Non-polynomial Spline Solution for a Fourth-Order Non-homogeneous Parabolic Partial Differential Equation with a Separat
In this paper, a fourth-order non-homogeneous parabolic partial differential equation with initial and separated boundary conditions is solved by using a non-polynomial spline method. In the solution of the problem, finite difference discretization in tim
- PDF / 178,401 Bytes
- 7 Pages / 439.36 x 666.15 pts Page_size
- 65 Downloads / 172 Views
Abstract In this paper, a fourth-order non-homogeneous parabolic partial differential equation with initial and separated boundary conditions is solved by using a non-polynomial spline method. In the solution of the problem, finite difference discretization in time, and parametric quintic spline along the spatial coordinate have been carried out. The result shows that the applied method in this paper is an applicable technique and approximates the exact solution very well.
1 Introduction Taking numerical aspects into consideration, the theory of spline functions should be viewed as an active field of approximation theory. This is also true for partial differential equations with some initial and boundary conditions including the separated conditions. We consider a fourth-order non-homogeneous partial differential equation which represents the undumped transverse vibrations of a flexible straight beam. @2 u @4 u C 4 D f .x; t/ ; 0 x 1; t > 0 (1) 2 @t @x subject to the initial conditions, u .x; 0/ D g .x/ ;
ut .x; 0/ D N .x/ ;
0 x 1;
(2)
and the boundary conditions at the spatial points x D 0, x D 1 including the separated boundary condition at x D 0; 5 are in the form uxx .0; t/ D p0 .t/ ;
uxx .1; t/ C u .0:5; t/ D p1 .t/ ;
t 0
(3)
N.F. Er () S. Yeniceri H. Caglar C. Akkoyunlu Mathematics - Computer Department, Istanbul Kultur University, 34156 Atakoy Istanbul, Turkey e-mail: [email protected]; [email protected]; [email protected]; [email protected] S.G. Stavrinides et al. (eds.), Chaos and Complex Systems, DOI 10.1007/978-3-642-33914-1 15, © Springer-Verlag Berlin Heidelberg 2013
117
118
N.F. Er et al.
where u is the transverse displacement of the beam, t and x are time and distance variables, respectively. The function f .x; t/ that makes the problem nonhomogeneous, is defined as a dynamic driving force per unit mass. The numerical analysis literature contains many other methods including polynomial and nonpolynomial spline functions, developed to generate an approximate solution for the fourth order parabolic partial differantial equations. E.A. Al-Said and M.A. Noor [1] studied on fourth order obstacle boundary value problems by using quartic spline methods. Usmani [8] studied on this problem and established convergent secondorder and fourth-order methods for this problem, but with a change in boundary conditions, using first-order instead of second order derivative. For this problem we studied on it, Rashidinia et al. [6] and Siddiqi et al. [7] developed a difference scheme via quintic spline functions. For this problem, Zhu [9] constructed an optimal quartic spline collocation method, and Loghmani et al. [4] generated an approximate solution as a combination of quartic B-splines. A family of B-spline methods is used to solve the same problem by Caglar [3]. A spline method based on a nonpolynomial spline function with a polynomial and trigonometric parts, is constructed by Ramadan et al. [5]. Aziz et al. [2] solved this problem, with a method based on parametric quintic spline
Data Loading...
