Volume 3 (2018): Issue 2 (July 2018)

In this paper, we establish two general transformation formulas for Exton’s quadruple hypergeometric functions K₅ and K₁₂ by application of the generalized Kummer’s summation theorem. Further, a number of generating functions for Jacobi polynomials are also derived as an applications of our main results.

In a previous (herein referred to as Ammar, Amin and Hassan Paper [1]) the statement of the problem was formulated and the basic visibility function between two satellites in terms of the orbital elements and time were derived. In this paper the perturbing effect due to drag force on the visibility function were derived explicitly up to O(e⁴), by using Taylor’s expansion for the visibility function about certain epoch. We determine the rise and set times of the satellites through the sign of the visibility function. Numerical examples were worked out for some satellites in order to check the validity of the work.

This paper presents an analytical method to determine the rise-set times of satellite-satellite visibility periods in different orbits. The Visibility function in terms of the orbital elements of the two satellites versus the time were derived explicitly up to e⁴. The line-of-sight corrected for Earth Oblateness up to J₂, were considered as a perturbation to the orbital elements. The visibility intervals of the satellites were calculated for some numerical examples in order to test the results of the analytical work.

In this manuscript, we shall apply the tools and methods from optimal control to analyze various minimally parameterized models that describe the dynamics of populations of cancer cells and elements of the tumor microenvironment under different anticancer therapies. In spite of their simplicity, the analysis of these models that capture the essence of the underlying biology sheds light on more general scenarios and, in many cases, leads to conclusions that confirm experimental studies and clinical data. We focus on four applications: optimal control applied to compartmental models, brain tumors, drug resistance and antiangiogenic treatment.

There is no mathematical solution to adding up transcendental functions other than numerical process. This paper put forward analytical method to model the sum of sigmoid like functions with an equivalent function. The Brillouin and Langevin as well as the error function, the tanh, sigmoid and the tan^-1 functions are investigated, their equivalent functions are calculated for four components and the error between the numerical (computer assisted) result and the equivalent function is tested for accuracy. The best modelling function, the most useful to include into mathematical operations, is pointed out finally, based on its performance and convenience. The paper intends to help people involved mostly in modelling hysteresis in Magnetism and other field of research in physics.

In this paper we study the modified equal-width equation, which is used in handling simulation of a single dimensional wave propagation in nonlinear media with dispersion processes. Lie point symmetries of this equation are computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter using an optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions are presented. The solutions obtained are cnoidal and snoidal waves. Furthermore, conservation laws for the modified equal-width equation are derived by employing two different methods, the multiplier method and Noether approach.

The product graph G_m *G_p of two given graphs G_m and G_p, defined by J.C. Bermond et al.[J Combin Theory, Series B 36(1984) 32-48] in the context of the so-called (Δ,D)-problem, is one interesting model in the design of large reliable networks. This work deals with sufficient conditions that guarantee these product graphs to be hamiltonian-connected. Moreover, we state product graphs for which provide panconnectivity of interconnection networks modeled by a product of graphs with faulty elements.

Let G be a graph and let m_ij(G), i, j ≥ 1, represents the number of edge of G, where i and j are the degrees of vertices u and v respectively. In this article, we will compute different polynomials of flower graph f(_n×m), namely M polynomial and Forgotten polynomial. These polynomials will help us to find many degree based topological indices, included different Zagreb indices, harmonic indices and forgotten index.

Balaban index is defined as J(G)=mm−n+2Σ1w(u)⋅w(v),$J\left( G \right)=\frac{m}{m-n+2}\Sigma \frac{1}{\sqrt{w\left( u \right)\cdot w\left( v \right)}},$ where the sum is taken over all edges of a connected graph G, n and m are the cardinalities of the vertex and the edge set of G, respectively, and w(u) (resp. w(v)) denotes the sum of distances from u (resp. v) to all the other vertices of G. In 2011, H. Deng found an interesting property that Balaban index is a convex function in double stars. We show that this holds surprisingly to general graphs by proving that attaching leaves at two vertices in a graph yields a new convexity property of Balaban index. We demonstrate this property by finding, for each n, seven trees with the maximum value of Balaban index, and we conclude the paper with an interesting conjecture.

In this paper, a comparative study on Haar wavelet method (HWM) and Hosoya Polynomial method(HPM) for the numerical solution of Fredholm integral equations. Illustrative examples are tested through the error analysis for efficiency. Numerical results are shown in the tables and figures.

In this paper we study a nonlinear multi-dimensional partial differential equation, namely, a generalized second extended (3+1)-dimensional Jimbo-Miwa equation. We perform symmetry reductions of this equation until it reduces to a nonlinear fourth-order ordinary differential equation. The general solution of this ordinary differential equation is obtained in terms of the Weierstrass zeta function. Also travelling wave solutions are derived using the simplest equation method. Finally, the conservation laws of the underlying equation are computed by employing the conservation theorem due to Ibragimov, which include conservation of energy and conservation of momentum laws.

Chaotic behaviour of dynamical systems, their routes to chaos, and the intermittency in particular are interesting and investigated subjects in nonlinear dynamics. The studying of these phenomena in non-smooth dynamical systems is of the special scientists’ interest. In this paper we study the type-III intermittency route to chaos in strongly nonlinear non-smooth discontinuous 2-DOF vibroimpact system. We apply relatively new mathematical tool – continuous wavelet transform CWT – for investigation this phenomenon. We show that CWT applying allows to detect and determine the chaotic motion and the intermittency with great confidence and reliability, gives the possibility to demonstrate intermittency route to chaos, to distinguish and analyze the laminar and turbulent phases.

In the paper we present results of accuracy evaluation of numerous numerical algorithms for the numerical approximation of the Inverse Laplace Transform. The selected algorithms represent diverse lines of approach to this problem and include methods by Stehfest, Abate and Whitt, Vlach and Singhai, De Hoog, Talbot, Zakian and a one in which the FFT is applied for the Fourier series convergence acceleration. We use C++ and Python languages with arbitrary precision mathematical libraries to address some crucial issues of numerical implementation. The test set includes Laplace transforms considered as difficult to compute as well as some others commonly applied in fractional calculus. Evaluation results enable to conclude that the Talbot method which involves deformed Bromwich contour integration, the De Hoog and the Abate and Whitt methods using Fourier series expansion with accelerated convergence can be assumed as general purpose high-accuracy algorithms. They can be applied to a wide variety of inversion problems.

The present study aims to explore the effects of an adapted classical dance intervention on the psychological and functional status of institutionalized elder people using a Bayesian network. All participants were assessed at baseline and after the 9 weeks period of the intervention. Measures included balance and gait, psychological well-being, depression, and emotional distress.

According to the Bayesian network obtained, the dance intervention increased the likelihood of presenting better psychological well-being, balance, and gait. Besides, it also decreased the probabilities of presenting emotional distress and depression. These findings demonstrate that dancing has functional and psychological benefits for institutionalized elder people. Moreover it highlights the importance of promoting serious leisure variety in the daily living of institutionalized elder adults.

This paper propose Noether symmetries and the conserved quantities of the relative motion systems on time scales. The Lagrange equations with delta derivatives on time scales are presented for the system. Based upon the invariance of Hamilton action on time scales, under the infinitesimal transformations with respect to the time and generalized coordinates, the Hamilton’s principle, the Noether theorems and conservation quantities are given for the systems on time scales. Lastly, an example is given to show the application the conclusion.

We summarize the known results on the integrability of the complex Hamiltonian systems with two degrees of freedom defined by the Hamiltonian functions of the form

H=12∑i=12pi2+V(q1,q2),$$\begin{array}{} \displaystyle H=\frac{1}{2}\sum_{i=1}^{2}p_i^2+V(q_1,q_2), \end{array} $$

where V(q₁,q₂) are homogeneous polynomial potentials of degree k.

Decomposition of the free (triaxial) rigid body Hamiltonian into a “main problem” and a perturbation term provides an efficient integration scheme that avoids the use of elliptic functions and integrals. In the case of short-axis-mode rotation, it is shown that the use of complex variables converts the integration of the torque-free motion by perturbations into a simple exercise of polynomial algebra that can also accommodate the gravity-gradient perturbation when the rigid body rotation is close enough to the axis of maximum inertia.

In this work, we use an Eulerian mathematical model to forecast the movement of oil spills in the open sea and we design objective functions to obtain optimal trajectories for skimmer ships to clean and recover the oil. Here, we first validate the ability of this mathematical model to forecast the fate of the oil by comparing our results with satellite images. Then, we create a synthetic study case based on real data, and we show that following optimal trajectories greatly improves the amount of oil recovered at the whole area of study.

The aim of this paper was to study an oscillatory flow of a Casson fluid through an elastic tube of variable cross section. The radial displacement of tube wall is taken into consideration. The problem is modelled under the assumption that the variation of the cross section of the tube is slow in the axial direction. Cylindrical coordinate system is chosen to study the problem. The analytical expressions for axial velocity and mass flux as a function of pressure gradient are obtained. The change in pressure distribution for various pressure****radius relationships is analyzed by considering different geometries. The effects of elastic parameter, Womersley parameter and Casson parameter on excess pressure and pressure gradient along axial direction are discussed through graphs. The results reveal that the elastic parameter plays a key role in the variation of pressure along the tube. Womersley parameter has significant effect on pressure distribution. Another important observation is that the amplitude of pressure increases for growing values of Casson parameter for both tapered and constricted tubes. In addition, the pressure oscillates more for the case of locally constricted tube when compared to other geometries.

In this paper, the high accuracy mass-conserved splitting domain decomposition method for solving the parabolic equations is proposed. In our scheme, the time extrapolation and local multi-point weighted average schemes are used to approximate the interface fluxes on interfaces of sub-domains, while the interior solutions are computed by one dimension high-order implicit schemes in sub-domains. The important feature is that the developed scheme keeps mass conservation and are of second-order convergent in time and fourth-order convergent in space. Numerical experiments confirm the convergence.

In this work, we deal with the predecessors existence problems in sequential dynamical systems over directed graphs. The results given in this paper extend those existing for such systems over undirected graphs. In particular, we solve the problems on the existence, uniqueness and coexistence of predecessors of any given state vector, characterizing the Garden-of-Eden states at the same time. We are also able to provide a bound for the number of predecessors and Garden-of-Eden state vectors of any of these systems.

Herpes simplex virus (HSV-2) triples the risk of acquiring human immunodeficiency virus (HIV) and contributes to more than 50% of HIV infections in other parts of the world. A deterministic mathematical model for the co-interaction of HIV and HSV-2 in a community, with all the relevant biological detail and poor HSV-2 treatment adherence is proposed. The threshold parameters of the model are determined and stabilities are analysed. Further, we applied optimal control theory. We proved the existence of the optimal control and characterized the controls using Pontryagin’s maximum principle. The controls represent monitoring and counselling of individuals infected with HSV-2 only and the other represent monitoring and counselling of individuals dually infected with HIV and HSV-2. Numerical results suggests that more effort should be devoted to monitoring and counselling of individuals dually infected with HIV and HSV-2 as compared to those infected with HSV-2 only. Overall, the study demonstrate that, though time dependent controls will be effective on controlling HIV cases, they may not be sustainable for certain time intervals.

In this paper a rotating two-fluid model for the propagation of internal waves is introduced. The model can be derived from a rotating-fluid problem by including gravity effects or from a nonrotating one by adding rotational forces in the dispersion balance. The physical regime of validation is discussed and mathematical properties of the new system, concerning well-posedness, conservation laws and existence of solitary-wave solutions, are analyzed.

In this article, we have presented a parametric finite difference method, a numerical technique for the solution of two point boundary value problems in ordinary differential equations with mixed boundary conditions. We have tested proposed method for the numerical solution of a model problem. The numerical results obtained for the model problem with constructed exact solution depends on the choice of parameters. The computed result of a model problem suggests that proposed method is efficient.

The purpose of this paper is to improve and complement the main results of Jiang and Gu (J. Nonlinear Sci. Appl., 10 (2017), 1881–1895). By using nontrivial methods, some common coupled fixed point results in metric spaces are obtained. Moreover, it is shown that some recent fixed point results in the setting of multiplicative metric spaces are actually equivalent to the counterpart of the standard metric spaces. In addition, an example to illustrate the presented theoretical result is also given.