International Journal of Dynamical Systems and Differential Equations (44 papers in press)

Regular Issues

• Matrix Mittag-Leffler function and solution of multi-term fractional differential equations  
  by Junsheng Duan 
  Abstract: In this paper, we first derive the solution of fractional differential equation system expressed in matrix Mittag-Leffler function by using the Adomian decomposition method. Then we consider the initial value problem (IVP) for multi-term fractional differential equation. By introducing new unknown functions, we rewrite the IVP for multi-term fractional differential equation into the IVP for a fractional differential equation system. Thus the solution can be given in terms of matrix Mittag-Leffler functions. We demonstrate the method using four numerical examples and the results are simulated using MATHEMATICA 8.
  Keywords: fractional calculus; fractional derivative; Mittag-Leffler function; fractional differential equation.
  
  
• Analysis of migration pattern of prey species with reserved zone  
  by JYOTIRMOY ROY, Shariful Alam 
  Abstract: In this article a generalized prey-predator system has been analyzed, where the whole habitat is divided into two different zones, namely free zone and reserved zone. It is assumed that in the reserved zone only prey species can access and predation is strictly prohibited, whereas in the free zone both the species can cohabit and naturally predation is allowed. The migration rates of the prey species from reserved zone to unreserved zone and vice-versa both depends on predator's availability and accordingly suitable functions has been incorporated in the model system. The local and global stability analysis of the model system have been performed in a systematic manner and system persistence criterion has been established. The role of prey migration rate from reserved zone to unreserved zone has been investigated and it is found that Hopf bifurcation occurs when the prey migration rate from reserved zone to unreserved zone crosses a certain threshold value. It is also found that the prey migration rate has stabilizing effect on the dynamics of the system and has significant effect on the coexistence of all the species. Finally numerical simulation has been carried out to support our analytical findings.
  Keywords: Prey-predator model; Reserved zone; Stability and persistence; Hopf bifurcation; Limit cycle.
  
  
• Space time fractional Boussinesq equation with singular and non singular kernels  
  by Ritu Agarwal, Mahaveer Yadav, Ravi P. Agarwal 
  Abstract: Recently, many authors have found analytical and numerical solutions of fractional Boussinesq equation by applying various fractional operators with singular kernels. Motivated by recently introduced fractional operators with non-singular kernels, in this paper a comparison of the solution of linearized fractional Boussinesq equation has been made for the fractional operators Caputo (with singular kernel) and Caputo-Fabrizio (with non-singular kernel). Linearized Boussinesq equation is derived by assuming that the average thickness of saturated layer of an aquifer is constant.
  Keywords: Linearized Boussinesq equation; Caputo fractional derivative; Caputo Fabrizio fractional derivative; Fractional Laplacian Operator; Mittag Leffler function.
  
  
• Dynamics of a predator-prey model with discrete and distributed delay  
  by Bootan Rahman, Muhammad Yau, Yuliya Kyrychko, Konstantin Blyuss 
  Abstract: This paper considers a predator-prey model with discrete time delay representing prey handling time and assumed equal to the predator maturation period, and a distributed time delay describing intra-species interactions. We show that due to the delayed logistic growth of the prey, it is impossible for the species to become extinct through predation. Conditions for existence and local stability of the co-existence equilibrium are derived in terms of system parameters. Using techniques of centre manifold reduction and the normal form theory, we establish the direction of Hopf bifurcation of the co-existence equilibrium, as well as the stability of the bifurcating period solution. Numerical bifurcation analysis and simulations are performed to illustrate regions of stability of the co-existence equilibrium, to investigate how the amplitude and the period of bifurcating periodic solutions depend on parameters, and to demonstrate different types of dynamics of the system.
  Keywords: Stability; discrete and distributed delay; predator-prey model; Hopf bifurcation; periodic solutions.
  
  
• On the Oscillation of Conformable Fractional Partial Delay Differential Systems  
  by George E. Chatzarakis, Muthusamy Deepa, Nagamanickam Nagajothi, Vadivel Sadhasivam 
  Abstract: In this article, we investigate the oscillation of a conformable fractional three dimensional nonlinear partial delay differential system. We establish some new oscillation criteria of the solutions of the differential system by using the generalized Riccati transformation and the integral averaging method. The obtained results are illustrated by various examples.
  Keywords: Oscillation; Delay; Partial differential system; Conformable fractional derivative.
  
  
• Global dynamics analysis of a stochastic SIRS epidemic model with vertical transmission and different periods of immunity  
  by Driss KIOUACH, Yassine SABBAR 
  Abstract: In this work, we analyze a stochastic SIRS (Susceptible-InfectedRecovered-Susceptible) epidemic model with vertical transmission and different periods of immunity. This model has a global positive solution. Firstly, we establish sufficient conditions for extinction and persistence in the mean of a disease. Then, we prove the global stability of the system under a suitable condition of perturbation intensity. In the case of the non-autonomous system, we show that there exists at least one positive periodic solution. Finally, some numerical examples are introduced to show the validity of our results.
  Keywords: Stochastic SIRS model; vertical transmission; global stability; extinction; persistence; periodic solution.
  
  
• EXISTENCE OF POSITIVE QUASI-HOMOCLINIC SOLUTIONS FOR DAMPED p-LAPLACIAN DIFFERENTIAL EQUATIONS  
  by Monia Boujlida 
  Abstract: In this paper we prove the existence of nontrivial homoclinic so-rnlutions for the damped $p$-Laplacian differential equationrnrn$$('u''^{p-2}u')' + c('u''^{p-2}u')+ a(t)'u'^{p-2}u + f(t,u)=0 ; t in matbb{R};$$rnwhere $p geq 2$, c geq 0 is a constant and the functions $a$ and $f$ are continuous andrnnot necessarily periodic in $t$. Using the Mountain-Pass Theorem, we obtainrnthe existence of positive homoclinic solution in both cases sub-quadratic andrnsuper-quadratic.
  Keywords: Quasi-homoclinic solution; the (PS)-condition; Mountain Pass The-rnorem; damped $p$-Laplacian equation.
  
  
• Cheap controls for disturbances compensation in hyperbolic delayed systems.  
  by Salma Souhaile, Larbi Afifi 
  Abstract: Thiswork applies to the remediability problem for a class of hyperbolic perturbed systems with constant or time-varying delays.With a convenient choice of input operator (control) and through the observation (output), we show how to remedy the effect of any disturbance f on the considered system. We give the main properties and characterizations of the concept according to the delay. Then, under the appropriate hypothesis, we prove howto find the optimal control ensures the compensation of a disturbance using the corresponding observation only. The usual case of actuators and sensors is examined. An application and numerical results for a one-dimensional wave equation with delay are also presented.
  Keywords: Hyperbolic systems; Disturbance; Control; Observation; Delay; Remediability.
  
  
• Oscillation of delay difference equations with finite non-monotone arguments  
  by Limei Feng, Zhenlai Han 
  Abstract: In this paper, the oscillation of delay difference equations with finite non-monotone delayrn$$triangle x(t)+sum_{i=1}^mp_i(t)x(tau_i(t))=0, tin mathbb{N}$$rnis studied. Three criteria of these equations are obtained for oscillation. And examples are given to show the meanings of the theorems.rn
  Keywords: delay difference equation; non-monotone argument; oscillatory solution.
  
  
• Delay feedback strategy for a fractional-order chaotic financial system  
  by Changjin Xu 
  Abstract: In this paper, we are concerned with a new fractional incommensurate order financial system\r\nwhich is a generalized version of the\r\nfinancial model investigated in earlier works. Designing a suitable time-delayed feedback controller, we have controlled the chaotic phenomenon of the\r\nfractional incommensurate order financial system. By analyzing the characteristic equation of the involved financial system and regarding the delay as the bifurcation\r\nparameter, we establish a set of new sufficient conditions to guarantee the stability and the existence of Hopf bifurcation\r\n for fractional incommensurate order financial system.\r\n The study reveals that the delay and the fractional order have an important influence on the stability and Hopf bifurcation of considered financial system.\r\n Computer simulations are presented to illustrate the correctness of the theoretical results.\r\n The theoretical findings of this paper are new and have important meanings in dealing with the economic and financial problems.
  Keywords: Chaos control; financial system; stability; Hopf bifurcation; fractional order; delay.
  
  
• Residual power series method for the time fractional Fornberg-Whitham equation  
  by Jianke Zhang, Luyang Yin 
  Abstract: The purpose of this paper is to solve the time fractional Fornberg-Whitham equation by the residual power series method, where the fractional derivatives are in Caputo sense. According to the definition of generalized fractional power series, the solutions of the fractional differential equations are approximatively expanded and substituted into the differential equations. The coefficients to be determined in the approximate solutions are calculated according to the residual functions and the initial conditions, and the approximate analytical solutions of the equations can be obtained. Finally, the approximate analytical solutions are compared with the exact solutions. The results show that the residual power series method is convenient and effective for solving the time fractional Fornberg-Whitham equation.
  Keywords: Residual power series method; Time-fractional Fornberg-Whitham equation; Caputo derivative.
  
  
• Numerical Solution of Time-Delay Systems by Hermite Wavelet  
  by Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri, Sohrab Effati 
  Abstract: This paper presents a direct numerical method based on Hermite wavelet to find the solution of time-delay systems. The operational matrices of integration, differentiation, production, and delay are derived and utilized to reduce the time-delay dynamical system to a set of algebraic equations. Thus, the problem is simplified greatly. The method is easy to implement. The illustrative examples with time-invariant and time-varying coefficients demonstrate the validity of the method.
  Keywords: Time-delay system; Hermite wavelet; Operational matrix; Direct method.
  
  
• Solving Nonlinear Fredholm integral equations with PQWs in complex plane  
  by Majid Erfanian 
  Abstract: In this article, we propose a numerical scheme to solve a kind of nonlinear Fredholm integral equations of the second kind in the complex plane. The periodic quasi-wavelets (PQWs) constructed on [0,2pi] are utilized as a basis of the iteration method. Using the Banach fixed point theorem, we obtain some results concerning the error analysis. Illustrative examples are included to demonstrate the validity and applicability of the technique.
  Keywords: Nonlinear Fredholm integral equation; Periodic quasi-wavelet; Complex plane; fixed point theorem; error analysis.
  
  
• A Discrete Viral Infection Model with Both Modes of Transmission and Distributed Delays  
  by E.L. Boukari Brahim, Hattaf Khalid, E.L. Ghordaf Jalila 
  Abstract: The aim of this work is to propose and analyze a discrete virus dynamics model with distributed delays and both modes of transmission, one by virus-to-cell infection and the other by cell-to-cell transfer. In the proposed model, the first distributed delays describes the time needed for infected cells to produce new virions, and the second portrays the time necessary for the newly produced virions to become mature and infectious. In addition, the infection transmission process is modeled by general incidence functions for both modes. Furthermore, we prove that the proposed discrete model has the same dynamics as the corresponding continuous model, such as positivity, boundedness and global behaviors of solutions with no restriction on the time step size. Moreover, numerical simulations are given to illustrate and confirm our main analytical results.
  Keywords: Viral infection; distributed delay; difference equation; global stability.
  
  
• Single controller for synchronization of coupled neural networks with distributed time-varying delays  
  by Cheng-De Zheng, Fan Xie 
  Abstract: This paper deals with global synchronization in arrays of delayed chaotic neural networks with nonlinear hybrid coupling. By constructing a new Lyapunov-Krasovskii functional, a novel synchronization criterion is presented in terms of matrix inequalities based on Chen's integral inequalities and reciprocal convex technique. These established conditions are heavily dependent on the bounds of both time-delay and its derivative. Through employing Matlab Toolbox and adjusting some matrix parameters in the derived results, the design and applications of the generalized networks can be realized. The effectiveness and applicability of the proposed methods is demonstrated by a numerical example with simulations.
  Keywords: synchronization; matrix inequality; hybrid coupled neural networks; reciprocal convex technique.
  
  
• Numerical approach for solving nonlinear stochastic It  
  by Rebiha Zeghdane 
  Abstract: In this paper, we give a new method for solving stochastic nonlinear Volterra integral equations by using shifted Legendre operational matrix. It is discussed that how the stochastic differential equations (SDE) could numerically be solved as matrix problems. By using this new operational matrix of integration and the so-called collocation method, nonlinear Volterra integral equations is reduced to systems of algebraic equations with unknown Legendre coefficients. Finally, the high accuracy of approximated solutions are illustrated by several experiment.
  Keywords: Stochastic Volterra integral equation; Brownian motion; Approximate solution; Best approximation; Legendre polynomials; Collocation method.
  
  
• Interval oscillation criteria for damped second-order delay differential equation with nonlinearities given by Riemann-Stieltjes integral  
  by MUTHULAKSHMI V, MANJURAM R 
  Abstract: The purpose of this paper is to investigate the oscillatory behavior of certain types of damped second-order forced delay differential equation with nonlinearities given by Riemann-Stieltjes integral. By using the Riccati transformation, some inequalitiess and integral averaging technique, interval oscillation criteria of both El-Sayed type and Kong type are established. Finally, two examples are presented to illustrate the theoretical results.
  Keywords: Interval criteria; Oscillation; Delay differential equation; Damping term; Riemann-Stieltjes integral.
  
  
• Eventually periodicity of solutions for some discrete max-type system of third order  
  by Huili Ma, Haixia Wang 
  Abstract: This paper is concerned with the eventually periodicity of the following max-type difference equation systemrn$$ x_{n+1}=maxleft{frac{A}{x_{n}y_{n-1}},x_{n-2}right},$$rn$$ y_{n+1}=maxleft{frac{A}{y_{n}x_{n-1}},y_{n-2}right},$$rnwhere $nin N$, $Ain R$, and the initial values $x_{-2}, x_{-1}, x_{0}, y_{-2}, y_{-1}, y_{0}$ are arbitrary non-zero numbers.
  Keywords: Periodic solutions; Difference equations; Max-type system.
  
  
• Optimal Control of Behaviour and Treatment in a Nonautonomous SIR Model  
  by Samhita Das, Pritha Das, Parthasakha Das 
  Abstract: In this paper we have considered a nonautonomous SIR (susceptible, infected, removed) model with saturation incidence rate for disease transmission. The global dynamical properties like permanence and global stability of the system as well as extinction of disease are analytically and numerically studied. The impact of behavioural patterns of individuals on disease control is validated along with possible applications. Further, Pontryagin's Maximum Principle is used to characterize optimal level of the two controls, treatment and awareness level. Our objective is to minimise the infected population as well as the cost of applied control. The controls at optimal level are found to achieve different levels of impact on infection. It is observed that the combined impact of treatment and awareness exhibits more effective result in disease control compared to their single application. Based on observation, the strategy regarding the implementation of awareness and treatment is suggested.
  Keywords: Nonautonomous SIR model ; Saturation incidence rate; Permanence; Extinction; Optimal control; Awareness; Treatment.
  
  
• Existence results on impulsive stochastic semilinear differential inclusions  
  by Mustapha Meghnafi, Mohamed Ali Hammami, Tayeb Blouhi 
  Abstract: In this paper, we present some existence results of mild solutions and studyrnthe topological structure of solution sets for the following first-order impulsivernstochastic semilinear differential inclusions driven by L ́vy noise with periodicrnernboundary conditions. We consider the cases in which the right hand side can berneither convex . The results are obtained by using fixed point theorems for multi-rnvalued mappings, more precisely, the technique is based on fixed point theoremrna nonlinear alternative of LeraySchauders fixed point theorem in generalizedrnmetric and Banach spaces.rn
  Keywords: Mild solutions; Periodic solutions; impulses; matrix conver-rngent to zero; generalized Banach space; Poisson jumps; fixed point; set-valued analysis,rndifferential inclusions.rn.
  
  
• Oscillation Criteria for First Order Forced Delay Dynamic Equations with Maxima on Time Scales  
  by H.A. Agwa, Heba A. Hassan, Esraa Magdy 
  Abstract: In this work, we establish some new oscillation criteria for forced first order dynamic equations with maxima. Our results not only complement and generalize some existing results, but also can be applied to some oscillation problems that were not covered before, we also give some examples to illustrate our main results.
  Keywords: Oscillation; forced dynamic equations; time scales; maxima.
  
  
• Oscillation theorems and asymptotic behavior of certain third-order neutral differential equations with distributed deviating arguments  
  by Yibing Sun, Yige Zhao 
  Abstract: The purpose of this paper is to study the oscillation criteria for a class of third-order neutral differential equations with distributed deviating arguments $$ big[b(t)((a(t)(z'(t))^{alpha_1})')^{alpha_2}big]'+int^d_c q(t,xi)f(x(sigma(t,xi)))dxi=0, tgeq t_0 $$ where $z(t)=x(t)+int^n_m p(t,xi)x(tau(t,xi))dxi$ and $alpha_i$ are ratios of positive odd integers, $i=1, 2$. By using a generalized Riccati transformation and an integral averaging technique, we establish some new theorems, which ensure that all solutions of this equation oscillate or converge to zero. Some examples are given to illustrate our main results.
  Keywords: third-order neutral differential equations; distributed deviating arguments; oscillation; asymptotic behavior; generalized Riccati transformation.
  
  
• Explosive tritrophic food chain model with herd behaviour of prey and finite time blow-up of the top predator  
  by Debaldev Jana, G.P. Samanta, Ashok Mondal, Sudeshna Mondal, A.K. Pal, Debasis Manna 
  Abstract: In this work, we have discussed the dynamical behaviours of a three species food chain model where the prey species exhibits herd behaviour and sexually reproductive top predator are of generalist type. Positivity and uniform boundedness of the system are studied to verify its well-posedness. Some conditions for extinction of prey and predators are derived. Feasibility criteria and stability analysis of all the equilibrium points are discussed here. Hopf-bifurcation condition for interior equilibrium point is carried out analytically. Mathematical conditions for finite time blow-up of top predator are established. Numerical simulations are carried out to validate our analytical findings.
  Keywords: Square root functional response; generalist predator; sexual reproduction; Hopf-bifurcation; finite time blow-up.
  
  
• Global stability of virus dynamics models with capsids and two routes of infection  
  by Ahamed Elaiw, Sami Almalki 
  Abstract: We study the global dynamics of within-host viral infection models with virus DNA- containing capsids. The effect of antibody immune response has been considered. The uninfected cell become infected due to its contacts with a virus or an infected cell. In the second model, the incidence rate is given by saturation. The well-posedness of the model is establised. We utilize Lyapunov method to prove the global stability of the equilibria. We support our theoretical results by numerical simulations.
  Keywords: Viral infection; global stability; Lyapunov function; capsids.
  
  
• On the initial value problem of impulsive differential equation involving Caputo-Katugampola fractional derivative of order q(1, 2)  
  by Xianmin Zhang 
  Abstract: This paper mainly focuses on the non-uniqueness of solution to the initial value problem (IVP) of impulsive fractional differential equation (IFrDE) with Caputo-Katugampola derivative (of order q (1, 2)). This impulsive higher order fractional differential equation may involve two inhomogeneous impulses, and the obtained result show that their equivalent integral equation include two arbitrary constants, which means that its solution is non-unique. Next, a numerical example is used to show the non-uniqueness of solution for the IVP for IFrDE.
  Keywords: fractional differential equation; impulsive fractional differential equation; impulse; Caputo-Katugampola fractional derivative.
  
  
• On a coupled nonlinear fractional integro-differential equations with coupled non-local generalized fractional integral boundary conditions  
  by Subramanian Muthaiah 
  Abstract: We investigate a coupled Liouville-Caputo fractional integrodifferential equations (CLCFIDEs) with nonlinearities that depend on the lower order fractional derivatives of the unknown functions, and also fractional integrals of the unknown functions supplemented with the coupled non-local generalized Riemann-Liouville fractional integral (GRLFI) boundary conditions. The existence and uniqueness results have endorsed by Leray-Schauder nonlinear alternative, and Banach fixed point theorem respectively. Sufficient examples have also been supplemented to substantiate the proof, and we have discussed some variants of the given problem.
  Keywords: Fractional differential equations; Liouville-Caputo derivatives; Coupled system; Generalized fractional integrals; Non-local; Existence; Fixedpoint.
  
  
• A new hybrid collocation method for solving nonlinear two-point boundary value problems  
  by R. Delpasand, Seyed Mohammad Mehdi Hosseini, F.M. Maalek Ghaini 
  Abstract: In this paper, numerical solution of boundary value problems of nonlinear ordinary differential equations by the collocation method is considered. Of course, to avoid solving systems of nonlinear algebraic equations resulting from the method, residual function of the boundary value problem is considered and an unconstrained optimization model is suggested. Particle Swarm Optimization algorithm is used for solving the unconstrained optimization problem. In addition, convergence properties of the Chebyshev expansion are studied. The scheme is tested on some interesting examples and the obtained results demonstrate reliability and efficiency of the proposed hybrid method.
  Keywords: Nonlinear boundary value problems; Pseudospectral method; Chebyshev polynomials; Particle Swarm Optimization; Convergence analysis.
  
  
• Qualitative analysis of a fractional model for HBV infection with capsids and adaptive immunity  
  by Moussa Bachraoui, Khalid Hattaf, Noura Yousfi 
  Abstract: This paper presents a mathematical model governed by fractional differential equations (FDEs) that describes the dynamics of hepatitis B virus (HBV) infection in within human body. The FDE model takes into account the HBV DNA-containing capsids, and the adaptive immunity mediated by cytotoxic T lymphocytes (CTL) cells and antibodies. Also, the incidence of infection is presented by Hattaf-Yousfi functional response that includes various forms existing in the literature. Moreover, the qualitative properties of the FDE model is rigorously established. Finally, numerical simulations are presented to support the theoretical results.
  Keywords: HBV infection; adaptive immunity; fractional differential equations; global dynamics.
  
  
• Stable RBF-RA method for solving fuzzy fractional kinetic equation  
  by H. Jafari, F. Fakhr Kazemi 
  Abstract: The direct method based on the flat radial basis functions for obtaining numerical solution of differential equations is highly ill-conditioned. Therefore, many studies have been dedicated to overcome this ill-conditioning by using different techniques.\ Here, the radial basis function algorithm based on vector-valued rational approximations is utilized to obtain the numerical solution of fuzzy fractional differential equations. This stable method can be applied with any sort of smooth radial basis function easily and accurately. To illustrate the accuracy and stability of the presented algorithm, we focus on solving the kinetic model with fuzzy fractional derivative.
  Keywords: Radial basis functions; Rational approximation; Kinetic fuzzy fractional model; Shape parameter; Caputo-fuzzy fractional derivative.
  
  
• A two-echelon supply chain model with deterioration and stock-dependent demand via forward and backward stocking policies  
  by GANESH KUMAR M 
  Abstract: We have developed an integrated inventory model for deteriorating items in a two echelon supply chain. In this model, we have assumed that the vendor produced a single product at a constant rate and transferred it in equal-sized batches to the buyers warehouse. Some of the products are presented to the customer in the buyer display area and the demand is assumed to be positively dependent on the products displayed. Shortages are not permitted, and instantaneous replenishment is made when the inventory level reaches zero. Due to deterioration, the vendor incurs a warranty cost for each deteriorated item produced. In this model, we incorporated unit time production costs. We compared the total profit for both forward and backward stock policy, and we show that the holding cost decreases as the stock moves downstream, the vendor has to adhere to the forward stock policy. The aim is to determine the number of deliveries needed to transfer the items from the vendor to the buyers warehouse and from the buyers warehouse to display area, lot size such that the average profit of the system attains its maximum. Numerical examples are provided for illustrating the model.
  Keywords: Discrete optimisation; inventory control; lot sizing; supply chain; stock dependent demand.
  
  
• Coronary Artery Disease Classification from Clinical Heart Disease Features using Deep Neural Network  
  by RAJESWARI D, THANGAVEL K 
  Abstract: Coronary artery disease (CAD) is the most dreadful clinical syndrome affecting a multitude of people globally and it increases the morbidity rate every year. Early detection of CAD is very important for appropriate treatment which can stop complications like heart failure. The clinical health data can effectively be used for the non-invasive detection of CAD. In this work, we employ Deep Neural Network (DNN) for developing a heart disease prediction model. The proposed model has been tested on Z-Alizadeh Sani dataset from UCI and the results show that the DNN classifier improves prediction accuracy significantly. The performance improvement of 75.7% using DNN architecture has been achieved when compared to K-Nearest Neighbour (KNN).
  Keywords: Coronary Artery Disease; Heart Disease; Data Mining; Machine Learning; Deep learning; Deep Neural Network; KNN; Classification.
  
  
• Existence and Boundary Behavior of Positive Solutions for a Coupled Fractional System  
  by Imen Ben Saad, Sameh Turki, Zagharide Zine El Abidine 
  Abstract: We consider the following semilinear fractional system begin{equation*} label{eq1} left{ begin{array}{ll} displaystyle D^{alpha} u=p(t)displaystyle u^{a}displaystyle v^{r}textrm{ in }(0,1) , displaystyle D^{beta} v=q(t)displaystyle u^{s }displaystyle v^{b}textrm{ in }(0,1) , displaystylelim_{t rightarrow 0^{+}}displaystyle t^{1-alpha}u(t) =displaystylelim_{t rightarrow 0^{+}} displaystyle t^{1-beta}v(t)=0, end{array} right. end{equation*}% where $ alpha, ; betain (0,1)$, $a,; b in(-1,1)$, $r, ;sinmathbb{R}$ such that $(1-'a')(1-'b')-'rs'>0$, $D^{alpha}$, $D^{beta}$ are the Riemann-Liouville fractional derivatives of orders $alpha, ; beta$ and the nonlinearities $p, ; q$ are positive measurable functions on $(0, 1)$. Applying the Sch"{a}uder fixed point theorem, we establish the existence and the boundary behavior of positive solutions in the space of weighted continuous functions.
  Keywords: System of fractional differential equations; Boundary behavior; Karamata class; Schauder's fixed point theorem.
  
  

Special Issue on: CDSM2CT-2019 Advances in Qualitative Behaviours of Dynamical Systems

• Analysing of Complementary Perfect Hop Domination Numeral of Corona Products of Graphs  
  by Mahadevan G, Vijayalakshmi V 
  Abstract: Recently, the authors introduced the concept of Complementary perfect hop domination number of a graph. A set S subset of V is a hop dominating set of G, if every vertex v belongs to V-S there exists u belongs to S such that d(u,v) = 2. A set S subset of V is said to be complementary perfect hop dominating set of G, if S is a hop dominating set and has atleast one perfect matching. The minimum cardinality of complementary perfect hop dominating sets is called complementary perfect hop domination number of G and it is denoted by CPHD(G). In this paper we explore the CPHD number for the Corona product of two distinct paths and cycles.
  Keywords: complementary perfect hop dominating set; hop dominating set.
  
  
• Bifurcation Analysis of Fractional-Order VD Model  
  by Ramesh P 
  Abstract: In this paper, we introduced the fractional order VD model. First, we established basic results such as existence, uniqueness, boundedness of the fractional order dynamical system. Next, we evaluate the local stability and Hopf bifurcation of the fractional order VD model. Finally, some numerical simulations evaluated with some examples.
  Keywords: Boundedness; Existence and uniqueness; Fractional dynamical system; Stability; Hopf bifurcation.
  
  
• 2 - Pebbling Property of Buttery Derived Graphs  
  by Sagaya Suganya 
  Abstract: For a graph G, f(G) is the least configuration of p pebbles on the vertices of G, so that we can move a pebble to any vertex by a sequence of moves and each move is taking two pebbles of one vertex and placing one pebble on an adjacent vertex. A graph G is said to satisfy 2 - pebbling property, if it is possible to move two pebbles to any arbitrarily chosen vertex with a possible configuration of 2f(G)- q + 1 pebbles, where q is the number of vertices with at least one pebble. This paper determines the pebbling number and the 2 - pebbling property of butterfly derived graphs.
  Keywords: pebbling; 2 - pebbling; butterfly graph; Benes graph; augmented butterfly graph; enhanced butterfly graph.
  
  
• Cartesian Product of the Extensions of Fuzzy Soft Ideals over Near-rings  
  by T. Manikantan, S. Ramkumar 
  Abstract: In this paper, the notions of fuzzy translation, fuzzy multiplication and fuzzy magnified translation of the cartesian product of fuzzy soft sets are introduced. The cartesian product of the extensions of fuzzy soft sets over a near-ring is defined. Using these notions, the concepts of fuzzy soft near-ring and fuzzy soft ideal over a near-ring are studied. Finally, the fuzzy translation, fuzzy multiplication and fuzzy magnified translation of the cartesian product of the extensions of two fuzzy soft sets over a regular commutative near-ring are equivalent for a fuzzy soft near-ring (resp. ideal) is proved.
  Keywords: Fuzzy magnified translation; Extension of fuzzy soft set; Cartesian product of the extensions of fuzzy soft sets; Fuzzy soft near-ring; Fuzzy soft ideal.
  
  
• Convergence results of K iteration process for nonexpansive mappings with an application  
  by Sankara Narayanan M, Anbukkarasi V, Marudai M 
  Abstract: This paper deals with the convergence theorems that approximate the fixed points of nonexpansive mappings via K iteration process under the framework of uniformly convex Banach space. One numerical example is provided to illustrate the derived result. Further, based on the proposed result, the existence of the mild solution for wave equation is discussed. In addition to that one new iterative scheme is proposed for finding the fixed points of nonexpansive and quasinonexpansive mappings.
  Keywords: K iteration process; uniformly convex Banach space; nonexpansive mapping.
  
  
• Dengue Outbreaks Prediction Model for Urban Colombo using Meteorological Data  
  by KKWH Erandi, S.S.N. Perera, A.C. Mahasinghe 
  Abstract: Dengue is a viral born disease with complex transmission dynamics. Disease outbreak can exert an increasing pressure on the health system with high mortality. Understanding and predicting the outbreaks of dengue transmission is vital in controlling the spread. In this work we propose a generalised linear regression model to understand the dynamics of the disease. Further, to moderate the model we analyse the correlation with meteorological parameters. Then we define a threshold value in order to capture the outbreak. Finally, we compare the proposed model with the existing methods.
  Keywords: Dengue; Climate Factors; Generalized Linear Model; Disease Outbreak; Threshold.
  
  
• Inventory control techniques in a two-echelon supply chain model with fuzzy demand and learning effect  
  by S. Ganesan, R. Uthayakumar 
  Abstract: The crucial part of decision-making in a two-echelon supply chain modelling is to decide the production quantity of the manufacturer to satisfy the demand of the retailers. In this paper, we develop a two-echelon supply chain model with one manufacturer and multiple retailers. The production quantity of the manufacturer and demand of each retailer are the uncertain components of the model, and they are quantified by fuzzy numbers. Wright's learning function is applied in the fuzzy limits to appertain the knowledge acquired through experience of supply chain leaders in decision-making. We determine the optimal order quantity of each retailer by calculus method. An approximate value of generalized harmonic numbers is applied for the derivation of optimal values in learning model. Numerical examples are supplied to demonstrate both fuzzy and learning models. The robustness of the learning model is explained using numerical examples and comparative study.
  Keywords: supply chain; inventory control; parabolic fuzzy number; Wright's learning curve; generalized harmonic number.
  
  
• Stability Result for Fractional Neutral Stochastic Differential System Driven by Mixed Fractional Brownian Motion  
  by Dhanalakshmi K, Balasubramaniam P. 
  Abstract: In this manuscript, stability results for fractional neutral stochastic integro-differential system is established subject to mixed fractional Brownian motion(fBm). Sufficient conditions for stability results are derived based on the pth mean square norm, fixed point theorem and help of new integral inequality. As, a final point an example is given to illustrate the effectiveness of the obtained theory.
  Keywords: Fractional differential equations; Mild solution; Neutral stochastic differential equation; Exponential stability.
  
  
• A Deep Learning Approach for Brain Tumor Detection System using Convolutional Neural Networks  
  by Kalaiselvi T, Padmapriya S.T 
  Abstract: The proposed work is aimed to develop convolution neural network (CNN) architecture based computer aided diagnostic system for human brain tumor detection process from magnetic resonance imaging (MRI) volumes. CNN is a class of Deep Learning networks that are commonly applied to analyze voluminous images. In the proposed method, a CNN model is constructed and trained using MRI volumes of BraTS2013 data. More than 4000 images of normal and tumor slices are used to train the proposed CNN system with 2-layers. The system is tested with about 1000 slices from BraTS and got very accurate results about 90-98% of accuracy. Further, the performance of proposed CNN system is tested by taking a set of clinical MRI volumes of popular hospital. The obtained results are discussed and focused for the future improvement of the proposed system.
  Keywords: Neural networks; MRI; brain tumor; deep learning; tumor detection;.
  
  
• Mean-square asymptotic stability of stochastic inertial neural networks with time-delay and Markovian jump parameters  
  by KRISHNASAMY RAMASAMY, Raju K. George 
  Abstract: This article investigates the stability of inertial neural networks which incorporates the effects of both intrinsic and extrinsic noises along with time-delay. These intrinsic and extrinsic noises are taken to be in the form of Markovian jump parameters and Brownian motion respectively. Required sufficient stability conditions are established in the form of linear matrix inequalities from the construction of Lyapunov-Krasovskii functional. Derived conditions will be delay-dependent which includes information about the bounds of the time-delay and also its derivatives. Theory of Lyapunov stability, Ito calculus and linear matrix inequality are used to derive the main results. Numerical example is given to demonstrate the validity of the derived theoretical results.
  Keywords: Inertial neural networks; Mean-square asymptotic stability; time-delay; Markovian jump; Lyapunov-Krasovskii functional.
  
  

Special Issue on: Differential, Difference and Dynamic Equations

• Analysis of the bilateral Laplace transform on time scales with applications  
  by Tom Cuchta, Svetlin Georgiev 
  Abstract: The bilateral Laplace transform on time scales is investigated analytically and its absolute convergence, uniform convergence, and inversion integral are proven . Afterwards, a Fourier transform is defined and used to solve partial dynamic equations.
  Keywords: time scales calculus; bilateral Laplace transform; Fourier transform; partial dynamic equations.
  
  
• A discrete SIS model of fractional order  
  by Tom Cuchta, Sabrina Streipert 
  Abstract: In this work, we introduce two epidemic fractional difference equation models and derive their explicit solutions. The presented model is of the Susceptible-Infected-Susceptible class, which assumes that the disease is spread from susceptible to infected individuals who join the group of susceptible after recovery. The model is constructed using the fractional difference operators defined in [14], which sets it apart from the few existing discrete fractional epidemic model formulations. The unique solution of the presented fractional difference epidemic models is derived and relations to existing discrete SIS models are discussed.
  Keywords: Fractional Difference Equations; Riemann-Louiville Derivative; Difference Equations; Nabla-Difference Equations; Epidemic Model; SIS; Explicit Solution; Unique Solution.