Browsing by Author "Savasaneril, NB"

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    A Review of Polynomial Matrix Collocation Methods in Engineering and Scientific Applications
    Çevik, M; Savasaneril, NB; Sezer, M
    Ordinary, partial, and integral differential equations are indispensable tools across diverse scientific domains, enabling precise modeling of natural and engineered phenomena. The polynomial collocation method, a powerful numerical technique, has emerged as a robust approach for solving these equations efficiently. This review explores the evolution and applications of the collocation method, emphasizing its matrix-based formulation and utilization of polynomial sequences such as Chebyshev, Legendre, and Taylor series. Beginning with its inception in the late 20th century, the method has evolved to encompass a wide array of differential equation types, including integro-differential and fractional equations. Applications span mechanical vibrations, heat transfer, diffusion processes, wave propagation, environmental pollution modeling, medical uses, biomedical dynamics, and population ecology. The method's efficacy lies in its ability to transform differential equations into algebraic systems using orthogonal polynomials at chosen collocation points, facilitating accurate numerical solutions across complex systems and diverse engineering and scientific disciplines. This approach circumvents the need for mesh generation and simplifies the computational complexity associated with traditional numerical methods. This comprehensive review consolidates theoretical foundations, methodological advancements, and practical applications, highlighting the method's pivotal role in modern computational mathematics and its continued relevance in addressing complex scientific challenges.
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    EULER AND TAYLOR POLYNOMIALS METHOD FOR SOLVING VOLTERRA TYPE INTEGRO DIFFERENTIAL EQUATIONS WITH NONLINEAR TERMS
    Elmaci, D; Savasaneril, NB; Dal, F; Sezer, M
    In this study, the first order nonlinear Volterra type integro-differential equations are used in order to identify approximate solutions concerning Euler polynomials of a matrix method based on collocation points. This method converts the mentioned nonlinear integro-differential equation into the matrix equation with the utilization of Euler polynomials along with collocation points. The matrix equation is a system of nonlinear algebraic equations with the unknown Euler coefficients. Additionally, this approach provides analytic solutions, if the exact solutions are polynomials. Furthermore, some illustrative examples are presented with the aid of an error estimation by using the Mean-Value Theorem and residual functions. The obtained results show that the developed method is efficient and simple enough to be applied. And also, convergence of the solutions of the problems were examined. In order to obtain the matrix equations and solutions for the selected problems, code was developed in MATLAB.
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    LUCAS POLYNOMIAL SOLUTION FOR NEUTRAL DIFFERENTIAL EQUATIONS WITH PROPORTIONAL DELAYS
    Gümgüm, S; Savasaneril, NB; Kürkçü, ÖK; Sezer, M
    This paper proposes a combined operational matrix approach based on Lucas and Taylor polynomials for the solution of neutral type differential equations with proportional delays. The advantage of the proposed method is the ease of its application. The method facilitates the solution of the given problem by reducing it to a matrix equation. Illustrative examples are validated by means of absolute errors. Residual error estimation is presented to improve the solutions. Presented in graphs and tables the results are compared with the existing methods in literature.
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    Lucas polynomial solution of nonlinear differential equations with variable delays
    Gümgüm, S; Savasaneril, NB; Kürkçü, ÖK; Sezer, M
    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.