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In the above article, a second-order convergence in the maximum norm of the two-dimensional regularized long-wave equation is proved. However, due to a wrong inequality in Lemma 3.3 used in X. Wang et al.[1], there are some fundamental errors in this paper, in particular the proof of Theorem 3.3 and the convergence Theorem are no longer correct. The present brief paper contains clarifying comments.

The difference between conventional replicator dynamics and pairwise (PW) nonlinear Fermi dynamics can be discerned by studying the evolutionary dynamics of the interactions between the symmetric cyclic structure in the rock–paper–scissors game and inter- and intraspecific competitions. Often, conventional replicator models presume that the payoff difference among species is a linear function (a linear benefit). This study introduces a PW contrast under the properties of the well-known Fermi rule, where species play against one another in pairs. To model a PW nonlinear evolutionary environment (a nonlinear benefit) within this framework, both analytical and numerical approaches are applied. It is determined that the dynamics of the linear and nonlinear benefits can present the same stability conditions at equilibrium. Moreover, it is also demonstrated that, even in an identical equilibrium condition for both dynamics, the numerical result run by a deterministic approach presents a faster stability state for nonlinear benefit dynamics. This study also suggests that introducing mutation as demographic noise can effectively disrupt the phase regions and show the different relationships between linear and nonlinear dynamics. The symmetric bidirectional mutation among all the species reduced to the stable limit cycle by an arbitrary small mutation rate is also explored. Due to the environmental noise, however, linear and nonlinear exhibit the same steady state. Nevertheless, non-linearity illustrates more stable and faster stability situations. Our result suggests that environmental and demographic noise on the evolutionary dynamic framework can serve as a mechanism for supporting PW nonlinear dynamics in multi-species games.

Two limit cycles for the continuous model of the rockscissorspaper (RSP) game is constructed.The Hopf bifurcation method is used for the construction of limit cycles.The results give a partial answer to an open question posed by Neumann and Schuster. In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rockscissorspaper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game.

In this paper, we first point out a gap in Yang (2012). Next, strong convergence theorem for the common solution for a finite family of ? -strongly accretive operator equations in Banach spaces is established without any bounded assumption, and we also give an example to show universality of the result of this paper, which improve and extend some recent results.

In the paper the second order half-linear delay differential equation ( r ( t ) | u ' ( t ) | α - 1 u ' ( t ) ) ' + p ( t ) | u ( ? ( t ) ) | α - 1 u ( ? ( t ) ) = 0 , α 1 is studied under the condition ? ∞ r - 1 / α ( t ) d t < ∞ . The oscillation criterion proved by Sun and Meng (2006, Theorem 2.2) is carefully examined and improved by simplifying and sharpening the estimates used in the original proof and removing unnecessary assumptions. Among others, using elementary arguments we show that an arbitrary positive nondecreasing function ? from this criterion can be safely and with no loss of generality replaced by a constant function. An extension to other related equations, such as neutral equations is also provided.

We show that the main results in the recent paper [4] by Sadiq Basha (2012) are not real generalizations but particular cases of existing fixed point theorems in the literature.

In the present paper we studied the Voronovskaya type theorem for modified Gamma operators presented by A. Izgi in the paper ''Voronovskaya type asymptotic approximation by modified Gamma operators'' [Applied Mathematics and Computation 217 (2011) 8061-8067]. We show that this theorem is not true. We present a correct form of this theorem and its proof. Furthermore, we obtain an error estimate for modified Gamma operators.

Using advanced machine learning techniques as an alternative to conventional double-entry volume equations known as classical allometric models , regression models of the inside-bark volume (dependent variable) for standing Eucalyptus globulus trunks (or main stems) have been built as a function of the following three independent variables: age, height and outside-bark diameter at breast height ( D ). The allometric models of volume, biomass or carbon support the estimation of carbon storage in forests and agroforestry systems. On one hand, this paper presents the construction of allometric models of the inside-bark volume for E. globulus trees. On the other hand, the experimental observed data (age, height, D and inside-bark volume) for 142 trees ( E. globulus ) were measured and a nonlinear model was built using a data-mining methodology based on multivariate adaptive regression splines (MARS) technique and multilayer perceptron networks (MLP) for regression problems. Coefficients of determination and Furnival’s indices indicate the superiority of the MARS technique over the allometric regression models and the MLP network. The agreement of the MARS model with observed data confirmed the good performance of the same one. Finally, conclusions of this innovative research are exposed.

In this paper, we focus on the proposed algorithms to solve a linear programming problem with the convex combination of the max–min and the max–average composition and the max–star composition, respectively. They have been proposed by Ghodousian and Khorram [A. Ghodousian, E. Khorram, Solving a linear programming problem with the convex combination of the max–min and the max–average fuzzy relation equations, Appl. Math. Comput. 180 (2006) 411–418] and Khorram et al. [E. Khorram, A. Ghodousian, A. Abbasi Molai, Solving linear optimization problems with max–star composition equation constraints, Appl. Math. Comput. 179 (2006) 654–661], respectively. Firstly, we show that the “Tabular method algorithm” in the first paper and the “First procedure” in the second paper may not lead to the optimal solutions of the two models in some cases. Secondly, we generalize the proposed algorithm by Abbasi Molai and Khorram [A. Abbasi Molai, E. Khorram, A modified algorithm for solving the proposed models by Ghodousian and Khorram and Khorram and Ghodousian, Appl. Math. Comput. 190 (2007) 1161–1167] to solve the two models. In fact, it modifies the presented algorithms in the two papers. Finally, some numerical examples are given to illustrate the purposes.

In this paper, some sufficient conditions are established for the oscillation of second-order neutral differential equations(r(t)@j(x(t))|Z^'(t)|^@a^-^1Z^'(t))^'+q(t)f(x(@s(t)))=0,t>=t"0>0, where Z(t)=x(t)+p(t)x(t-@t) and @a>0,0==t"0>0, where @!"t"""0^~dtr(t)

Comment on the paper “A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications” in Applied Mathematics and Computation 174 (2006) 1169–1180. The paper considered a class of two-point boundary-value problems of the form (1) d 4 y d x 4 + f ( x ) y = g ( x ) , a ⩽ x ⩽ b , (2) y ( a ) = A 1 , y ( b ) = A 2 , y ′ ( a ) = B 1 , y ′ ( b ) = B 2 , where f ( x ) and g ( x ) are continuous on [ a , b ], and A i and B i ( i = 1, 2) are finite real constants. Here we correct some mistake in derivation of non-polynomial spline, boundary formulas, truncation errors, convergence analysis and computational experiments.

In this short note an addition to the paper [A.G. Bratsos, M. Ehrhardt, I.Th. Famelis, A discrete Adomian decomposition method for discrete nonlinear Schrodinger equations, Appl. Math. Comput. 197(1) (2008) 190-205] using the modulus of the terms evaluated from the Adomian decomposition method on p. 194 and their relation to the convergence of the resulting series is presented. Conclusions for the accuracy of the approximated solution are derived.

One of the desirable attributes of real-life examination timetabling solutions is the maximization of paper spread, which is a measure of the amount of study time that each student has between examinations. In this study, we face with a predefined examination schedule that must be modified in order to maximize paper spread in course of examination. We will show how the integer programming can be employed to achieve this aim. The model presents constraints for the quality of feasible examination timetables and all the requirements found in most academic institutions. The approach is tested on real-world exam timetabling problems. The computational experiments and results will be reported.

In this paper we will establish some new oscillation criteria for the second-order retarded differential equation of the form ( r ( t ) | u ′ ( t ) | α - 1 u ′ ( t ) ) ′ + p ( t ) | u [ τ ( t ) ] | α - 1 u [ τ ( t ) ] = 0 . The results obtained essentially improve and extend those of Džurina and Stavroulakis [Oscillation criteria for second-order delay differential equations, Appl. Math. Comput., 140 (2003) 445–453]. An open problem is proposed at the end of this paper.

The aim of this note is to show that the optimization algorithm proposed in [6] may not lead to the optimal solution in some cases. In fact, the optimization problem remains open if we do not apply the branch-and-bound method to solve it or compute all the objective values of minimal solutions of the feasible domain.

This paper considers certain neutral differential equation with the distributed delayddt[x(t)+c(t)x([email protected])]+p(t)x(t)+q(t)x([email protected])+r(t)@!"t"-"@d^tx(s)ds=0.Without imposing a nonnegative restriction on coefficients of the equation, we establish a delay-dependent asymptotic stability condition for the equation by introducing a new Lyapunov function. The condition presented here is different from that of Park [J.H. Park, LMI optimization approach to asymptotic stability of certain neutral delay differential equation with time-varying coefficients, Appl. Math. Comput. 160 (2005) 355-361] in the sense that it is dependent on all delays of the equation. Numerical examples illustrate effectiveness and sharpness of our theoretical result.

A counterexample is provided in this short note to show that some of local convergence theorems established in [D.K.R. Babajee, M.Z. Dauhoo, An analysis of the properties of the variants of Newton’s method with third order convergence, Appl. Math. Comput. 183 (2006) 659–684] are not always true. Some mistakes in the proofs of these theorems are pointed out.

In this technical note, we give a short extension application for nonlinear sum of ratios problem (P) considered in [Y.J. Wang, K.C. Zhang, Global optimization of nonlinear sum of ratios problem, Applied Mathematics and Computation 158 (2004) 319–330]. Actually our result is slightly more general, since we do not specify additional positive coefficient for each ratio. In this note, we use different equivalent problem as done in Wang and Zhang (2004). Our method introduce p variables less than other method (Wang and Zhang, 2004), and our approach need not additional special program to obtain the upper and lower bound of numerator and denominator for each ratio in the objective function.

In this note, we will modify several gaps in Gurudwan and Sharma [N. Gurudwan, B.K. Sharma, Convergence theorem for the common solution for a finite family of ϕ -strongly accretive operator equations, Appl. Math. Comput. 217 (2011) 6748–6754].

The recent paper mentioned in the title contains a confusing statement on computing conservation laws corresponding to symmetries of nonlinearly self-adjoint differential equations. The present brief article contains clarifying comments.

The aim of this short note is to show on a numerical example that one of the two optimization algorithms proposed in [E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition, Appl. Math. Comput. 173 (2006) 872–886] may not lead to the optimal solution in some cases. Besides it will be pointed out that the other algorithm (correct, but unfortunately ineffective for larger problems) proposed in [E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition, Appl. Math. Comput. 173 (2006) 872–886] can be extended to a wider class of non-convex optimization problems.

In this short paper, we introduce and analyze a generalized preconditioned parameterized inexact Uzawa method for solving singular saddle point problems, this method is a generalization of the PPIU method (see, Gao and Kong, 2010), then we give the semi-convergent conditions of this method, which are more practical for the applications.

The Wiener polarity index Wp(G) of a graph G is the number of unordered pairs of vertices {u,?v} in G such that the distance between u and v is equal to 3. Very recently, Zhang and Hu studied the Wiener polarity index in Y. Zhang, Y. Hu, 2016 38. In this short paper, we establish an upper bound on the Wiener polarity index in terms of Hosoya index and characterize the corresponding extremal graphs. Moreover, we obtain Nordhaus-Gaddum-type results for Wp(G). Our lower bound on W p ( G ) + W p ( G ? ) is always better than the previous lower bound given by Zhang and Hu.

Recently, some results on the trigonometric approximation of Lip(ω(t), p)-class have been given by Srivastava and Singh (2014). In this note, we point out that one of the assumption conditions in their theorems is invalid in some cases. Thus, all conclusions in the paper based on the condition lose the foundation. Moreover, we obtain the similar results for a class of more extensive functions after deleting the mentioned condition, and weakening another condition in their theorems.

In this short paper we study the counterpart, within the context of a general class of fluids, of two famous unsteady flows originally studied by Stokes, within the context of Navier-Stokes fluid, namely Stokes' first and second problems. The class of fluids considered, stress power-law fluids, are capable of stress thinning or stress thickening and can describe phenomena that the classical power-law fluids are incapable of modeling. Within the context of the problems considered, we are able to find solutions wherein stress boundary layers develop.

In this short paper, we present time scales version of mean value theorems for integrals in the single variable case.

Divider method is widely used to obtain the fractal dimension of curves embedded into a 2-dimensional space. This method suffers from a number of problems. The major problem is choosing initial and final step length of dividers. In this short paper we would like to demonstrate that the criterion for the selection of dividers (minimum and maximum value) in the calculation of fractal dimension of the irregular curves with minimum time complexity.

Investigation of the asymptotic properties of solutions to systems of discrete equations is a topic of permanent interest. Numerous papers analyze the so-called prescribed behaviour of solutions giving sufficient conditions for the existence of at least one solution having a given asymptotic behaviour. To a smaller extent the problem is considered of determining the initial data generating such solutions. The present paper finds its niche being concerned with the case of a triangular system of linear discrete equations. The existence of solutions with a prescribed asymptotic behaviour is proved and algorithms suggested for computing stepwise the initial data defining such solutions and eventually suggesting these. Illustrative examples (supported with a MATLAB program) are considered and some open problems are formulated as well.

In 1984, Kyurkchiev et al. constructed an infinite sequence of iterative methods for simultaneous approximation of polynomial zeros (with known multiplicity). The first member of this sequence of iterative methods is the famous root-finding method derived independently by Farmer and Loizou (1977) and Gargantini (1978). For every given positive integer N, the Nth method of this family has the order of convergence 2 N + 1 . In this paper, we prove two new local convergence results for this family of iterative methods. The first one improves the result of Kyurkchiev et al. (1984). We end the paper with a comparison of the computational efficiency, the convergence behavior and the computational order convergence of different methods of the family.

This paper has attended to obtain the approximate analytical expressions of periodic solutions in the delayed non-smooth Internet TCP-RED congestion control system employing the semi-analytical method named as the harmonic balance method with alternating frequency/time (HB-AFT) domain technique. For the first time, the methodology presented herein provides the accurate approximate analytical expressions of periodic solutions, including the non-impacting periodic solutions and impacting periodic solutions, in the non-smooth dynamical system with time-delays. They agree very well with the results of numerical simulations by MATLAB. It implies that the proposed method in this paper is accurate and effective. Furthermore, rich dynamics of this delayed non-smooth system is discovered in the present paper. Three kinds of bi-stability, i.e. coexistence of asymptotically stable equilibrium and equilibrium, equilibrium and periodic solution, periodic solution and periodic solution, have been found in the system with the variation of the delay. And the grazing solution, the impacting solution traversing both the lower threshold Tmin and the larger threshold Tmax, and a route to chaos, i.e., intermittent to chaos, have been also obtained by varying the delay employing numerical method.

On the basis of the previous work by Tang and Zhang [37], in this paper we present a more effective way to construct high-order symplectic integrators for solving second order Hamiltonian equations. Instead of analyzing order conditions step by step as shown in the previous work, the new technique of this paper is using Legendre expansions to deal with the simplifying assumptions for order conditions. With the new technique, high-order symplectic integrators can be conveniently devised by truncating an orthogonal series.

The solution of Bernoulli differential equation can be described as a generalized Logistic curve function. Weibull cumulative distribution function is also an analytic solution of one variable coefficient differential equation. But, however because of the exact analytical solution problem of the equation, up to now, the developing coefficient of all the grey forecasting models is still defined as a constant. So, the aim of this paper is just to construct a novel grey differential equation model by combining NGBM(1,1) and Weibull cumulative distribution function. The proposed model(WBGM(1,1)) has the advantages of NGBM(1,1) and Weibull cumulative distribution. Where there are double shape parameters, the developing coefficient of the grey forecasting model is extended to be a variant. Property analysis of WBGM(1,1) shows that the fitting accuracy is higher and the applicable confines are wider. Finally, this paper gives an optimization method for the parameters of WBGM(1,1). A classic example and a practical case are studied for confirming the effectiveness of WBGM(1,1). The case study is the prediction for the number of invention patents of integrated circuit(IC) filed in China from 2007 to 2017. Results of the example and case study are compared to other forecasting models, including GM(1,1), NGBM(1,1), Holt exponential smoothing and ARIMA. Results show that WBGM(1,1) is a more general and more efficient model in grey prediction theory.

This paper considers the problems on the existence and uniqueness, the pth(p ≥ 1)-moment and the almost sure stability with a general decay for the global solution of stochastic delay differential equations with Markovian switching, when the drift term and the diffusion term satisfy the locally Lipschitz condition and the monotonicity condition. By using the Lyapunov function approach, the Barbalat Lemma and the nonnegative semi-martingale convergence theorem, some sufficient conditions are proposed to guarantee the existence and uniqueness as well as the stability with a general decay for the global solution of such equations. It is mentioned that, in this paper, the time-varying delay is a bounded measurable function. The derived stability results are more general, which not only include the exponential stability but also the polynomial stability as well as the logarithmic one. At last, two examples are given to show the effectiveness of the theoretical results obtained.

This paper studies European call option pricing problem under a hard-to-borrow stock model where stock price and buy-in rate are fully coupled. Avellaneda and Lipkin (2009) proposed a simplified solution approach with an independence assumption, and then derived a semi-explicit pricing formula. However, such an approach has limited its application to more general cases. In this paper, we propose a partial differential equation (PDE) approach for pricing European call options, regardless of the independence assumption. A two-dimensional PDE is derived first with a set of appropriate boundary conditions. Then, two numerical schemes are provided with different treatments of the jump term. Through our numerical results, we find that the semi-explicit formula is a good approximate solution when the coupling parameter is small. However, when the stock price and the buy-in rate are significantly coupled, the PDE approach is preferred to solve the option pricing problem under the full hard-to-borrow model.

Delay-dependent stability analysis of linear systems with a time-varying delay is investigated in this paper. Firstly, instead of developing a Lyapunov–Krasovskii functional (LKF) including augmented non-integral quadratic terms as usual, this paper proposes two augmented-integral-function based LKFs, which can reflect closer relationships among system states. Secondly, to bound the derivative of LKF more accurately, a new integral inequality with several free matrices is developed. Compared with the free-matrix-based integral inequality, this inequality can provide additional freedom due to the free matrices introduced. Then, by employing those LKFs and the improved integral inequality, several delay-dependent stability criteria are established for two types of delays. Finally, two numerical examples are given to demonstrate the superiority of the proposed method.

In this paper, a pricing formula for volatility swaps is delivered when the underlying asset follows the stochastic volatility model with jumps and stochastic intensity. By using Feynman-Kac theorem, a partial integral differential equation is obtained to derive the joint moment generating function of the previous model. Moreover, discrete and continuous sampled volatility swap pricing formulas are given by employing transform techniques and the relationship between two pricing formulas is discussed. Finally, some numerical simulations are reported to support the results presented in this paper., 15PAGES

It is well known that a good approximate sampled-data model is very important in sampled-data controller design of nonlinear continuous-time system because the exact sampled-data model of nonlinear continuous-time system is often unavailable. It is also a truth that unstable zero dynamics of sampled-data model greatly limit the control performance of the system. This paper investigates the properties of approximate sampled-data model and its zero dynamics, which generated during the process of discretization with backward triangle sample and hold (BTSH). Further, a more accurate sampled-data model is obtained for nonlinear systems in BTSH case. The zero dynamics form and stable conditions are also derived. Finally, numerical examples are provided to better show the results developed in this paper.

In this paper, the stability problem is investigated for a class of discrete-time switched systems with unstable subsystems under the mode-dependent average dwell time (MDADT) switching. A multiple convex Lyapunov function (MCLF) and a multiple piecewise convex Lyapunov function (MPCLF) are firstly proposed, which are formulated in a convex combination form of positive definite matrices with quasi-time-dependent coefficients. It is pointed out in the paper that the multiple Lyapunov function (MLF) and the multiple discontinuous Lyapunov function (MDLF) can be regarded as special cases of the proposed MCLF and MPCLF, respectively. Then, the exponential stability conditions are derived by the new Lyapunov functions. Both slow switching and fast switching are exerted on stable modes and unstable modes, respectively. Finally, two numerical examples are given to demonstrate that larger stability regions and tighter MDADT bounds can be obtained by using our developed techniques compared with some recent results.

In this paper, we propose the split-step theta method for stochastic delay integro-differential equations by the Lagrange interpolation technique and investigate the mean square exponential stability of the proposed scheme. It is shown that the split-step theta method can inherit the mean square exponential stability of the continuous model under the linear growth condition and the proposed stability condition by the delayed differential and difference inequalities established in the paper. A numerical example is given at the end of the paper to illustrate the method and conclusion of the paper. In addition, the convergence of the split-step theta method is proved in the Appendix.

A multi-dimensional uncertain differential equation is a type of differential equation. Stability of a multi-dimensional means insensitivity of the state of a system to small changes in the initial state. This paper presents a concept of stability in mean for multi-dimensional uncertain differential equation. Some stability theorems for the solution of multi-dimensional uncertain differential equation are given, in which some sufficient conditions for a multi-dimensional uncertain differential equation being stable in mean. In addition, this paper discusses their relationships between stability in measure and stability in mean.

This paper aims to extend the scheme proposed in Jentzen et al. (2011) for stochastic partial differential equations (SPDEs) with global Lipschitz coefficients to non-global Lipschitz coefficients. We investigate the pathwise convergence of the scheme for a class of semilinear parabolic SPDEs with non-globally Lipschitz nonlinearity. We show first that the scheme is convergent uniformly in time and space under Lipschitz assumption on the nonlinearity of the SPDE, then we obtain the convergence in the case of non-globally Lipschitz nonlinearity via a localization technique. Compared to the scheme introduced in Jentzen (2009) for SPDEs under non-global Lipschitz coefficients, the scheme considered in this paper is simpler in the sense that the former uses two linear functionals of the noise while the latter uses one.

Irrational transfer function has been widely used in modelling and identification. But time response analysis of systems with irrational transfer functions is hard to be achieved in comparison with the rational ones. One of the main reasons is that irrational transfer function generally has infinite poles or zero. In this paper, the closed-loop time response of fractional-system with irrational transfer function is analyzed based on numerical inverse Laplace transform. The numerical solutions and stability evaluation of irrational fractional-order systems are presented. Several examples of fractional-order systems with irrational transfer functions are shown to verify the effectiveness of the proposed algorithm in both time and frequency domain analysis. The MATLAB codes developed to solve the fractional differential equations using numerical Laplace transform are also provided. The results of this paper can be used on analysis and design of control system described by irrational fractional-order or integer-order transfer function.

The paper is concerned with a qualitative analysis for a nonlinear second-order parabolic problem, subject to non-homogeneous Cauchy–Neumann boundary conditions, extending the types already studied. Under some certain assumptions, we prove the existence, estimate, regularity and uniqueness of a classical solution. The considered nonlinear second-order anisotropic diffusion model is then particularized for an image restoration task. The resulted PDE-based model is solved numerically by constructing a finite-difference based approximation algorithm that is consistent to the model and converges fast to its solution. An effective detail-preserving image filtering scheme that removes successfully the white additive Gaussian noise while overcoming the unintended effects is thus obtained. Our successful image restoration and method comparison results are also discussed in this paper.

The main purpose of this paper is to study the traveling wave solutions of the diffusive Lotka–Volterra systems with boundary conditions. With the help of Grobner bases elimination method, a series of new traveling wave solutions have been obtained through symbolic computation. In particular, it is worth noting that these traveling wave solutions may inspire us to explore new phenomena in this system. The obtained results in this paper substantially improve the corresponding results in the known literatures. Finally, we summarize the current study and give the future work.

This paper systematically investigates a stochastic Crowley-Martin predator-prey model. Firstly, we derive sufficient conditions for the existence of positive T -periodic solution for the impulsive case. Secondly, distinguished from the previous papers, asymptotical stability in probability is investigated by combining Khasminskii theory of stability with Lyapunov method. Our conclusion also improves and extends the corresponding existing ones. Thirdly, we establish the sufficient criteria for the existence of a unique ergodic stationary distribution for the Markovian switching case. Finally, two numerical examples are presented to demonstrate the effectiveness and feasibility of our analytical results and reveal the respective effect of white noises and Markovian switching.

This paper investigates convergence properties of gradient neural network (GNN) and GNN-based dynamical systems for computing generalized inverses. The main results are exact analytical solutions for the state matrices of the corresponding GNN and GNN-based dynamical systems. The exact solutions are given in each time instant, which enables to express final results as appropriate limiting expressions. This enables more rigorous convergence analysis in terms of exact solutions. Finally, trajectories of state variables in considered dynamical systems can be generated by avoiding time-consuming numerical solving matrix differential equations inside the selected time interval. Using the fact that the stated dynamical systems are in the essence systems of linear equations, explicit solutions can be obtained using known techniques of ordinary differential equations. Main results of the paper are further transformations of obtained exact solutions using main properties of generalized inverses and linear algebra tools. It is important to mention that the convergence results are expressed in terms of expressions involving outer inverses. Several examples are presented and a closed-form expressions for the solutions are given and implemented in package Mathematica. These are compared with the numerical solutions obtained by Matlab Simulink implementation.

The star chromatic index of a multigraph G, denoted χ s ′ ( G ) , is the minimum number of colors needed to properly color the edges of G such that no path or cycle of length four is bi-colored. A multigraph G is star k-edge-colorable if χ s ′ ( G ) ≤ k . Dvořak et al. (2013) proved that every subcubic multigraph is star 7-edge-colorable. They conjectured in the same paper that every subcubic multigraph should be star 6-edge-colorable. In this paper, we consider this problem in a more general setting, we investigate star list edge coloring of general graph G and obtain an upper bound for the choice number of star edge coloring of graphs, namely, we proved that χ s l ′ ≤ ⌈ 2 Δ 3 2 ( 1 Δ + 2 ) 1 2 + 2 Δ ⌉ .