In this study, a novel matrix method based on Lucas series and collocation points has been used to solve nonlinear differential equations with variable delays. The application of the method converts the nonlinear equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Lucas coefficients. The method is tested on three problems to show that it allows both analytical and approximate solutions.
In this paper, a Laguerre matrix method is developed to find an approximate solution of linear differential, integral and integro-differential equations with variable coefficients under mixed conditions in terms of Laguerre polynomials. For this purpose, Laguerre polynomials are used in the interval [0,b]. The proposed method converts these equations into matrix equations, which correspond to systems of linear algebraic equations with unknown Laguerre coefficients. The solution function is obtained easily by solving these matrix equations. The examples of these kinds of equations are solved by using this new method and the results are discussed and it is seen that the present method is accurate, efficient and applicable.
In this paper, a new numerical matrix-collocation technique is considered to solve functional integrodifferential equations involving variable delays under the initial conditions. This technique is based essentially on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points. Some descriptive examples are performed to observe the practicability of the technique and the residual error analysis is employed to improve the obtained solutions. Also, the numerical results obtained by using these collocation points are compared in tables and figures.
With increasing technologies, applications of delayed models are more frequently encountered in biology, physics and various fields of engineering. The single degree-of-freedom oscillator, on the other hand, is one of the fundamental systems in many physics and engineering problems;
thus, solving the equation of this problem would serve for many other sophisticated problems. In this study, a novel and simple numerical method for the solution of this system is introduced in the matrix form based on Laguerre polynomials. The method is exemplified through a numerical application
and the results obtained are compared with those of another method. In addition, an error analysis technique based on residual function is developed and applied to this problem to demonstrate the validity and applicability of the method. The convenience of the method is that it is quite simple
to employ by using computer programs.
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