Multilevel Multi-Integration with the influence of different load and speed. The result shows that the use of these oils has the potential to substitute the function of common lubricant so as to reduce dependence on conventional oil lubricants. The results obtained are comparable and the pressure spikes are smooth as compared to the earlier findings which are shown in terms of graphs and tables.

The forgotten topological index of a graph G is defined as the sum of the cube of the degrees of its vertices. In the recent paper [6], [W. Gao et al. (2016), Forgotten topological index of chemical structure in drugs, Saudi Pharmaceutical Journal, 24, 258-264], the forgotten topological index of some chemical structures has been obtained. In this note, we correct their result regarding triangular benzenoid. Also, we have given the expression for the forgotten topological index of graphene structure which is more compact than the one was obtained in the paper above.

In this paper we prove that the global attractor generated by strong solutions of a reaction-diffusion equation without uniqueness of the Cauchy problem is bounded in suitable L^r-spaces. In order to obtain this result we prove first that the concepts of weak and mild solutions are equivalent under an appropriate assumption.

Also, when the nonlinear term of the equation satisfies a supercritical growth condition the existence of a weak attractor is established.

Most of problems in Topological Dynamics in the theory of general autonomous discrete dynamical systems have been addressed in the non-autonomous setting. In this paper we will review some of them, giving references and stating open questions.

We present an analysis of a mathematical model describing the key features of the most frequent and aggressive type of primary brain tumor: glioblastoma. The model captures the salient physiopathological characteristics of this type of tumor: invasion of the normal brain tissue, cell proliferation and the formation of a necrotic core. Our study, based on phase space analysis, geometric perturbation theory, exact solutions and numerical simulations, proves the existence of bright solitary waves in the tumor coupled with kink and anti-kink fronts for the normal tissue and the necrotic core. Finally, we study the linear stability of the solutions to calculate the time of tumor recurrence.

Applied mathematics and nonlinear sciences have an enormous potential for application in cancer. Mathematical models can be used to raise novel hypotheses to test, develop optimized treatment schedules and personalize therapies. However. this potential is yet to be proven in real-world applications to specific cancer types. In this paper we discuss how we think mathematical knowledge may be better used to improve cancer patients’ outcome.

Along the years, the foundations of Fractal Geometry have received contributions starting from mathematicians like Cantor, Peano, Hilbert, Hausdorff, Carathéodory, Sierpiński, and Besicovitch, to quote some of them. They were some of the pioneers exploring objects having self-similar patterns or showing anomalous properties with respect to standard analytic attributes. Among the new tools developed to deal with this kind of objects, fractal dimension has become one of the most applied since it constitutes a single quantity which throws useful information concerning fractal patterns on sets. Several years later, fractal structures were introduced from Asymmetric Topology to characterize self-similar symbolic spaces. Our aim in this survey is to collect several results involving distinct definitions of fractal dimension we proved jointly with Prof.M.A. Sánchez-Granero in the context of fractal structures.

This paper is devoted to studying Hamiltonian oscillators in 1:1:1:1 resonance with symmetries, which include several models of perturbed Keplerian systems. Normal forms are computed in Poisson and symplectic formalisms, by mean of invariants and Lie-transforms respectively. The first procedure relies on the quadratic invariants associated to the symmetries, and is carried out using Gröner bases. In the symplectic approach, hinging on the maximally superintegrable character of the isotropic oscillator, the normal form is computed a la Delaunay, using a generalization of those variables for 4-DOF systems. Due to the symmetries of the system, isolated as well as circles of stationary points and invariant tori should be expected. These solutions manifest themselves rather differently in both approaches, due to the constraints among the invariants versus the singularities associated to the Delaunay chart.

Taking the generalized van der Waals family as a benchmark, the explicit expression of the Delaunay normalized Hamiltonian up to the second order is presented, showing that it may be extended to higher orders in a straightforward way. The search for the relative equilibria is used for comparison of their main features of both treatments. The pros and cons are given in detail for some values of the parameter and the integrals.

In this work, we study the controllability for a class of nonlinear neutral stochastic functional integrodifferential equations with infinite delay in a real separable Hilbert space. Sufficient conditions for the controllability are established by using Nussbaum fixed point theorem combined with theories of resolvent operators. As an application, an example is provided to illustrate the obtained result.

In this article, the lubrication of a long porous slider in which the fluid is injected into the porous bottom is considered. The similarity transformations reduce the governing problem of Navier-Stokes equations to coupled nonlinear ordinary differential equations which are solved by HAM. Solutions are obtained for much larger values of Reynolds number compared to analytical and numerical methods. The results comprise good agreement between approximate and numerical solutions. HAM gives rapid convergent series solutions which show that this method is efficient, accurate and has advantages over other methods. Further, homotopy-pade’ technique is used to accelerate the convergence of series solution.

In this paper we revisit the problem of monotonicity preservation of curves and surfaces and we provide some new proofs and open problems. In particular, we prove a new formula for the derivation of rational Bézier curves. We also deal with the rational monotonicity preservation of rational Bézier surfaces and a related conjecture is presented.

In this paper, modified wavelet full-approximation scheme is introduced for the numerical solution of nonlinear Volterra integral and integro-differential equations. Wavelet Prolongation and Restriction operators are developed using Daubechies wavelet filter coefficients. Results show that the proposed scheme offers an efficient and good accuracy with faster convergence in less computation cost, which is justified through the error analysis and CPU time.

In this paper a new technique for stepsize changing in the numerical solution of Initial Value Problems for ODEs by means of Adams type methods is considered. The computational cost of the new technique is equivalent to those of the well known interpolation technique (IT). It is seen that the new technique has better stability properties than the IT and moreover, its leading error term is smaller. These facts imply that the new technique can outperform the IT.

The study of everyday phenomena involving friction continues to maintain a high level of difficulty despite its long history. The causes of this problem lie in the different scale of the characteristics of the phenomenon, macroscopic and microscopic. Thus, very different models, valid in a narrow scope which prevents generalization, have been appearing. This survey presents the application of network simulation method to the numerical solution to the study of friction at very different scales. On the one hand, on a microscopic scale an atomic force microscope model has been studied, related to the analysis of soft surfaces at the atomic scale. Furthermore, on a macroscopic scale model related to the analysis of an industrial device, such as a brake mechanism has been studied. After presenting herein is a review of the different formulations of the friction force, the nature of the surfaces involved in the phenomenon, as well as the definition of the problems to be analyzed. The design of network models and the implementation of the initial conditions are explained. The results of the application of network models to selected problems are presented. In order to verify the reliability of the proposed models, their results are compared with the solutions obtained by other numerical methods or experimental results, one from a device developed during the preparation of this report.

Asymptotic and global dynamics of weak solutions for a damped nonlinear wave equation with a critical growth exponent on the unbounded domain ℝⁿ(n ≥ 3) is investigated. The existence of a global attractor is proved under typical dissipative condition, which features the proof of asymptotic compactness of the solution semiflow in the energy space with critical nonlinear exponent by means of Vitali-type convergence theorem.

This paper concentrates on the vibrations attenuation of a rotor driven by a DC motor and its frame flexibly coupled with a baseplate by linear cylindrical helical springs and damped by an element that can work either in inertia or impact regime. The system oscillation is governed by three mutually coupled second-order ordinary differential equations. The nonlinear behaviour occurs if the impact regime is adjusted. The damping element operating in inertia mode reduces efficiently the oscillations amplitude only in a narrow frequency interval. In contrast, the damping device working in impact regime attenuates vibrations of the rotor frame in a wider range of the excitation frequencies and it can be easily extended if the clearances between the rotor casing and the damping element are controlled. The development of a computational procedure for investigation of vibration of a flexibly supported rotor and for its attenuation by the inertia and impact dampers; learning more on efficiency of the individual damping regimes; finding possibilities of extension of the frequency intervals of applicability of the damping device; and obtaining more information on the character of the vibration induced by impacts are the main contributions of this research work.

We make a survey of results published by the authors about the backward and forward unilateral weighted shift operators in Kóthe spaces, the so-called generalized derivation and integration operators, extending well-known results for spaces of analytic functions.