, Vol. 15, No. 3, pp. 334343, 2010.
© Association for Scientific Research
Mehmet Çevik
Izmir Vocational School
Dokuz Eylül University, 35150
Buca, Izmir, Turkey
mehmet.cevik@deu.edu.tr
The present study introduces a novel and simple matrix method for the
solution of longitudinal vibration of rods in terms of Taylor polynomials. The proposed
method converts the governing partial differential equation of the system into a matrix
equation, which corresponds to a system of linear algebraic equations with unknown
Taylor coefficients. Then the solution is obtained easily by solving these matrix
equations. Both free and forced vibrations of the system are studied; particular and
general solutions are determined. The method is demonstrated by an illustrative
example using symbolic computation. Comparison of the numerical solution obtained in
this study with the exact solution is quite good.
!"# Taylor matrix method, Longitudinal vibration, Numerical solution, Partial
differential equation.
$%
Longitudinal vibration of a bar or rod is one of the fundamental problems in
mechanical vibrations since the transition from this classical problem to the more
contemporary problems is logical and direct. The governing equation of motion of this
problem, which is also known as the wave equation, is a partial differential equation of
second order in both space and time. The solution of this equation by separation of
variables can be found in textbooks [1,2,3].
In the present study, a novel and simple matrix method in terms of Taylor
polynomials is introduced for the solution of this partial differential equation. This
method is based on first taking the truncated Taylor series of the function in the
equation and then substituting the matrix forms into the given equation. The result
matrix equation can be solved and the unknown Taylor coefficients can be found
approximately. Taylor polynomials have been used by many researchers for the solution
of differential and integral equations. Everitt et al. [4] gave orthogonal polynomial
solutions of linear ordinary differential equations. Gülsu and Sezer [5] and Sezer and
Da;çıoğlu [6] gave Taylor polynomial approximations for the solution of m
th
order and
higher order linear differentialdifference equations with variable coefficients under the
mixed conditions about any point. Taylor matrix method has been used to solve the
Riccati differential equation by Gülsu and Sezer [7] and to solve the generalized
pantograph equations with linear functional argument by Sezer and Da;çıoğlu [8]. In
both of the studies, the methodology is compared with some other known techniques to
show that the present approach is relatively easy and highly accurate. Li [9] proposed a
simple yet efficient method to approximately solve linear ordinary differential equations