• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.27 no.3
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• pp.194-206
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• 2023

We present families of nonlinear transformations useful for numerical evaluation of weakly singular integrals. First, for end-point singular integrals, we define a prototype function with some appropriate features and then suggest a family of transformations. In addition, for interior-point singular integrals, we develop a family of nonlinear transformations based on the aforementioned prototype function. We take some examples to explore the efficiency of the proposed nonlinear transformations in using the Gauss-Legendre quadrature rule. From the numerical results, we can find the superiority of the proposed transformations compared to some existing transformations, especially for the integrals with high singularity strength.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.20 no.1
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• pp.121-127
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• 2008

Numerical experiments with weakly nonlinear MIKE21 BW module and fully nonlinear FUNWAVE model are performed to identify the nonlinear characteristics of Boussinesq models with varying nonlinearity. Generation of waves with varying amplitudes, nonlinear shoaling and wave propagation over submerged bar experiments showed the importance of nonlinear model in shallow water where nonlinearity becomes prominent. Fully nonlinear model showed the nonsymmetrical wave form more clearly and gave larger shoaling coefficients than those of weakly nonlinear model.

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.97-106
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• 2012

In this paper, the main purpose is to study existence of the global attractors for the weakly damped wave equation with nonlinear boundary conditions. To this end, we first show that the existence o a bounded absorbing set by the perturbed energy method. Secondly, we utilize the decomposition of the solution operator to verify the asymptotic compactness.

• Nonlinear Functional Analysis and Applications
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• v.26 no.3
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• pp.565-580
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• 2021

In this paper, first of all we prove a fixed point theorem for 𝜓_∫𝜑-weakly contractive mapping. Next, we prove some common fixed point theorems for a pair of weakly compatible self maps along with E.A. property and (CLR) property. An example is also given to support our results.

• Journal of the Korean Mathematical Society
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• v.41 no.6
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• pp.957-976
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• 2004

It is well known that the application of the nonlinear coordinate transformations is useful for efficient numerical evaluation of weakly singular integrals. In this paper, we consider the trapezoidal rule combined with a nonlinear transformation $\Omega$$_{m}$(b;$\chi$), containing a parameter b, proposed first by Yun [14]. It is shown that the trapezoidal rule with the transformation $\Omega$$_{m}$(b;$\chi$), like the case of the Gauss-Legendre quadrature rule, can improve the asymptotic truncation error by using a moderately large b. By several examples, we compare the numerical results of the present method with those of some existing methods. This shows the superiority of the transformation $\Omega$$_{m}$(b;$\chi$).TEX>).

• Journal of Advanced Research in Ocean Engineering
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• v.3 no.2
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• pp.83-101
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• 2017

In this study, the added resistances of the large container ship in head and oblique seas are evaluated using a time-domain Rankine panel method. The mean forces and moments are computed by the near-field method, namely, the integration of the second-order pressure directly on the ship surface. Furthermore, a weakly nonlinear approach in which the nonlinear restoring and Froude-Krylov forces on the exact wetted surface of a ship are included in order to examine the effects of amplitudes of waves on ship motions and added resistances. The computation results for various advance speeds and heading angles are validated by comparing with the experimental data, and the validation shows reasonable consistency. Nevertheless, there exist discrepancies between the numerical and experimental results, especially for a shorter wave length, a higher advance speed, and stern quartering seas. Therefore, the accuracies of the linear and weakly nonlinear methods in the evaluation of the mean drift forces and moments are also discussed considering the characteristics of the hull such as the small incline angle of the non-wall-sided stern and the fine geometry around the high-nose bulbous bow.

• Nonlinear Functional Analysis and Applications
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• v.27 no.2
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• pp.271-287
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• 2022

In this article, we shall prove a common fixed point theorem for two weakly compatible self-maps 𝒫 and 𝔔 on a dislocated metric space (M, d^*) satisfying the following ξ-weakly expansive condition: d^*(𝒫c, 𝒫d) ≥ d^* (𝔔c, 𝔔d) + ξ(∧(𝔔c, 𝔔d)), ∀ c, d ∈ M, where $${\wedge}(Qc,\;Qd)=max\{d^*(Qc,\;Qd),\;d^*(Qc,\;\mathcal{P}c),\;d^*(Qd,\;\mathcal{P}d),\;\frac{d^*(Qc,\;\mathcal{P}c){\cdot}d^*(Qd,\;\mathcal{P}d)}{1+d^*(Qc,\;Qd)},\;\frac{d^*(Qc,\;\mathcal{P}c){\cdot}d^*(Qd,\;\mathcal{P}d)}{1+d^*(\mathcal{P}c,\;\mathcal{P}d)}\}$$. Also, we have proved common fixed point theorems for the above mentioned weakly compatible self-maps along with E.A. property and (CLR) property. An illustrative example is also provided to support our results.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.877-888
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• 2008

This paper deals with some new nonlinear delay and weakly singular integral inequalities of Gronwall-Bellman type. These results generalize the inequalities discussed by Xiang and Kuang [19]. Several other inequalities proved by $Medve{\check{d}}$ [15] and Ou-Iang [17] follow as special cases of this paper. This work can be used in the analysis of various problems in the theory of certain classes of differential equations, integral equations and evolution equations. A modification of the Ou-Iang type inequality with delay is also treated in this paper.

• Proceedings of the Korea Water Resources Association Conference
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• 2005.09b
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• pp.1319-1320
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• 2005

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.257-265
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• 2011

In this paper, we first prove a common fixed point theorem for generalized nonlinear contraction mappings in complete metric spaces there by generalizing and extending some known results. Then we present this result in the context of ordered metric spaces by using monotone non-decreasing mapping.