• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.629-644
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• 2012

In this work, the authors study the blow-up properties of solutions to a parabolic system with nonlocal boundary conditions and nonlocal sources. Conditions for the existence of global or blow-up solutions are given. Global blow-up property and precise blow-up rate estimates are also obtained.

• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1435-1446
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• 2009

In this paper we investigate the blow up properties of the positive solutions to a semi linear parabolic system with coupled nonlocal sources $u_t={\Delta}u+k_1{\int}_{\Omega}u^{\alpha}(y,t)v^p(y,t)dy,\;v_t={\Delta}_v+k_2{\int}_{\Omega}u^q(y,t)v^{\beta}(y,t)dy$ with non local Dirichlet boundary conditions. We establish the conditions for global and non-global solutions respectively and obtain its blow up set.

• Kyungpook Mathematical Journal
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• v.51 no.2
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• pp.165-175
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• 2011

The purpose of this note is to give Opial type inequalities with nonlocal conditions. Also, a reverse of the original inequality with y(a) = y(b) = 0 is derived. We apply these inequalities to second-order differential equations with nonlocal conditions to derive several necessary conditions for the existence of solutions.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.497-502
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• 2022

Existence and uniqueness for fractional differential equations satisfying a general nonlocal initial or boundary condition are proven by means of Schauder's fixed point theorem. The nonlocal condition is given as an integral with respect to a signed measure, and includes the standard initial value condition and multi-point boundary value condition.

• Steel and Composite Structures
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• v.17 no.5
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• pp.641-665
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• 2014

This paper presents the analytical solutions for the size-dependent static analysis of the functionally graded (FG) beams with various boundary conditions based on the nonlocal continuum model. The nonlocal behavior is described by the differential constitutive model of Eringen, which enables to this model to become effective in the analysis and design of nanostructures. The elastic modulus of beam is assumed to vary through the thickness or longitudinal directions according to the power law. The governing equations are derived by using the nonlocal continuum theory incorporated with Euler-Bernoulli beam theory. The explicit solutions are derived for the static behavior of the transversely or axially FG beams with various boundary conditions. The verification of the model is obtained by comparing the current results with previously published works and a good agreement is observed. Numerical results are presented to show the significance of the nonlocal effect, the material distribution profile, the boundary conditions, and the length of beams on the bending behavior of nonlocal FG beams.

• Communications of the Korean Mathematical Society
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• v.14 no.1
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• pp.223-231
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• 1999

In this paper we prove the existence of solutions of a nonlinear integrodifferential equation of Sobolev type with nonlocal conditions. The results are obtained by using compact semigroups and the Schauder fixed point theorem.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1765-1780
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• 2013

This paper deals with some types of semilinear parabolic systems with localized or nonlocal sources and nonlocal boundary conditions. The authors first derive some global existence and blow-up criteria. And then, for blow-up solutions, they study the global blow-up property as well as the precise blow-up rate estimates, which has been seldom studied until now.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.14 no.2
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• pp.79-91
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• 2010

In this paper, a class of fractional integrodifferential equations of mixed type with nonlocal conditions is considered. First, using contraction mapping principle and Krasnoselskii's fixed point theorem via Gronwall's inequailty, the existence and uniqueness of mild solution are given. Second, the existence of optimal pairs of systems governed by fractional integrodifferential equations of mixed type with nonlocal conditions is also presented.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.23 no.4
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• pp.324-331
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• 2013

In this paper, static and oscillatory instability of a nanotube conveying fluid and modeled as a thin-walled beam is investigated. Analytically nonlocal effect, effects of boundary conditions, transverse shear and rotary inertia are incorporated in this study. The governing equations and boundary conditions are derived through Hamilton's principle. Numerical analysis is performed by using extended Galerkin method which enables us to obtain more accurate results compared with conventional Galerkin method. Variations of critical flow velocity of carbon nanopipes with two different boundary conditions based on the analytically nonlocal theory and partially nonlocal theory are investigated and pertinent conclusions are outlined.

• Structural Engineering and Mechanics
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• v.65 no.4
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• pp.465-476
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• 2018

This article investigates buckling behavior of a multi-phase nanocrystalline nanobeam resting on Winkler-Pasternak foundation in the framework of nonlocal couple stress elasticity and a higher order refined beam model. In this model, the essential measures to describe the real material structure of nanocrystalline nanobeams and the size effects were incorporated. This non-classical nanobeam model contains couple stress effect to capture grains micro-rotations. Moreover, the nonlocal elasticity theory is employed to study the nonlocal and long-range interactions between the particles. The present model can degenerate into the classical model if the nonlocal parameter, and couple stress effects are omitted. Hamilton's principle is employed to derive the governing equations and the related boundary conditions which are solved applying an analytical approach. The buckling loads are compared with those of nonlocal couple stress-based beams. It is showed that buckling loads of a nanocrystalline nanobeam depend on the grain size, grain rotations, porosities, interface, elastic foundation, shear deformation, surface effect, nonlocality and boundary conditions.