Journal of Mathematical Modeling
https://jmm.guilan.ac.ir/
Journal of Mathematical Modelingendaily1Mon, 01 Jul 2024 00:00:00 +0330Mon, 01 Jul 2024 00:00:00 +0330Stochastic dynamics of Izhikevich-Fitzhugh neuron model
https://jmm.guilan.ac.ir/article_7436.html
This paper is concerned with stochastic stability and stochastic bifurcation of the Fitzhug-Nagumo model with multiplicative white noise. We employ largest Lyapunov exponent and singular boundary theory to investigate local and global stochastic stability at the equilibrium point. In the rest, the solution of averaging the Ito diffusion equation and extreme point of steady-state probability density function provide sufficient conditions that the stochastic system undergoes pitchfork and phenomenological bifurcations. These theoretical results of the stochastic neuroscience model are confirmed by some numerical simulations and stochastic trajectories. Finally, we compare this approach with Rulkov approach and explain how pitchfork and phenomenological bifurcations describe spiking limit cycles and stability of neuron's resting state.Targeted drug delivery in multi-layer capsules: an analytical and numerical study
https://jmm.guilan.ac.ir/article_7657.html
Recently, polymeric multi-layer capsules have gained a great deal of attention from the life science community. Furthermore, myriad &nbsp;interesting systems have appeared in the literature with biodegradable components and biospecific functionalities. In the present work, we presented a mathematical model of drug release from a multi-layer capsule into a target tissue. The diffusion problem was described by a system of coupled partial differential equations, Fickian and non-Fickian, which we solved numerically via nonuniform finite differences method. Energy estimates were further established for the coupled system and also, the convergence properties of the proposed numerical method were justified. We ultimately demonstrated the qualitative behavior of the system.A novel fitted numerical scheme for time-fractional singularly perturbed convection-diffusion problems with a delay in time via cubic $B$-spline approach
https://jmm.guilan.ac.ir/article_7441.html
This paper presents a uniformly convergent numerical scheme for time-fractional &nbsp;singularly perturbed convection-diffusion problem with delay in time. The time-fractional derivative is considered in the Caputo sense and treated using the implicit Euler method. Then, a uniformly convergent numerical scheme based on cubic $B$-spline method is developed along the spatial direction. The technique is proved rigorously for parameter-uniform convergence. By a numerical experimentation, it is also validated that the computational result agrees with the theoretical expectation and it is also more accurate than some existing numerical methods.An efficient numerical method based on cubic B--splines for the time--fractional Black--Scholes European option pricing model
https://jmm.guilan.ac.ir/article_7677.html
In this study, we develop a precise and effective numerical approach to solve the time--fractional Black--Scholes equation, which is used to calculate European options. The method employs cubic B-spline collocation for spatial discretization and a finite difference method for time discretization. An &nbsp;analysis of the method's stability is conducted. Finally, two numerical examples are included to show the effectiveness and applicability of the suggested method.A compact discretization of the boundary value problems of the nonlinear Fredholm integro-differential equations
https://jmm.guilan.ac.ir/article_7445.html
In this paper, we propose a &nbsp;fourth-order compact discretization method &nbsp; for solving a second-order boundary value problem governed by the nonlinear Fredholm integro-differential equations. &nbsp;Using an efficient approximate polynomial, &nbsp;a &nbsp;compact numerical integration method is first designed. Then by applying the derived numerical integration formulas, the original problem is converted into a nonlinear system of algebraic equations. &nbsp;It is shown that the proposed method is easy to implement and has the third order of accuracy in the infinity norm. Some &nbsp;numerical examples are presented to demonstrate its &nbsp;approximation accuracy and computational efficiency, &nbsp; as well as to compare the derived results with those &nbsp;obtained in the literature.Dynamics and bifurcations of a discrete-time neural network model with a single delay
https://jmm.guilan.ac.ir/article_7678.html
In the present study, we analyze dynamics and bifurcations of a discrete-time &nbsp; &nbsp;Hopfield neural network &nbsp;based on two neurons and the same time delay. We determine stability and bifurcations of the system consisting flip, pitchfork and Neimark-Sacker &nbsp;bifurcations. The normal form coefficients for the all bifurcations are calculated using reducing to the corresponding &nbsp;center manifold, then these coefficients are &nbsp;numerically obtained using MatContM. Numerical analysis validates our analytical results and reveals more complex dynamical behaviors.Computational treatment of a convection-diffusion type nonlinear system of singularly perturbed differential equations
https://jmm.guilan.ac.ir/article_7495.html
In this article, a nonlinear system of singularly perturbed differential equations of convection-diffusion type with Dirichlet boundary conditions is considered on the interval $[0,1].$ Both components of the solution of the system exhibit boundary layers near $t = 0.$ A new computational method involving classical finite difference operators, a piecewise-uniform Shishkin mesh and an algorithm based on the continuation method is developed to compute the numerical approximations. The computational method is proved to be first order convergent uniformly with respect to the perturbation parameters. &nbsp;Numerical experiments &nbsp;support the theoretical results.Application of compact local integrated RBFs technique to solve fourth-order time-fractional diffusion-wave system
https://jmm.guilan.ac.ir/article_7679.html
The main aim of the current paper is to apply the compact local integrated RBFs technique to the numerical solution of the fourth-order time-fractional diffusion-wave system. A finite difference formula is employed to obtain a time-discrete scheme. The stability and convergence rate of the semi-discrete plan are proved by the energy method. A new unknown variable is defined to obtain a second-order &nbsp; system of PDEs. Then, the compact local integrated radial basis functions (RBFs) is used to approximate the spatial derivative. The utilized numerical method is a truly meshless technique. &nbsp;The numerical approach put forth is genuinely meshless, allowing for the utilization of irregular physical domains in obtaining numerical solutions.Complexity analysis of primal-dual interior-point methods for convex quadratic programming based on a new twice parameterized kernel function
https://jmm.guilan.ac.ir/article_7543.html
In this paper, we present primal-dual interior-point methods (IPMs) for convex quadratic programming (CQP) based on a new twice parameterized kernel function (KF) with a hyperbolic barrier term. &nbsp;To our knowledge, this is the first KF with a twice parameterized hyperbolic barrier term. By using some conditions and simple analysis, we derive the currently best-known iteration bounds for large- and small-update methods, namely, $\textbf{O}\big(\sqrt{n}\log n\log\frac{n}{\epsilon}\big)$ and $\textbf{O}\big(\sqrt{n}\log\frac{n}{\epsilon}\big)$, respectively, with &nbsp;special choices of the parameters. Finally, some numerical results regarding the practical performance of the new proposed KF are reported.On differential-integral optimal control problems
https://jmm.guilan.ac.ir/article_7705.html
In this paper, we will study the &nbsp;optimal control problem of a system containing a differential integral (D-I) operator. We will deduce the necessary optimality conditions and apply it first to the problem of minimum energy to find the lowest energy for an electrical circuit containing a resistor, a coil and a capacitor (RLC circuit), and second to the problem of the minimum time to transfer electrical current in &nbsp;RLC circuit from one state to another in the shortest possible time.&nbsp;On the blow up of solutions for hyperbolic equation involving the fractional Laplacian with source terms
https://jmm.guilan.ac.ir/article_7544.html
In this paper, we study the blow-up of solutions for hyperbolic equations involving the fractional Laplacian operator with damping and source terms. &nbsp;We obtain the global existence results. Then, we observe the blow-up of solutions using the concavity method. Finally, we present some numerical simulation results.Introducing three new smoothing functions: Analysis on smoothing-Newton algorithms
https://jmm.guilan.ac.ir/article_7706.html
In this paper, we focus on solving the system of absolute value equations (AVE), which is one of the most popular classes of nonlinear equations. First, a new smoothing technique with three different smoothing functions is introduced, and the AVE is transformed into a family of parametrized smooth equations with the help of these smoothing functions. Then, a smoothing Newton-type algorithm with hybridized inexact line search is developed based on the proposed smoothing technique. The numerical experiments have been carried out on some well-known and randomly generated test problems, and the results are analyzed in terms of line search techniques. The numerical results show that the proposed hybrid approach is more efficient than the other algorithms.Tau algorithm for fractional delay differential equations utilizing seventh-kind Chebyshev polynomials
https://jmm.guilan.ac.ir/article_7562.html
Herein, we present an algorithm for handling fractional delay differential equations (FDDEs). Chebyshev polynomials (CPs) class of the seventh kind is a subclass of the generalized Gegenbauer (ultraspherical) polynomials. The members of this class make up the basis functions in this paper. Our suggested numerical algorithm is derived using new theoretical findings about these polynomials and their shifted counterparts. We will use the Tau method to convert the FDDE with the governing conditions into a linear algebraic system, which can then be solved numerically using a suitable procedure. We will give a detailed discussion of the convergence and error analysis of the shifted Chebyshev expansion. Lastly, some numerical examples are provided to verify the precision and applicability of the proposed strategy.Tensor splitting preconditioners for multilinear systems
https://jmm.guilan.ac.ir/article_7737.html
In this paper, we propose some new preconditioners &nbsp;for solving multilinear system $\mathcal{A}\mathbf{x}^{m-1}=\mathbf{b}$. These preconditioners are based on tensor splitting. We also present some theorems for analyzing and convergence of the preconditioned &nbsp;Jacobi-, Gauss-Seidel-, and SOR-type iterative methods. Numerical examples are presented to verify the efficiency of the proposed preconditioned methods.A hybrid CG algorithm for nonlinear unconstrained optimization with application in image restoration
https://jmm.guilan.ac.ir/article_7567.html
This paper presents a new hybrid conjugate gradient method for solving&nbsp; nonlinear unconstrained optimization problems; it is based on a combination of $RMIL$&nbsp; (Rivaie-Mustafa-Ismail-Leong)&nbsp; and $hSM$&nbsp; (hybrid Sulaiman- Mohammed) methods. The proposed algorithm enjoys the sufficient descent condition without depending on any line search; moreover, it is globally convergent under the usual and strong Wolfe line search assumptions. &nbsp;The performance of the algorithm is demonstrated through numerical experiments on a set of 100 test functions from [1] and four image restoration problems with two noise levels. The numerical comparisons with four existing methods show that the proposed method is promising and effective.A uniformly convergent numerical scheme for singularly perturbed parabolic turning point problem
https://jmm.guilan.ac.ir/article_7789.html
A uniformly convergent numerical scheme is developed for solving a singularly perturbed parabolic turning point problem. The properties of continuous solutions and the bounds of the derivatives are discussed. Due to the presence of a small parameter as a multiple of the diffusion coefficient, it causes computational difficulty when applying classical numerical methods. As a result, the scheme is formulated using the Crank-Nicolson method in the temporal discretization and an exponentially fitted finite difference method in the space on a uniform mesh. The existence of a unique discrete solution is guaranteed by the comparison principle. The stability and convergence analysis of the method are investigated. Two numerical examples are considered to validate the applicability of the scheme. The numerical results are displayed in tables and graphs to support the theoretical findings. The scheme converges uniformly with order one in space and two in time.A nonautonomous delayed viscoelastic wave equation with a linear damping: well-posedness and exponential stability
https://jmm.guilan.ac.ir/article_7591.html
In this paper, we consider a nonautonomous viscoelastic wave equation with linear damping and delayed terms. Under some appropriate assumptions, we prove the global existence using the semi-group theory. Furthermore, for a small enough coefficient of delay, we obtained a stability result via a suitable Lyapunov function where the kernel function decays exponentially.Lions's partial derivatives with respect to probability measures for general mean-field stochastic control problem
https://jmm.guilan.ac.ir/article_7804.html
In this paper, a necessary stochastic maximum principle for stochastic model governed by mean-field nonlinear controlled It$\rm{\ddot{o}}$-stochastic differential equations is proved. The coefficients of our model are nonlinear and depend explicitly on the control variable, the state process as well as of its probability distribution. The control region is assumed to be bounded and convex. Our main result is derived by applying the Lions's partial-derivatives with respect to random measures in Wasserstein space. The associated It$\rm{\ddot{o}}$-formula and convex-variation approach are applied to establish the optimal control.Radial polynomials as alternatives to flat radial basis functions
https://jmm.guilan.ac.ir/article_7592.html
Due to the high approximation power and simplicity of computation of smooth radial basis functions (RBFs), in recent decades they have received much attention for function approximation. These RBFs contain a shape parameter that regulates their approximation power and stability but its optimal selection is challenging. To avoid this difficulty, this paper follows a novel and computationally efficient strategy to propose a space of radial polynomials with even degree that well approximates flat RBFs. The proposed space, $\mathcal{H}_n$, is the shifted radial polynomials of degree $2n$. By obtaining the dimension of $\mathcal{H}_n$ and introducing a basis set, it is shown that $\mathcal{H}_n$ is considerably smaller than $\mathcal{P}_{2n}$ while the distances from RBFs to both $\mathcal{H}_n$ and $\mathcal{P}_{2n}$ are nearly equal. For computation, by introducing new basis functions, two computationally efficient approaches are proposed. Finally, the presented theoretical studies are verified by the numerical results.Unconditionally stable finite element method for the variable-order fractional Schrödinger equation with Mittag-Leffler kernel
https://jmm.guilan.ac.ir/article_7805.html
The Schr&ouml;dinger equation with variable-order fractional operator is a challenging problem to be solved numerically. In this study, an implicit fully discrete continuous Galerkin finite element method is developed to tackle this equation while the fractional operator is expressed with a nonsingular Mittag-Leffler kernel. To begin with, the finite difference scheme known as the L1 formula is employed to discretize the temporal term. Next, the continuous Galerkin method is used for spatial discretization. This combination ensures accuracy and stability of the numerical approximation. Our next step is to conduct a stability and error analysis of the proposed scheme. Finally, some numerical results are carried out to validate the theoretical analysis.Numerical treatment for a multiscale nonlinear system of singularly perturbed differential equations of convection-diffusion type
https://jmm.guilan.ac.ir/article_7593.html
In this article, a multiscale nonlinear system of singularly perturbed differential equations of convection-diffusion type is considered. A numerical technique combined with the continuation method is constructed to obtain the numerical computations. The newly developed numerical method is shown to be first order convergent uniformly with respect to the perturbation parameter.Stochastic permanence and extinction of a hybrid predator-prey system with jumps
https://jmm.guilan.ac.ir/article_7832.html
This paper concerns the dynamics of a stochastic Holling-type II predator-prey system with Markovian switching and L{e}vy noise. First, the existence and uniqueness of global positive solution to the system with the given initial value is proved.Then, sufficient conditions for extinction and stochastic permanence of the system are obtained. Finally, an example and its numerical simulations are given to support the theoretical results.Robust exponential concurrent learning adaptive control for systems preceded by dead-zone input nonlinearity
https://jmm.guilan.ac.ir/article_7654.html
A concurrent learning (CL) adaptive control is proposed for a class of nonlinear systems in the presence of dead-zone input nonlinearity to guarantee the exponential convergence of the tracking and the parameter estimation errors. The proposed method enriches and encompasses the conventional filtering-based CL by proposing robust and optimal terms. The optimal term is designed by solving a suitable quadratic programming optimization problem based on control Lyapunov function theory which also meets the need for prescribed control bounds. A suitable robust term is proposed to tackle the presence of the dead-zone input nonlinearity. Recent methods of adaptive CL tune the control parameters using trial and error, which is a tedious task. In this paper, by some analysis and proposing two nonlinear optimization problems, the values of the control parameters are derived. The nonlinear optimization problems are solved using the time-varying iteration particle swarm optimization algorithm. The simulation results indicate the effectiveness of the proposed method.Evaluating cost efficiency of decision-making units in an uncertain environment
https://jmm.guilan.ac.ir/article_7867.html
The efficiency evaluation of organizational units provides managers with a perspective on the current state of the organization and solutions for their improvement. One of the methods of organizational evaluation is to determine the organization's minimum cost or cost efficiency. Cost efficiency in practice can be calculated when the input prices are available. In traditional models of cost efficiency, input and output data are crisp. However, there are situations where input and/or output may be imprecise. For such cases, experts are invited to model their opinion. Then uncertainty theory can be applied which is introduced by Liu as a mathematical branch rationally dealing with belief degrees. In this paper, a model is proposed to estimate the cost of decision-making units in the uncertain environment, where inputs and outputs are uncertain but the input prices are crisp. Several theorems are presented to discuss some features of the introduced model. When the data has a linear distribution, the cost efficiencies of the decision-making units are calculated. Also, the model is implemented on two numerical examples. The obtained results are compared with previous results. Finally, in the presence of input prices, a different cost efficiency score for the decision-making units is obtained. The proposed model helps decision-makers to improve their performance by using experts' opinions.A new approach to numerical solution of the time-fractional KdV-Burgers equations using least squares support vector regression
https://jmm.guilan.ac.ir/article_7884.html
The evolution of the waves on shallow water surfaces is described by a mathematical model given by nonlinear KdV and KdV-Burgers equations. These equations have many other applications and have been simulated by classical numerical methods in recent decades. In this paper, we develop a machine learning algorithm for the time-fractional KdV-Burgers equations. The proposed method implements a linearization of the problem and a time reduction by a Crank-Nicolson scheme. The least squares support vector regression (LS-SVR) is proposed to seek the approximate solution in a finite-dimensional polynomial kernel space. The Bernstein polynomials are used as the kernel of the proposed algorithm to handle the homogeneous boundary conditions easily in the framework of the Petrov-Galerkin spectral method. The proposed LS-SVR implements the orthogonal system of Bernstein-dual polynomials in the learning process, which gives quadratic programming in the primal form and provides a linear system of equations in dual variables with sparse positive definite matrices. It is shown that the involving mass and stiffness matrices are sparse. Some new theorems for the introduced basis are provided. Also, numerical results are presented to support the spectral convergence and accuracy of the method.&nbsp;Multi--objective model for architecture optimization and training of radial basis function neural networks
https://jmm.guilan.ac.ir/article_7891.html
Radial Basis Function Neural Network (RBFNN) is a type of artificial neural networks used for supervised learning. They rely on radial basis functions (RBFs), nonlinear mathematical functions employed to approximate complex nonlinear data. Determining the architecture of the network is challenging, impacting the achievement of optimal learning and generalization capacities. This paper presents a multi--objective model for optimizing and training RBFNN architecture. The model aims to fulfill three objectives: the first is the summation of distances between the input vector and the corresponding center for the neurons in the hidden layer. The second objective is the global error of the RBFNN, defined as the discrepancy between the calculated output and the desired output. The third objective is the complexity of the RBFNN, quantified by the number of neurons in the hidden layer. This innovative approach utilizes multiple objective simulated annealing to identify optimal parameters and hyperparameters for neural networks. The numerical results provide accuracy and reliability of the theoretical results discussed in this paper, as well as advantages of the proposed approach.Alternative views on fuzzy numbers and their application to fuzzy differential equations
https://jmm.guilan.ac.ir/article_7893.html
&nbsp;We consider fuzzy valued functions from two parametric representations of $\alpha$-level sets. New concepts are introduced and compared with available notions. Following the two proposed approaches, we study fuzzy differential equations. Their relation with Zadeh's extension principle and the generalized Hukuhara derivative is discussed. Moreover,we prove existence and uniqueness theorems for fuzzy differential equations. Illustrative examples are given.New general integral transform on time scales
https://jmm.guilan.ac.ir/article_7894.html
In this paper, we introduce &nbsp;a single integral transform that &nbsp;defines all known time scales generalized integral transforms in the family of Laplace transform as the new general integral transform on time scales. As a result, a unified approach is developed for the use of integral transforms representing the family of Laplace transform for solving problems on continuous and discrete cases dynamics. The convergence &nbsp;conditions and some principal properties accompanying the convolution theorem are given. It is shown that all generalized integral transforms on time scales included in the family of the Laplace transform are special cases of a new general integral transform. The applicability of this transform is demonstrated by solving certain ordinary dynamic equations and integral equations.Numerical stability of discrete energy for a thermoelastic-Bresse system with second sound
https://jmm.guilan.ac.ir/article_7895.html
Our contribution consists of studying numerical methods based on finite element space and finite difference schema in time of the linear one-dimensional thermoelastic Bresse system with second sound. We establish some a priori error estimates, and present some numerical analysis results of discrete energy under different decay rate profiles. Moreover, we study the behaviors of discrete energy with respect to the system parameters and the initial data. Some numerical simulations will be given in order to validate the theoretical results.A numerical method based on the radial basis functions for solving nonlinear two-dimensional Volterra integral equations of the second kind on non-rectangular domains
https://jmm.guilan.ac.ir/article_7904.html
In this investigation, a numerical method for solving nonlinear two-dimensional Volterra integral equations is presented. This method uses radial basis functions (RBFs) constructed on scattered points as a basis in the discrete collocation method. Therefore, the method does not need any background mesh or cell structure of the domain. All the integrals that appear in this method are approximated by the composite Gauss-Legendre integration formula. This method transforms the source problem into a system of nonlinear algebraic equations. Error analysis is presented for this method. Finally, numerical examples are included to show the validity and efficiency of this technique.Nonstandard finite difference method for solving singularly perturbed time-fractional delay parabolic reaction-diffusion problems
https://jmm.guilan.ac.ir/article_7933.html
This work addresses the singularly perturbed time-fractional delay parabolic reaction-diffusion of initial boundary value problems. The temporal derivative&rsquo;s discretization is handled by the Caputo fractional derivative combined with the implicit Euler technique with uniform step size. It also utilizes the nonstandard finite difference approach for the spatial derivative. The scheme has been demonstrated to converge and has an accuracy of &nbsp;$O(h^{2}+(\Delta t^{2-\alpha}))$. To assess the suitability of the approach, two model examples are taken into consideration. The findings, which are provided in tables and figures, illustrate that the system has twin layers at the end of space domain and is uniformly convergent.On the preconditioning of the Schur complement matrix of a class of two-by-two block matrices
https://jmm.guilan.ac.ir/article_7944.html
We consider &nbsp;a class of two-by-two block complex system of linear equations obtained from finite element discretization of the distributed optimal control with time-periodic parabolic equations. Using the Schur complement technique we transform the obtained system to two subsystems. We propose a preconditioner to the subsystem with the Schur complement matrix. Spectral properties of the preconditioned matrix are analyzed. Some numerical results are presented to show the effectiveness of the preconditioner.Using a modified Sinc neural network to identify the chaotic systems with an application in wind speed forecasting
https://jmm.guilan.ac.ir/article_7985.html
Continuous-time models for dynamic nonlinear systems offer greater reliability than their discrete-time counterparts. Discrete-time models can suffer from information loss and increased noise susceptibility. Chaotic and hyperchaotic systems pose significant challenges due to their unpredictable nature. These systems are prevalent in various fields, including weather, climate, finance, and biology. Artificial neural networks, inspired by the human nervous system, are effective in approximating complex nonlinear systems. A recent innovation, the Sinc neural network (SNN), leverages the properties of the Sinc function, which is smooth and oscillatory, making it suitable for approximation tasks. Despite limited research, SNNs have shown promising results in applications like speech recognition, human motion recognition, and fractional optimal control problems. This study introduces a modified Sinc neural network (MSNN) to enhance the performance of SNN in identifying continuous-time nonlinear systems. The MSNN employs a stable online training algorithm based on Lyapunov stability theory. It is utilized to identify several chaotic systems, including the Duffing-Van der Pol oscillator, the Lorenz system, and a financial hyperchaotic system. Additionally, the MSNN is used for forecasting wind speed, an important factor in renewable energy generation. Data from Khorramabad, Iran, is utilized for this purpose. The MSNN's simple structure and strong performance in identifying nonlinear systems and forecasting wind speed demonstrate its potential.A neuro-fuzzy approach to compute the solution of a $Z$-numbers system with Trapezoidal fuzzy data
https://jmm.guilan.ac.ir/article_7986.html
Linear systems of equations with $Z$-numbers have recently attracted some interest. Some approaches have been developed for solving these systems. Since, there are many ambiguities and uncertainties in such issues, there is no analytic solution for these kinds of systems. Therefore, numerical schemes are usually used to estimate the solution of them. &nbsp;In this research, a computational scheme for solving linear systems involving trapezoidal $Z$-numbers is presented. The proposed approach is designed in such a way that it is firstly converted the $Z$-numbers coefficients to the corresponding fuzzy numbers and then using a ranking function, the fuzzy coefficients are converted to real coefficients. In this trend, after two stages, firstly, the original $Z$-numbers system becomes a fuzzy linear system and the fuzzy system is converted to a real system. Then, the obtained crisp linear system is solved based on the artificial neural network algorithm. Finally, two sample trapezoidal $Z$-numbers systems are solved based on the given approach to illustrate the process of the proposed algorithm.Stabilization by delay feedback control for highly nonlinear HSDDEs driven by Lévy noise
https://jmm.guilan.ac.ir/article_7987.html
This research aims to investigate the stabilization of highly nonlinear hybrid stochastic differential delay equations (HSDDEs) with L\'evy noise by delay feedback control. The coefficients of these systems satisfy a more general polynomial growth condition instead of classical linear growth condition. Precisely, an appropriate Lyapunov functional is constructed to analyze the stabilization of such systems in the sense of $H_{\infty}$-stability and asymptotic stability. The theoretical analysis indicates that the delay can affect the stability of highly nonlinear hybrid stochastic systems. &nbsp;Special approximation method for solving system of ordinary and fractional integro-differential equations
https://jmm.guilan.ac.ir/article_8008.html
This paper concerns with some special approximate methods in order to solve the system of ordinary and the fractional integro-differential &nbsp;equations. The approach that we use begins by a method of converting the fractional integro-differential equations into an integral equation including both Volterra and the Fredholm parts. Then a specific successive approximation technique is &nbsp;applied to the Volterra part. Due to the presence of the factorial factor in the denominator of its kernel, the Volterra part tends to zero in the next iterations, leading us to discard the Volterra's sentence as an error of the &nbsp;method that we use. The analytical-approximate solution to the problem is then obtained by solving the resulting equation, as a Fredholm integral equation of the second kind. This method is applied to the boundary value problems in two distinct cases involving system of ordinary and fractional differential equations.Blow-up phenomena for a couple of parabolic equations with memory and source terms: Analytical and simulation
https://jmm.guilan.ac.ir/article_8011.html
In this paper, the focus is on investigating the asymptotic behavior of the solution for a system of parabolic equations with memory terms acting in both equations. This system has many applications in various scientific fields, including heat conduction in materials with memory effects and the study of biological systems exhibiting memory phenomena. The system of parabolic equations with a memory term provides a powerful framework for understanding and predicting the behavior of such complex systems, with emphasis on the role of the memory term in capturing the system's history-dependent behavior. Firstly, we assume that the relaxation functions $\mu_{2}\left( t\right) \leq\mu_{1}\left( t\right),\text{ for all} t\geq0$, and under certain conditions regarding the function p($ \cdot $) we prove that the solution with positive initial energy blows up in finite time. Finally, we present the theoretical results as numerical findings in the form of figures that illustrate and confirm the results by studying examples in two dimensions.