• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.26 no.4
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• pp.353-361
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• 2022

For an initial value problem, using a weighted average between two adjacent approximates, we propose a simple one-step method based on the Euler method. This method is useful for solving stiff initial value problem, even when the step size is not very small. Moreover, it can be seen that the proposed method with some selected weights results in improved approximation errors.

• Kyungpook Mathematical Journal
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• v.51 no.4
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• pp.435-456
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• 2011

The Chebyshev collocation method in [21] to solve stiff initial-value problems is generalized by using arbitrary degrees of interpolation polynomials and arbitrary collocation points. The convergence of this generalized Chebyshev collocation method is shown to be independent of the chosen collocation points. It is observed how the stability region does depend on collocation points. In particular, A-stability is shown by taking the mid points of nodes as collocation points.

• Kyungpook Mathematical Journal
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• v.52 no.2
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• pp.167-177
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• 2012

In this article, we propose exponentially fitted error correction methods(EECM) which originate from the error correction methods recently developed by the authors (see [10, 11] for examples) for solving nonlinear stiff initial value problems. We reduce the computational cost of the error correction method by making a local approximation of exponential type. This exponential local approximation yields an EECM that is exponentially fitted, A-stable and L-stable, independent of the approximation scheme for the error correction. In particular, the classical explicit Runge-Kutta method for the error correction not only saves the computational cost that the error correction method requires but also gives the same convergence order as the error correction method does. Numerical evidence is provided to support the theoretical results.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.26 no.3
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• pp.138-155
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• 2022

To solve the initial value problem we present a new single-step implicit method based on the Euler method. We prove that the proposed method has convergence order 2. In practice, numerical results of the proposed method for some selected examples show an error tendency similar to the second-order Taylor method. It can also be found that this method is useful for stiff initial value problems, even when a small number of nodes are used. In addition, we extend the proposed method by using weighted averages with a parameter and show that its convergence order becomes 2 for the parameter near $\frac{1}{2}$. Moreover, it can be seen that the extended method with properly selected values of the parameter improves the approximation error more significantly.

• Kyungpook Mathematical Journal
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• v.57 no.2
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• pp.287-299
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• 2017

In this paper, we develop an enhanced Chebyshev collocation method based on an integration scheme of the generalized Chebyshev interpolations for solving stiff initial value problems. Unlike the former error embedded Chebyshev collocation method (CCM), the enhanced scheme calculates the solution and its truncation error based on the interpolation of the derivative of the true solution and its integration. In terms of concrete convergence and stability analysis, the constructed algorithm turns out to have the $7^{th}$ convergence order and the A-stability without any loss of advantages for CCM. Throughout a numerical result, we assess the proposed method is numerically more efficient compared to existing methods.