• 충청수학회지
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• 제27권4호
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• pp.763-769
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• 2014

In this paper, we illustrate a counter example for the converse of a certain conjecture proposed by De Giorgi. De Giorgi suggested a series of conjectures, in which a certain integral condition for singularity or degeneracy of an elliptic operator is satisfied, the solutions are continuous. We construct some singular elliptic operators and solutions such that the integral condition does not hold, but the solutions are continuous.

• 대한수학회논문집
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• 제13권4호
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• pp.875-888
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• 1998

The use of the zeros of Chebyshev polynomial of the first kind $T_{4n＋4(x}$ ) and second kind $U_{2n＋1}$ (x) for Gauss-Chebyshev quad-rature and collocation of singular integral equations of Cauchy type yields computationally accurate solutions over other combinations of $T_{n}$ /(x) and $U_{m}$(x) as in [8]. We show that the coefficient matrix of the overdetermined system has the generalized inverse. We estimate the residual error using the norm of the generalized inverse.e.

• East Asian mathematical journal
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• 제39권1호
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• pp.29-36
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• 2023

In this paper, we consider φ-Laplacian nonlocal boundary value problems with singular weight function which may not be in L¹(0, 1). The existence and nonexistence of positive solutions to the given problem for parameter λ belonging to some open intervals are shown. Our approach is based on the fixed point index theory.

• 대한수학회논문집
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• 제10권3호
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• pp.751-758
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• 1995

We prove the existence of solution for a Dirichlet singular boundary value problem. The proofs are based on the method of upper and lower solutions.

• 충청수학회지
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• 제31권1호
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• pp.161-166
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• 2018

This paper deals an impulsive fractional integral inequality with singular kernel which can be used in getting the explicit estimate of solutions of impulsive fractional differential equations.

• Journal of applied mathematics & informatics
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• 제28권3_4호
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• pp.697-710
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• 2010

In this paper, we prove existence of infinitely many positive and concave solutions, by means of a simple approach, to $3^{th}$ order three-point singular boundary value problem {$x^{\prime\prime\prime}(t)=\alpha(t)f(t,x(t))$, 0 < t < 1, $x(0)=x'(\eta)=x^{\prime\prime}(1)=0$, (1/2 < $\eta$ < 1). Moreover with respect to multiplicity of solutions, we don't assume any monotonicity on the nonlinearity. We rely on a combination of the analysis of the corresponding vector field on the phase-space along with Knesser's type properties of the solutions funnel and the well-known Krasnosel'ski$\breve{i}$'s fixed point theorem. The later is applied on a new very simple cone K, just on the plane $R^2$. These extensions justify the efficiency of our new approach compared to the commonly used one, where the cone $K\;{\subset}\;C$ ([0, 1], $\mathbb{R}$) and the existence of a positive Green's function is a necessity.

• East Asian mathematical journal
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• 제39권3호
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• pp.355-367
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• 2023

Extending [14], we establish the existence of multiple positive solutions for a Schrödinger-type singular elliptic equation: $$\{{-{\Delta}u+V(x)u={\lambda}{\frac{f(u)}{u^{\beta}}},\;x{\in}{\Omega}, \atop u=0,\;x{\in}{\partial}{\Omega},$$ where 0 ∈ Ω is a bounded domain in ℝ^N, N ≥ 1, with a smooth boundary ∂Ω, β ∈ [0, 1), f ∈ C[0, ∞), V : Ω → ℝ is a bounded function and λ is a positive parameter. In particular, when f(s) > 0 on [0, σ) and f(s) < 0 for s > σ, we establish the existence of at least three positive solutions for a certain range of λ by using the method of sub and supersolutions.

• East Asian mathematical journal
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• 제36권5호
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• pp.583-591
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• 2020

In [8, 9] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities. They consider the Poisson equations with corner singularities, compute the finite element solutions using standard Finite Element Methods and use the extraction formula to compute the stress intensity factor(s), then they posed new PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor(s), which converges with optimal speed. From the solution they could get an accurate solution just by adding the singular part. The error analysis was given in [5]. In their approaches, the singular functions and the extraction formula which give the stress intensity factor are the basic elements. In this paper we consider the biharmonic problems with the cramped and/or simply supported boundary conditions and get the singular functions and its duals and find properties of them, which are the cornerstones of the approaches of [8, 9, 10].

• International Journal of Computer Science & Network Security
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• 제21권12호
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• pp.223-227
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• 2021

A treatment based on the algebraic operations on fuzzy numbers is used to replace the fuzzy problem into an equivalent crisp one. The finite difference technique is used to replace the continuous boundary value problem (BVP) of arbitrary order 1<α≤2, with fuzzy boundary parameters into an equivalent crisp (algebraic or differential) system. Three numerical examples with different behaviors are considered to illustrate the treatment of the singular perturbed case with different fractional orders of the BVP (α=1.8, α=1.9) as well as the classical second order (α=2). The calculated fuzzy solutions are compared with the crisp solutions of the singular perturbed BVP using triangular membership function (r-cut representation in parametric form) for different values of the singular perturbed parameter (ε=0.8, ε=0.9, ε=1.0). Results are illustrated graphically for the different values of the included parameters.

• 호남수학학술지
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• 제40권4호
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• pp.785-794
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• 2018

In [8] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities. They consider the Poisson equations with homogeneous Dirichlet boundary condition with one corner singularity at the origin, and compute the finite element solution using standard FEM and use the extraction formula to compute the stress intensity factor, then pose a PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor, which converges with optimal speed. From the solution they could get an accurate solution just by adding the singular part. This approach uses the polar coordinate and the cut-off function to control the singularity and the boundary condition. In this paper we consider Poisson equations with multiple singular points, which involves different cut-off functions which might overlaps together and shows the way of cording in FreeFEM++ to control the singular functions and cut-off functions with numerical experiments.