174 research outputs found
Free Boundary Formulation for BVPs on a Semi-Infinite Interval and Non-Iterative Transformation Methods
This paper is concerned with two examples on the application of the free
boundary formulation to BVPs on a semi-infinite interval. In both cases we are
able to provide the exact solution of both the BVP and its free boundary
formulation. Therefore, these problems can be used as benchmarks for the
numerical methods applied to BVPs on a semi-infinite interval and to free BVPs.
Moreover, we emphasize how for two classes of free BVPs, we can define
non-iterative initial value methods, whereas BVPs are usually solved
iteratively. These non-iterative methods can be deduced within Lie's group
invariance theory. Then, we show how to apply the non-iterative methods to the
two introduced free boundary formulations in order to obtain meaningful
numerical results. Finally, we indicate several problems from the literature
where our non-iterative transformation methods can be applied.Comment: 30 pages, 7 figures, 4 table
Numerical Methods for a Nonlinear BVP Arising in Physical Oceanography
In this paper we report and compare the numerical results for an ocean
circulation model obtained by the classical truncated boundary formulation, the
free boundary approach and a quasi-uniform grid treatment of the problem. We
apply a shooting method to the truncated boundary formulation and finite
difference methods to both the free boundary approach and the quasi-uniform
grid treatment. Using the shooting method, supplemented by the Newton's
iterations, we show that the ocean circulation model cannot be considered as a
simple test case. In fact, for this method we are forced to use as initial
iterate a value close to the correct missing initial condition in order to be
able to get a convergent numerical solution. The reported numerical results
allow us to point out how the finite difference method with a quasi-uniform
grid is the less demanding approach and that the free boundary approach
provides a more reliable formulation than the classical truncated boundary
formulation.Comment: 25 pages, 12 figures, 5 table
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