• 대한수학회논문집
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• 제31권3호
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• pp.567-573
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• 2016

We obtain solution formulas for some nonlocal heat equations. As a simple application, we prove finite time blow-up of the solution.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제16권4호
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• pp.235-242
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• 2012

In this paper, we investigate a nonlocal semilinear heat equation with homogeneous Dirichlet boundary condition in a bounded domain, and prove that there exist solutions with positive initial energy that blow up in finite time.

• Kyungpook Mathematical Journal
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• 제64권2호
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• pp.353-369
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• 2024

In this study, we investigate the unknown time-dependent heat source function in inverse problems. We consider three general nonlocal conditions; two classical boundary conditions and one nonlocal over-determination, condition, these genereate six different cases. The finite integration method (FIM), based on numerical integration, has been adapted to solve PDEs, and we use it to discretize the spatial domain; we use backward differences for the time variable. Since the inverse problem is ill-posed with instability, we apply regularization to reduce the instability. We use the first-order Tikhonov's regularization together with the minimization process to solve the inverse source problem. Test examples in all six cases are presented in order to illustrate the accuracy and stability of the numerical solutions.

• Structural Engineering and Mechanics
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• 제73권3호
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• pp.287-302
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• 2020

This investigation deals with a size-dependent coupled thermoelasticity analysis based on Green-Naghdi (GN) theory in nano scale using a new modified nonlocal model of heat conduction, which is based on the GN theory and nonlocal Eringen theory of elasticity. In the analysis based on the proposed model, the nonlocality is taken into account in both heat conduction and elasticity. The governing equations including the equations of motion and the energy balance equation are derived using the proposed model in a nano beam resonator. An analytical solution is proposed for the problem using the Laplace transform technique and Talbot technique for inversion to time domain. It is assumed that the nano beam is subjected to sinusoidal thermal shock loading, which is applied on the one of beam ends. The transient behaviors of fields' quantities such as lateral deflection and temperature are studied in detail. Also, the effects of small scale parameter on the dynamic behaviors of lateral deflection and temperature are obtained and assessed for the problem. The proposed GN-based model, analytical solution and data are verified and also compared with reported data obtained from GN coupled thermoelasticity analysis without considering the nonlocality in heat conduction in a nano beam.

• 한국산업융합학회 논문집
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• 제5권3호
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• pp.219-224
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• 2002

We consider the problem y"=a(x,y)(y-b), y(0)=0, y'(1)=g(y(${\xi}$), y'(${\xi}$)), (0${\xi}$ fixed in(0,1)) as a model of steady-slate heat conduction in a rod when the heat flux at the end x = 1 is determined by observation of the temperature and heat flux at some interior point ${\xi}$. We establish conditions sufficient for existence, uniqueness.

• 한국산업융합학회 논문집
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• 제5권4호
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• pp.303-307
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• 2002

We consider the problem y"=a(x,y)(y-b), (0$$y(0)=0,\;y^{\prime}(1)=g(y({\xi}),\;y^{\prime}({\xi})),\;{\xi}$$ fixed in (0,1). This is a model of steady-state heat conduction in a rod when the heat flux at the end x=1 is determined by observation of the temperature and heat flux at some interior point ${\xi}$. We establish conditions sufficient for existence, uniqueness, and positivity of solutions.

• 한국산업융합학회 논문집
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• 제5권4호
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• pp.309-312
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• 2002

Motivated by the problem of steady-state heat conduction in a rod whose heat flux at one end is determined by observation of the temperature and heat flux at some point ${\xi}$ in the interior of the rod, we consider the problem y"(x)=a(x, y(x))y(x) (0$${\lim_{x{\rightarrow}{\infty}}}y(x)=0,\;y^{\prime}(0)=g(y({\xi}),\;y^{\prime}({\xi}))$$ for some fixed ${\xi}{\in}(0,{\infty})$. We establish conditions guaranteeing existence and uniqueness for this problem on the semi-infinite interval [0,${\infty}$).