Journal of Mathematical Modeling
Year:2024 | Volume:12 | Issue:2|Papers Count:12

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.

This paper presents a uniformly convergent numerical scheme for time-fractional  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.

In this paper, we propose a  fourth-order compact discretization method   for solving a second-order boundary value problem governed by the nonlinear Fredholm integro-differential equations.  Using an efficient approximate polynomial,  a  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.  It is shown that the proposed method is easy to implement and has the third order of accuracy in the infinity norm. Some  numerical examples are presented to demonstrate its  approximation accuracy and computational efficiency,   as well as to compare the derived results with those  obtained in the literature.

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.  Numerical experiments  support the theoretical results.

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.  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  special choices of the parameters. Finally, some numerical results regarding the practical performance of the new proposed KF are reported.

In this paper, we study the blow-up of solutions for hyperbolic equations involving the fractional Laplacian operator with damping and source terms.  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.

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.

This paper presents a new hybrid conjugate gradient method for solving  nonlinear unconstrained optimization problems; it is based on a combination of $RMIL$  (Rivaie-Mustafa-Ismail-Leong)  and $hSM$  (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.  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.

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.

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.

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.

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.