Abstract:
The boundary and initial boundary value problems have always played a vital
character in the fields of science and technology. Different numerical techniques are
used to obtain numerical approximations of such problems. We present and illustrate
novel numerical techniques for the numerical approximations of higher order
boundary and initial boundary value problems. The numerical techniques derived in
this research work are based upon the fact of employing polynomial cubic spline
(PCS) scheme and non polynomial cubic spline (NPCS) scheme in conjunction with
the decomposition procedure. In the case for ordinary differential equations, the
decomposition procedure is used to reduce the higher order boundary value problems
(BVPs) into the corresponding system of second order boundary value problems. Then
PCS and NPCS schemes are constructed for each second order ordinary differential
equation. The first order derivatives are approximated by the central finite differences
of (ℎ ). For partial differential equations, the second order time derivatives are
decomposed into the first order derivatives. The process of decomposition generates a
linear system of partial differential equations, where the first order time derivatives are
approximated by the central finite differences.
The performance of the new derived schemes is illustrated by numerical tests
that involve comparing numerical approximations with analytical solutions on a
collection of carefully selected problems from the literature. These problems range
from those involving higher order ordinary differential equations, for example, fifth,
sixth, seventh, twelfth, and thirteenth order ordinary differential equations and partial
differential equations, like fourth order parabolic equations, one dimensional
hyperbolic telegraph equations, and one dimensional wave equations. In addition,
Adomian decomposition method is used to construct the boundary conditions for the
solution of fourth order parabolic equations.
