Unknown Taylor Coefficients Research Articles

Taylor polynomial solutions of second order linear partial differential equations

The purpose of this study is to give a Taylor polynomial approximation for the solution of second order linear partial differential equations with two variables and variable coefficients. For this purpose, Taylor matrix method for the approximate solution of second order linear partial differential equations with specified associated conditions in terms of Taylor polynomials about any point. The method is based on, first, taking truncated Taylor series of the functions in equation and then substituting their matrix forms into the given equation. Thereby the equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. Hence, the result matrix equation can be solved and the unknown Taylor coefficients can be found approximately.

Taylor polynomial solutions of linear differential equations

A matrix method, which is called Taylor-matrix method, for the approximate solution of linear differential equations with specified associated conditions in terms of Taylor polynomials about any point. The method is based on, first, taking truncated Taylor series of the functions in equation and then substituting their matrix forms into the given equation. Thereby the equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. To illustrate the method, it is applied to certain linear differential equations and the generalized Hermite, Laguerre, Legendre and Chebyshev equations given by Costa and Levine [Int. J. Math. Edu. Sci. Technol. 20 (1989) 1].

A Taylor Collocation Method for the Solution of Linear Integro-Differential Equations

In this study, a matrix method called the Taylor collocation method is presented for numerically solving the linear integro-differential equations by a truncated Taylor series. Using the Taylor collocation points, this method transforms the integro-differential equation to a matrix equation which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. Also the method can be used for linear differential and integral equations. To illustrate the method, it is applied to certain linear differential, integral, and integro-differential equations and the results are compared.