In this paper, we use polynomial splines of eleventh degree with three collocation
points to develop a method for computing approximations to the solution and its
derivatives up to ninth order for general linear and nonlinear ninth-order boundary-v
alue
problems (BVPs). The study shows that the spline method with three collocation points
when is applied to these problems is existent and unique. We prove that the proposed
method if applied to ninth-order BVPs is stable and consistent of order eleven, and it
possesses convergence rate greater than six.
Finally, some numerical experiments are presented for illustrating the theoretical
results and by comparing the results of our method with the other methods, we reveal that
the proposed method is better than others.
In this paper, spline approximations with five collocation points are used for the
numerical simulation of stochastic of differential equations(SDE). First, we have modeled
continuous-valued discrete wiener process, and then numerical asymptotic st
ochastic
stability of spline method is studied when applied to SDEs. The study shows that the
method when applied to linear and nonlinear SDEs are stable and convergent.
Moreover, the scheme is tested on two linear and nonlinear problems to illustrate
the applicability and efficiency of the purposed method. Comparisons of our results with
Euler–Maruyama method, Milstein’s method and Runge-Kutta method, it reveals that the
our scheme is better than others.
In this paper, spline technique with five collocation parameters for finding the
numerical solutions of delay differential equations (DDEs) is introduced. The presented
method is based on the approximating the exact solution by C4-Hermite spline
i
nterpolation and as well as five collocation points at every subinterval of DDE.The study
shows that the spline solution of purposed technique is existent and unique and has
strongly stable for some collocation parameters. Moreover, this method if applied to test
problem will be consistent, p-stable and convergent from order nine .In addition ,it
possesses unbounded region of p-stability. Numerical experiments for four examples are
given to verify the reliability and efficiency of the purposed technique. Comparisons show
that numerical results of our method are more accurate than other methods.