Murat Karaçayır

A numerical scheme to solve boundary value problems involving singular perturbation

ABSTRACT In this study, a numerical method is presented in order to approximately solve singularly perturbed second-order differential equations given with boundary conditions. The method uses the set of monomials whose degrees do not exceed a prescribed as the set of base functions, resulting from the supposition that the approximate solution is a polynomial of degree whose coefficients are to be determined. Then, following Galerkin’s approach, inner product with the base functions are applied to the residual of the approximate solution polynomial. This process, with a suitable incorporation of the boundary conditions, gives rise to an algebraic linear system of size . The approximate polynomial solution is then obtained from the solution of this resulting system. Additionally, a technique, called residual correction, which exploits the linearity of the problem to estimate the error of any computed approximate solution is discussed briefly. The numerical scheme and the residual correction technique are illustrated with two examples.

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

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

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 numerical method for solutions of Lotka–Volterra predator–prey model with time-delay

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 taking inner product of a set of monomials with a vector obtained from the problem under consideration. Doing this, the problem is transformed to a nonlinear system of algebraic equations. This system is then solved, yielding coefficients of the approximate polynomial solutions. In addition, the technique of residual correction, which aims to increase the accuracy of the approximate solution by estimating its error, is discussed in some detail. The method and the residual correction technique are illustrated with two examples.

Application of homotopy perturbation method to solve two models of delay differential equation systems

In this paper, two delay differential systems are considered, namely, a famous model from mathematical biology about the spread of HIV viruses in blood and the advanced Lorenz system from mathematical physics. We then apply the homotopy perturbation method (HPM) to find their approximate solutions. It turns out that the method gives rise to easily obtainable solutions. In addition, residual error functions of the solutions are graphed and it is shown that increasing the parameter n in the method improves the results in both cases.

A Galerkin-like scheme to solve Riccati equations encountered in quantum physics

In this study, a Galerkin-like approach is applied to numerically solve Riccati differential equations. In this method, inner product is applied to a set of monomials and an expression obtained from the equation in question. The resulting nonlinear system is then solved, yielding a polynomial as the approximate solution. Additionally, the technique of residual correction, whose aim is to increase the accuracy of the approximate solution, is discussed briefly. Lastly, the method and the residual correction technique are illustrated with two examples.