• The Pure and Applied Mathematics
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• v.13 no.1 s.31
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• pp.19-38
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• 2006

We analyze the spectral collocation approximation for a parabolic partial integrodifferential equations(PIDE) with a weakly singular kernel. The space discretization is based on the spectral collocation method and the time discretization is based on Crank-Nicolson scheme with a graded mesh. We obtain the stability and second order convergence result for fully discrete scheme.

• Bulletin of the Korean Mathematical Society
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• v.53 no.1
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• pp.247-262
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• 2016

We propose and analyze spectral and pseudo-spectral Jacobi-Galerkin approaches for weakly singular Volterra integral equations (VIEs). We provide a rigorous error analysis for spectral and pseudo-spectral Jacobi-Galerkin methods, which show that the errors of the approximate solution decay exponentially in $L^{\infty}$ norm and weighted $L^2$-norm. The numerical examples are given to illustrate the theoretical results.

• Nonlinear Functional Analysis and Applications
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• v.27 no.2
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• pp.363-379
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• 2022

In this paper, we carried out a new numerical approach for solving integral algebraic equations with weakly singular kernels. The novel method is based on the construction of B-spline tight framelets using the unitary and oblique extension principles. Some numerical examples are given to provide further explanation and validation of our method. The result of this study introduces a new technique for solving weakly singular integral algebraic equation and thus in turn will contribute to providing new insight into approximation solutions for integral algebraic equation (IAE).

• 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 the Korean Mathematical Society
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• v.48 no.1
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• pp.133-146
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• 2011

We employ a hyperbolic tangent function to construct nonlinear transformations which are useful in numerical evaluation of weakly singular integrals and Cauchy principal value integrals. Results of numerical implementation based on the standard Gauss quadrature rule show that the present transformations are available for the singular integrals and, in some cases, give much better approximations compared with those of existing non-linear transformation methods.

• Korean Journal of Mathematics
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• v.7 no.2
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• pp.215-218
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• 1999

We investigate characterizations of rings whose simple singular right R-modules are GP-injective. It is proved that if R is a semiprime ring whose simple singular right R-modules are GP-injective, then the center $Z(R)$ of R is a von Neumann regular ring. We consider the condition ($^*$): R satisfies $l(a){\subseteq}r(a)$ for any $a{\in}R$. Also it is shown that if R satisfies ($^*$) and every simple singular right R-module is GP-injective, then R is a reduced weakly regular ring.

• Journal of the Korean Mathematical Society
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• v.48 no.2
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• pp.397-420
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• 2011

In this paper we study the structure of closed weakly dense ideals in Privalov spaces $N^p$ (1 < p < $\infty$) of holomorphic functions on the disk $\mathbb{D}$ : |z| < 1. The space $N^p$ with the topology given by Stoll's metric [21] becomes an F-algebra. N. Mochizuki [16] proved that a closed ideal in $N^p$ is a principal ideal generated by an inner function. Consequently, a closed subspace E of $N^p$ is invariant under multiplication by z if and only if it has the form $IN^p$ for some inner function I. We prove that if $\cal{M}$ is a closed ideal in $N^p$ that is dense in the weak topology of $N^p$, then $\cal{M}$ is generated by a singular inner function. On the other hand, if $S_{\mu}$ is a singular inner function whose associated singular measure $\mu$ has the modulus of continuity $O(t^{(p-1)/p})$, then we prove that the ideal $S_{\mu}N^p$ is weakly dense in $N^p$. Consequently, for such singular inner function $S_{\mu}$, the quotient space $N^p/S_{\mu}N^p$ is an F-space with trivial dual, and hence $N^p$ does not have the separation property.

• Journal of electromagnetic engineering and science
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• v.17 no.2
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• pp.91-97
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• 2017

An accurate and efficient evaluation for hypersingular integrals (HIs), strongly singular integrals (SSIs), and weakly singular integrals (WSIs) plays an essential role in the numerical solutions of 3D electromagnetic scattering problems. We derive analytic formulas for WSIs based on Stokes' theorem, which can be expressed in elementary functions. Several numerical examples are presented to validate these analytic formulas. Then, to show the feasibility of the proposed formulations for numerical methods, these formulations are used with the existing analytical expressions of HIs and SSIs to correct the near-field interaction in an iterative physical optics (IPO) scheme. Using IPO, the scattering caused by a dihedral reflector is analyzed and compared with the results of the method of moments and measurement data.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.2 no.2
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• pp.31-40
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• 1998

Spectral analysis by energy method is given for a fully discrete methods for hyperbolic integro-differential equations with a weakly singular kernel. Stability and error estimates in $H^1$-norm are derived.

• Journal of applied mathematics & informatics
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• v.6 no.2
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• pp.503-512
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• 1999

The purpose of this paper is to introduce the (Toeplitz) quadrature method for solving fredholm integral equations of the second kind with mildly singular kernels. We are presented some numerical examples for the computation of the error estimate using the MathCad package.