Abstract:We consider direct solution to third order ordinary differential equations in this paper. Collocation and interpolation approach was adopted to generate a continuous hybrid multistep method. We adopted the use of power series as a basis function for approximate solution. We evaluated at off-grid points to get a continuous hybrid multistep method. Block method was later adopted to generate the independent solution at selected off-grid points. The properties of the block viz: order, zero stability and region of absolute stability are investigated. Our method was tested on third order ordinary differential equation and found to give better result when compared with existing methods.
In this paper, we developed a new continuous block method by the method of interpolation and collocation to derive new scheme. We adopted the use of power series as a basis function for approximate solution. We evaluated at off grid points to get a continuous hybrid multistep method. The continuous hybrid multistep method is solved for the independent solution to yield a continuous block method which is evaluated at selected points to yield a discrete block method. The basic properties of the block method were investigated and found to be consistent, zero stable and convergent. The results were found to compete favorably with the existing methods in terms of accuracy and error bound. In particular, the scheme was found to have a large region of absolute stability. The new method was tested on real life problem namely: Dynamic model.
Let H be a real Hilbert space. Let $F:H\rightarrow 2^{H}$
F
:
H
→
2
H
and $K:H\rightarrow 2^{H}$
K
:
H
→
2
H
be two maximal monotone and bounded operators. Suppose the Hammerstein inclusion $0\in u+KFu$
0
∈
u
+
K
F
u
has a solution. We construct an inertial-type algorithm and show its strong convergence to a solution of the inclusion. As far as we know, this is the first inertial-type algorithm for Hammerstein inclusions in Hilbert spaces. We also give numerical examples to compare the new algorithm with some existing ones in the literature.
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