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Biological Delay Systems Linear Stability Theory

Collocation Methods For Volterra Integral And Related Functional Equations

We present a detailed quantitative comparison between a finite-difference traveling wave (FDTW) model and a delayed differential equation (DDE) approach for the simulation of passive mode-locking in quantum dot lasers with both ring and Fabry-Perot (FP) cavities. Modifications with respect to the standard DDE models available in the literature are proposed. The new DDE approach improves the quantitative agreement with the FDTW model when applied to the simulation of passive mode-locking in FP lasers, preserving a very high computational efficiency. The modifications proposed in the DDE model also apply to the simulation of quantum-well and bulk devices.

Modelling passive mode-locking in Quantum-Dot Lasers: a comparison between a Finite-Difference Travelling-Wave model and a Delayed Differential Equation approach

A fractional model for population dynamics of two interacting species by using spectral and Hermite wavelets methods

Viruses have different mechanisms in causing a disease in an organism, which largely depend on the viral species. The recent advancement, through coupling data analysis and mathematical modeling, has allowed the identification and characterization of the nature of the virus. In the present paper, the homotopy analysis method is applied to provide an approximate solution of the basic HIV viral dynamic model describing the viral dynamics in a susceptible population. The proposed method allows for the solution of the governing system of differential equations to be calculated in the form of an infinite series with components which can be easily calculated. The homotopy analysis method utilizes a simple method to adjust and control the convergence region of the infinite series solution by using an auxiliary parameter. By using the homotopy series solutions, firstly, several β − curves using an appropriate ratio are plotted to demonstrate the regions of convergence and the optimum value of ℏ, then the residual and absolute errors are obtained for different values of these regions. Secondly, the residual error functions are applied to show the accuracy of the applied homotopy analysis method. Also, the convergence theorem of homotopy analysis method for the HIV viral dynamic model is proved. Mathematica software is used for the calculations and numerical results. The results obtained show the effectiveness and strength of the homotopy analysis method.

Estimating the approximate analytical solution of HIV viral dynamic model by using homotopy analysis method

In this paper, we present a numerical scheme to obtain polynomial approximations for the solutions of continuous time-delayed population models for two interacting species. The method includes taki...

A numerical method for solutions of lotka-volterra predator-prey model with time-delay

In this study, a numerical method is proposed to solve high-order linear Volterra delay integro-differential equations. In this approach, we assume that the exact solution can be expressed as a power series, which we truncate after the (N + 1)-st term so that it becomes a polynomial of degree N. Substituting the unknown function, its derivatives and the integrals by their matrix counterparts, we obtain a vector equivalent of the equation in question. Applying inner product to this vector with a set of monomials, we are left with a linear algebraic equation system of N unknowns. The approximate solution of the problem is then computed from the solution of the resulting linear system. In addition, the technique of residual correction, whose aim is to increase the accuracy of the approximate solutions by estimating the error of those solutions, is discussed briefly. Both the method and this technique are illustrated with several examples. Finally, comparison of the present scheme with other methods is made w...

A Numerical Approach for Solving High-Order Linear Delay Volterra Integro-Differential Equations

In this study, a Galerkin-type approach is presented in order to numerically solve one-dimensional hyperbolic telegraph equation The method includes taking inner product of a set of bivariate monomials with a vector obtained from the equation in question The initial and boundary conditions are also taken into account by a suitable utilization of collocation points The resulting linear system is then solved, yielding a bivariate polynomial as the approximate solution Additionally, the technique of residual correction, which aims to increase the accuracy of the approximate solution, is discussed briefly The method and the residual correction technique are illustrated with four examples Lastly, the results obtained from the present scheme are compared with other methods present in the literature

A Galerkin-Type Method to Solve One-Dimensional Telegraph Equation Using Collocation Points in Initial and Boundary Conditions

In this paper, a Galerkin-like approach is presented to numerically solve multi-pantograph type delay differential equations. The method includes taking inner product of a set of monomials with a vector obtained from the equation under consideration. The resulting linear system is then solved, yielding a polynomial as the approximate solution. We also provide an error analysis and discuss the technique of residual correction, which aims to increase the accuracy of the approximate solution. Lastly, the method, error analysis and the residual correction technique are illustrated with several examples. The results are also compared with numerous existing methods from the literature.

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A Galerkin-like approach to solve multi-pantograph type delay differential equations

The topic of this study is one of the most important models of mathematical biology; namely, a nonlinear first order delayed model about the infection of CD
 $$4^+$$
 T-cells by HIV. In this model, the effect of virus on human immune system is described through depletion of healthy CD
 $$4^+$$
 T-cells, where the delayed term representing the eclipse phase of virus makes the model more realistic. To obtain approximate solutions of this model, we develop a method reminiscent of the discrete Galerkin method. Considering the approximate solutions in the form of polynomials, we first substitute these approximate solutions in the original model. Some relations are thus obtained, which we express in terms of matrices. Following Galerkin’s path, we take inner product of a set of monomials with these matrix expressions, thus obtaining a nonlinear system of algebraic equations. The solution of this system gives the approximate solutions of the model. Additionally, the technique of residual correction, which aims to reduce the error of the approximate solution by estimating this error, is discussed in some detail. The method and the residual correction technique are illustrated with an example problem.

Murat Karaçayır

Papers

A numerical method for solutions of lotka-volterra predator-prey model with time-delay

A Numerical Approach for Solving High-Order Linear Delay Volterra Integro-Differential Equations

A Galerkin-Type Method to Solve One-Dimensional Telegraph Equation Using Collocation Points in Initial and Boundary Conditions

A Galerkin-like approach to solve multi-pantograph type delay differential equations

A Galerkin-Type Method for Solving a Delayed Model on HIV Infection of CD $$\mathbf{4^+}$$ 4 + T-cells