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In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $. The derivation is extended from a recently published paper by Huseynov et al. in [1], which is limited for incommensurate fractional order $ 0 < \alpha, \beta < 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 < \alpha, \beta < 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.

The mean curvature-based image deblurring model is widely used to enhance the quality of the deblurred images. However, the discretization of the associated Euler-Lagrange equations produce a nonlinear ill-conditioned system which affect the convergence of the numerical algorithms like Krylov subspace methods. To overcome this difficulty, in this paper, we present two new symmetric positive definite (SPD) preconditioners. An efficient algorithm is presented for the mean curvature-based image deblurring problem which combines a fixed point iteration (FPI) with new preconditioned matrices to handle the nonlinearity and ill-conditioned nature of the large system. The eigenvalues analysis is also presented in the paper. Fast convergence has shown in the numerical results by using the proposed new preconditioners.

In this paper we consider an optimal control problem governed by a space-time fractional diffusion equation with non-homogeneous initial conditions. A spectral method is proposed to discretize the problem in both time and space directions. The contribution of the paper is threefold: (1) A discussion and better understanding of the initial conditions for fractional differential equations with Riemann-Liouville and Caputo derivatives are presented. (2) A posteriori error estimates are obtained for both the state and the control approximations. (3) Numerical experiments are performed to verify that the obtained a posteriori error estimates are reliable.

The B-spline curves have been grasped tremendous achievements inside the widely identified field of Computer Aided Geometric Design (CAGD). In CAGD, spline functions have been used for the designing of various objects. In this paper, new Quadratic Trigonometric B-spline (QTBS) functions with two shape parameters are introduced. The proposed QTBS functions inherit the basic properties of classical B-spline and have been proved in this paper. The proposed scheme associated with two shape parameters where the classical B-spline functions do not have. The QTBS has been used for designing of different parts of airplane like winglet, airfoil, turbo-machinery blades and vertical stabilizer. The designed part can be controlled or changed using free parameters. The effect of shape parameters is also expressed.

This paper proposes a simple, accurate and effective empirical formula to determine the number of supporting nodes in a newly-developed method, the localized method of fundamental solutions (LMFS). The LMFS has the merits of meshless, high-accuracy and easy-to-simulation in large-scale problems, but the number of supporting nodes has a certain impact on the accuracy and stability of the scheme. By using the curve fitting technique, this study established a simple formula between the number of supporting nodes and the node spacing. Based on the developed formula, the reasonable number of supporting nodes can be determined according to the node spacing. Numerical experiments confirmed the validity of the proposed methodology. This paper perfected the theory of the LMFS, and provided a quantitative selection strategy of method parameters.

In this paper, we discuss a novel numerical algorithm for solving boundary value problems. We introduce an orthonormal basis generated from compressed Legendre polynomials. This basis can avoid Runge phenomenon caused by high-order polynomial approximation. Based on the new basis, a numerical algorithm of two-point boundary value problems is established. The convergence and stability of the method are proved. The whole analysis is also applicable to higher order equations or equations with more complex boundary conditions. Four numerical examples are tested to illustrate the accuracy and efficiency of the algorithm. The results show that our algorithm have higher accuracy for solving linear and nonlinear problems.

Aiming at the problem of solving nonlinear ordinary differential equations with variable coefficients, this paper introduces the elastic transformation method into the process of solving ordinary differential equations for the first time. A class of first-order and a class of third-order ordinary differential equations with variable coefficients can be transformed into the Laguerre equation through elastic transformation. With the help of the general solution of the Laguerre equation, the general solution of these two classes of ordinary differential equations can be obtained, and then the curves of the general solution can be drawn. This method not only expands the solvable classes of ordinary differential equations, but also provides a new idea for solving ordinary differential equations with variable coefficients.

In this paper, we introduce a generalized fractional operator in the setting of Hilfer fractional derivatives, the $ \psi $-Hilfer generalized proportional fractional derivative of a function with respect to another function. The proposed operator can be viewed as an interpolator between the Riemann-Liouville and Caputo generalized proportional fractional operators. The properties of the proposed operator are established under some classical and standard assumptions. As an application, we formulate a nonlinear fractional differential equation with a nonlocal initial condition and investigate its equivalence with Volterra integral equations, existence, and uniqueness of solutions. Finally, illustrative examples are given to demonstrate the theoretical results.

In this paper, we present new Lyapunov-type inequalities for Hilfer-Katugampola fractional differential equations. We first give some unique properties of the Hilfer-Katugampola fractional derivative, and then by using these new properties we convert the multi-point boundary value problems of Hilfer-Katugampola fractional differential equations into the equivalent integral equations with corresponding Green's functions, respectively. Finally, we make use of the Banach's contraction principle to derive the desired results, and give a series of corollaries to show that the current results extend and enrich the previous results in the literature.

In this paper, we study a boundary value problem consisting of Hahn integro-difference equation supplemented with four-point fractional Hahn integral boundary conditions. The novelty of this problem lies in the fact that it contains two fractional Hahn difference operators and three fractional Hahn integrals with different quantum numbers and orders. Firstly, we convert the given nonlinear problem into a fixed point problem, by considering a linear variant of the problem at hand. Once the fixed point operator is available, we make use the classical Banach's and Schauder's fixed point theorems to establish existence and uniqueness results. An example is also constructed to illustrate the main results. Several properties of fractional Hahn integral that will be used in our study are also discussed.

This paper considers a class of nonlinear implicit Hadamard fractional differential equations with impulses. By using Banach's contraction mapping principle, we establish some sufficient criteria to ensure the existence and uniqueness of solution. Furthermore, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of this system are obtained by applying nonlinear functional analysis technique. As applications, an interesting example is provided to illustrate the effectiveness of main results.

In this paper, a flexible version of the Fréchet distribution called the extended Weibull–Fréchet (EWFr) distribution is proposed. Its failure rate has a decreasing shape, an increasing shape, and an upside-down bathtub shape. Its density function can be a symmetric shape, an asymmetric shape, a reversed-J shape and J shape. Some mathematical properties of the EWFr distribution are explored. The EWFr parameters are estimated using several frequentist estimation approaches. The performance of these methods is addressed using detailed simulations. Furthermore, the best approach for estimating the EWFr parameters is determined based on partial and overall ranks. Finally, the performance of the EWFr distribution is studied using two real-life datasets from the medicine and engineering sciences. The EWFr distribution provides a superior fit over other competing Fréchet distributions such as the exponentiated-Fréchet, beta-Fréchet, Lomax–Fréchet, and Kumaraswamy Marshall–Olkin Fréchet.

In this paper, a new modified hybrid explicit group (MHEG) iterative method is presented for the efficient and accurate numerical solution of a time-fractional diffusion equation in two space dimensions. The time fractional derivative is defined in the Caputo sense. In the proposed method, a Laplace transformation is used in the temporal domain, and, for the spatial discretization, a new finite difference scheme based on grouping strategy is considered. The unique solvability, unconditional stability and convergence are thoroughly proved by the matrix analysis method. Comparison of numerical results with analytical and other approximate solutions indicates the viability and efficiency of the proposed algorithm.

In this paper, we consider stochastic approximation algorithms for least-square and logistic regression with no strong-convexity assumption on the convex loss functions. We develop two algorithms with varied step-size motivated by the accelerated gradient algorithm which is initiated for convex stochastic programming. We analyse the developed algorithms that achieve a rate of $ O(1/n^{2}) $ where $ n $ is the number of samples, which is tighter than the best convergence rate $ O(1/n) $ achieved so far on non-strongly-convex stochastic approximation with constant-step-size, for classic supervised learning problems. Our analysis is based on a non-asymptotic analysis of the empirical risk (in expectation) with less assumptions that existing analysis results. It does not require the finite-dimensionality assumption and the Lipschitz condition. We carry out controlled experiments on synthetic and some standard machine learning data sets. Empirical results justify our theoretical analysis and show a faster convergence rate than existing other methods.

In this present paper, the basic properties of an extended Mittag-Leffler function are studied. We present some fractional integral and differential formulas of an extended Mittag-Leffler function. In addition, we introduce a new extension of Prabhakar type fractional integrals with an extended Mittag-Leffler function in the kernel. Also, we present certain basic properties of the generalized Prabhakar type fractional integrals.

In this paper, the maximal and minimal iterative positive solutions are investigated for a singular Hadamard fractional differential equation boundary value problem with a boundary condition involving values at infinite number of points. Green's function is deduced and some properties of Green's function are given. Based upon these properties, iterative schemes are established for approximating the maximal and minimal positive solutions.

Differential operators based on convolution have been recognized as powerful mathematical operators able to depict and capture chaotic behaviors, especially those that are not able to be depicted using classical differential and integral operators. While these differential operators have being applied with great success in many fields of science, especially in the case of dynamical system, we have to confess that they were not able depict some chaotic behaviors, especially those with additionally similar patterns. To solve this issue new class of differential and integral operators were proposed and applied in few problems. In this paper, we aim to depict chaotic behavior using the newly defined differential and integral operators with fractional order and fractal dimension. Additionally we introduced a new chaotic operators with strange attractors. Several simulations have been conducted and illustrations of the results are provided to show the efficiency of the models.

In this paper, we construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations. Firstly, we use the piecewise Quadratic Lagrange interpolation method to construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations, and then analyze the local truncation error of the high order numerical scheme. Secondly, based on the local truncation error, the convergence order of $ 3-\theta $ order is obtained. And the convergence are strictly analyzed. Finally, the numerical simulation of the high order numerical scheme is carried out. Through the calculation of typical problems, the effectiveness of the numerical algorithm and the correctness of theoretical analysis are verified.

This paper proposes a local semi-analytical meshless method for simulating heat conduction in nonlinear functionally graded materials. The governing equation of heat conduction problem in nonlinear functionally graded material is first transformed to an anisotropic modified Helmholtz equation by using the Kirchhoff transformation. Then, the local knot method (LKM) is employed to approximate the solution of the transformed equation. After that, the solution of the original nonlinear equation can be obtained by the inverse Kirchhoff transformation. The LKM is a recently proposed meshless approach. As a local semi-analytical meshless approach, it uses the non-singular general solution as the basis function and has the merits of simplicity, high accuracy, and easy-to-program. Compared with the traditional boundary knot method, the present scheme avoids an ill-conditioned system of equations, and is more suitable for large-scale simulations associated with complicated structures. Three benchmark numerical examples are provided to confirm the accuracy and validity of the proposed approach.

In this paper, we aim to explore the oscillation of solutions for a class of second-order neutral functional differential equations. We propose new criteria to ensure that all obtained solutions are oscillatory. The obtained results can be used to develop and provide theoretical support for and further develop the oscillation study for a class of second-order neutral differential equations. Finally, an illustrated example is given to demonstrate the effectiveness of our new criteria.

The solvability of some $ p $-Laplace boundary value problems with Caputo fractional derivative are discussed. By using the fixed-point theory and analysis techniques, some existence results of one or three non-negative solutions are obtained. Two examples showed that the conditions used in this paper are somewhat easy to check.

In this paper, we derive the escape criteria for general complex polynomial $ f(x) = \sum_{i = 0}^{p}a_{i}x^{i} $ with $ p\geq2 $, where $ a_{i} \in \mathbb{C} $ for $ i = 0, 1, 2, \dots, p $ to generate the fractals. Moreover, we study the orbit of an implicit iteration (i.e., Jungck-Ishikawa iteration with $ s $-convexity) and develop algorithms for Mandelbrot set and Multi-corn or Multi-edge set. Moreover, we draw some complex graphs and observe how the graph of Mandelbrot set and Multi-corn or Multi-edge set vary with the variation of $ a_{i} $'s.

The generalized time fractional Fisher equation is one of the significant models to describe the dynamics of the system. The study of effective numerical techniques for the equation has important scientific significance and application value. Based on the alternating technique, this article combines the classical explicit difference scheme and the implicit difference scheme to construct a class of explicit implicit alternating difference schemes for the generalized time fractional Fisher equation. The unconditional stability and convergence with order $ O\left({\tau }^{2-\alpha }+{h}^{2}\right) $ of the proposed schemes are analyzed. Numerical examples are performed to verify the theoretical analysis. Compared with the classical implicit difference scheme, the calculation cost of the explicit implicit alternating difference schemes is reduced by almost $ 60 $%. Numerical experiments show that the explicit implicit alternating difference schemes are also suitable for solving the time fractional Fisher equation with initial weak singularity and have an accuracy of order $ O\left({\tau }^{\alpha }+{h}^{2}\right) $, which verify that the methods proposed in this paper are efficient for solving the generalized time fractional Fisher equation.

In this paper, we consider numerical methods for the linear complementarity problem (LCP). By introducing a positive diagonal parameter matrix, the LCP is transformed into an equivalent fixed-point equation and the equivalence is proved. Based on such equation, the general fixed-point (GFP) method with two cases are proposed and analyzed when the system matrix is a $ P $-matrix. In addition, we provide several concrete sufficient conditions for the proposed method when the system matrix is a symmetric positive definite matrix or an $ H_{+} $-matrix. Meanwhile, we discuss the optimal case for the proposed method. The numerical experiments show that the GFP method is effective and practical.

The purpose of this paper is to establish a general stability result for a one-dimensional linear swelling porous-elastic system with past history, irrespective of the wave speeds of the system. First, we establish an explicit and general decay result under a wider class of the relaxation (kernel) functions. The kernel in our memory term is more general and of a broader class. Further, we get a better decay rate without imposing some assumptions on the boundedness of the history data considered in many earlier results in the literature. We also perform several numerical tests to illustrate our theoretical results. Our output extends and improves some of the available results on swelling porous media in the literature.

In this paper, two-person zero-sum Markov games with Borel state space and action space, unbounded reward function and state-dependent discount factors are studied. The optimal criterion is expected discount criterion. Firstly, sufficient conditions for the existence of optimal policies are given for the two-person zero-sum Markov games with varying discount factors. Then, the existence of optimal policies is proved by Banach fixed point theorem. Finally, we give an example for reservoir operations to illustrate the existence results.

In this paper, a delayed predator-prey system with additional food and asymmetric functional response is investigated. We discuss the local stability of equilibria and the existence of local Hopf bifurcation under the influence of the time delay. By using the normal form theory and center manifold theorem, the explicit formulas which determine the properties of bifurcating periodic solutions are obtained. Further, we prove that global periodic solutions exist after the second critical value of delay via Wu's theory. Finally, the correctness of the previous theoretical analysis is demonstrated by some numerical cases.

This article deals with a class of variational inequalities known as absolute value variational inequalities. Some new merit functions for the absolute value variational inequalities are established. Using these merit functions, we derive the error bounds for absolute value variational inequalities. Since absolute value variational inequalities contain variational inequalities, absolute value complementarity problem and system of absolute value equations as special cases, the findings presented here recapture various known results in the related domains. The conclusions of this paper are more comprehensive and may provoke futuristic research.

In this paper, we consider a market model where the risky asset is a jump diffusion whose drift, volatility and jump coefficients are influenced by market regimes and history of the asset itself. Since the trajectory of the risky asset is discontinuous, we modify the delay variable so that it remains defined in this discontinuous setting. Instead of the actual path history of the risky asset, we consider the continuous approximation of its trajectory. With this modification, the delay variable, which is a sliding average of past values of the risky asset, no longer breaks down. We then use the resulting stochastic process in formulating the state variable of a portfolio optimization problem. In this formulation, we obtain the dynamic programming principle and Hamilton Jacobi Bellman equation. We also provide a verification theorem to guarantee the optimal solution of the corresponding stochastic optimization problem. We solve the resulting finite time horizon control problem and show that close form solutions of the stochastic optimization problem exist for the cases of power and logarithmic utility functions. In particular, we show that the HJB equation for the power utility function is a first order linear partial differential equation while that of the logarithmic utility function is a linear ordinary differential equation.

In this paper, we devoted to deal with the rotational periodic problem of some fractional iterative systems in the sense of Caputo fractional derivative. Under one sided-Lipschtiz condition on nonlinear term, the existence and uniqueness of solution for a fractional iterative equation is proved by applying the Leray-Schauder fixed point theorem and topological degree theory. Furthermore, the well posedness for a nonlinear control system with iteration term and a multivalued disturbance is established by using set-valued theory. The existence of solutions for a iterative neural network system is demonstrated at the end.

In this paper we review the applications of fractional differential equation in economic growth models. This includes the theories about linear and nonlinear fractional differential equation, including the Fractional Riccati Differential Equation (FRDE) and its applications in economic growth models with memory effect. The method used in this study is by comparing related literatures and evaluate them comprehensively. The results of this study are the chronological order of the applications of the Fractional Differential Equation (FDE) in economic growth models and the development on theories of the FDE solutions, including the FRDE forms of economic growth models. This study also provides a comparative analysis on solutions of linear and nonlinear FDE, and approximate solution of economic growth models involving memory effects using various methods. The main contribution of this research is the chonological development of the theory to find necessary and sufficient conditions to guarantee the existence and uniqueness of the FDE in economic growth and the methods to obtain the solution. Some remarks on how further researches can be done are also presented as a general conclusion.

This paper proposes a new definition of the nonlinear Fredholm integro-differential equation of the second kind with continuous kernel in two-dimensional (NT-DFIDE). Furthermore, the work is concerned to study this new equation numerically. The existence of a unique solution of the equation is proved. In addition, the approximate solutions of NT-DFIDE are obtained by two powerful methods Adomian Decomposition Method (ADM) and Homotopy Analysis Method (HAM). The given numerical examples showed the efficiency and accuracy of the introduced methods.

The conjugate gradient (CG) method is a method to solve unconstrained optimization problems. Moreover CG method can be applied in medical science, industry, neural network, and many others. In this paper a new three term CG method is proposed. The new CG formula is constructed based on DL and WYL CG formulas to be non-negative and inherits the properties of HS formula. The new modification satisfies the convergence properties and the sufficient descent property. The numerical results show that the new modification is more efficient than DL, WYL, and CG-Descent formulas. We use more than 200 functions from CUTEst library to compare the results between these methods in term of number of iterations, function evaluations, gradient evaluations, and CPU time.

In this paper, we investigate blow-up conditions for the semilinear generalized Tricomi equation with a general nonlinear memory term in $ \mathbb{R}^n $ by using suitable functionals and employing iteration procedures. Particularly, a new combined effect from the relaxation function and the time-dependent coefficient is found.

In this paper, we study a three step iterative scheme to estimate fixed points of non-expansive mappings in the framework of Banach spaces. Further, some convergence results are proved for such mappings. A nontrivial numerical example is presented to verify our assertions and main results. Finally, we approximate the solution of a boundary value problem of second order differential equation.

In this paper, a new derivative-free approach for solving nonlinear monotone system of equations with convex constraints is proposed. The search direction of the proposed algorithm is derived based on the modified scaled Davidon-Fletcher-Powell (DFP) updating formula in such a way that it is sufficiently descent. Under some mild assumptions, the search direction is shown to be bounded. Subsequently, the convergence result of the proposed method is established. The performance of the proposed algorithm on a collection of some test problems as well as signal recovery problems is demonstrated in comparison with some existing algorithms with similar characteristics. The results of the numerical experiments confirm the efficiency as well as the robustness of the proposed algorithm by comparing it with some existing methods in the literature.

This paper deals with some existence of random solutions and the Ulam stability for a class of Katugampola random fractional differential equations in Banach spaces. A random fixed point theorem is used for the existence of random solutions, and we prove that our problem is generalized Ulam-Hyers-Rassias stable. An illustrative example is presented in the last section.

This paper deals with the attractor problem of Hopfield neural networks with multiple time-varying delays. The mathematical expression of the networks cannot be expressed in the vector-matrix form due to the existence of the multiple delays, which leads to the existence condition of the attractor cannot be easily established by linear matrix inequality approach. We try to derive the existence conditions of the linear matrix inequality form of pullback attractor by employing Lyapunov-Krasovskii functional and inequality techniques. Two examples are given to demonstrate the effectiveness of our theoretical results and illustrate the conditions of the linear matrix inequality form are better than those of the algebraic form.

In this paper, a mathematical model of generalized fractional-order is constructed to study the problem of human immunodeficiency virus (HIV) infection of CD$ 4^+ $ T-cells with treatment. The model consists of a system of four nonlinear differential equations under the generalized Caputo fractional derivative sense. The existence results for the fractional-order HIV model are investigated via Banach's and Leray-Schauder nonlinear alternative fixed point theorems. Further, we also established different types of Ulam's stability results for the proposed model. The effective numerical scheme so-called predictor-corrector algorithm has been employed to analyze the approximated solution and dynamical behavior of the model under consideration. It is worth noting that, unlike many discusses recently conducted, dimensional consistency has been taken into account during the fractionalization process of the classical model.

In this paper, we establish some new Hermite-Hadamard type inequalities for differential exponential type convex functions and discuss several special cases. Moreover, in order to give the efficient of our main results, some applications for special means and error estimations are obtain.

Many problems in engineering and social sciences can be transformed into system of nonlinear equations. As a result, a lot of methods have been proposed for solving the system. Some of the classical methods include Newton and Quasi Newton methods which have rapid convergence from good initial points but unable to deal with large scale problems due to the computation of Jacobian matrix or its approximation. Spectral and conjugate gradient methods proposed for unconstrained optimization, and later on extended to solve nonlinear equations do not require any computation of Jacobian matrix or its approximation, thus, are suitable to handle large scale problems. In this paper, we proposed a spectral conjugate gradient algorithm for solving system of nonlinear equations where the operator under consideration is monotone. The search direction of the proposed algorithm is constructed by taking the convex combination of the Dai-Yuan (DY) parameter and a modified conjugate descent (CD) parameter. The proposed search direction is sufficiently descent and under some suitable assumptions, the global convergence of the proposed algorithm is proved. Numerical experiments on some test problems are presented to show the efficiency of the proposed algorithm in comparison with an existing one. Finally, the algorithm is successfully applied in signal recovery problem arising from compressive sensing.

In this paper, we consider the statistical inferences for varying coefficient partially nonlinear model with missing responses. Firstly, we employ the profile nonlinear least squares estimation based on the weighted imputation method to estimate the unknown parameter and the nonparametric function, meanwhile the asymptotic normality of the resulting estimators is proved. Secondly, we consider empirical likelihood inferences based on the weighted imputation method for the unknown parameter and nonparametric function, and propose an empirical log-likelihood ratio function for the unknown parameter vector in the nonlinear function and a residual-adjusted empirical log-likelihood ratio function for the nonparametric component, meanwhile construct relevant confidence regions. Thirdly, the response mean estimation is also studied. In addition, simulation studies are conducted to examine the finite sample performance of our methods, and the empirical likelihood approach based on the weighted imputation method (IEL) is further applied to a real data example.

In this paper, we propose a new framework of weighted generalized proportional fractional integral operator with respect to a monotone function $ \Psi, $ we develop novel modifications of the aforesaid operator. Moreover, contemplating the so-called operator, we determine several notable weighted Chebyshev and Gruss type inequalities with respect to increasing, positive and monotone functions $ \Psi $ by employing traditional and forthright inequalities. Furthermore, we demonstrate the applications of the new operator with numerous integral inequalities by inducing assumptions on $ \omega $ and $ \Psi $ verified the superiority of the suggested scheme in terms of efficiency. Additionally, our consequences have a potential association with the previous results. The computations of the proposed scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.

The present article is a continuation of the work done by Aral and Erbay [ 1 ]. We discuss the rate of convergence of the generalized Baskakov operators considered in the above paper with the aid of the second order modulus of continuity and the unified Ditzian Totik modulus of smoothness. A bivariate case of these operators is also defined and the degree of approximation by means of the partial and total moduli of continuity and the Peetre's K-functional is studied. A Voronovskaya type asymptotic result is also established. Further, we construct the associated Generalized Boolean Sum (GBS) operators and investigate the order of convergence with the help of mixed modulus of smoothness for the Bogel continuous and Bogel differentiable functions. Some numerical results to illustrate the convergence of the above generalized Baskakov operators and its comparison with the GBS operators are also given using Matlab algorithm.

In this paper, we propose a novel criterion on the partial synchronization in a generalized linearly coupled network by employing Lyapunov stability theory and linear matrix inequality. The obtained criterion is only dependent on intra-community connections, and the information of inter-community connections is not necessary. Therefore, it provides more convenience in reducing network sizes in practice. Compared with the previous classical criterion, the threshold derived from the obtained criterion is no less than the classical threshold. We give some particular cases in which the obtained threshold is equal to the classical threshold. Finally, we show numerical simulations to verify the validity of the proposed criteria and comparisons.

In this paper, we propose a three step iteration process and analyze the performance of the process for a contractive-like operators. It is observed that this iterative procedure is faster than several iterative methods in the existing literature. To support the claim, a numerical example is presented using Maple 13. Some images are generated by using this iteration method for complex cubic polynomials. We believe that our presented work enrich the polynomiography software.

In this paper, we discuss existence and stability results for a new class of impulsive fractional boundary value problems with non-separated boundary conditions containing the Caputo proportional fractional derivative of a function with respect to another function. The uniqueness result is discussed via Banach's contraction mapping principle, and the existence of solutions is proved by using Schaefer's fixed point theorem. Furthermore, we utilize the theory of stability for presenting different kinds of Ulam's stability results of the proposed problem. Finally, an example is also constructed to demonstrate the application of the main results.

In this paper, we investigate stochastic comparisons of the second largest order statistics of homogeneous samples coupled by Archimedean copula, and we establish the reversed hazard rate and likelihood ratio orders, and we further generalize the corresponding results to the case of random sample size. Also, we derive some results for relative ageing between parallel systems and $ 2 $-out-of-$ n $/$ 2 $-out-of-$ (n+1) $ systems. Finally, some examples are given to illustrate the obtained results.

This paper studies Langevin equation with nonlocal boundary conditions involving a $ \psi $-Caputo fractional operators of different orders. By the aid of fixed point techniques of Krasnoselskii and Banach, we derive new results on existence and uniqueness of the problem at hand. Further, a new $ \psi $-fractional Gronwall inequality and $ \psi $-fractional integration by parts are employed to prove Ulam-Hyers and Ulam-Hyers-Rassias stability for the solutions. Examples are provided to demonstrate the advantage of our major results. The proposed results here are more general than the existing results in the literature which can be obtained as particular cases.