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Berikbol T Torebek - One of the best experts on this subject based on the ideXlab platform.
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A method for solving ill-posed nonlocal problem for the elliptic equation with data on the whole boundary
Journal of Pseudo-Differential Operators and Applications, 2019Co-Authors: Tynysbek Sh. Kal’menov, Berikbol T TorebekAbstract:In this paper a nonlocal problem for the elliptic equation in a Cylindrical Domain is considered. It is shown that this problem is ill-posed as well as the Cauchy problem for the Laplace equation. The method of spectral expansion in eigenfunctions of the nonlocal problem for equations with involution establishes a criterion of the strong solvability of the considered nonlocal problem. It is shown that the ill-posedness of the nonlocal problem is equivalent to the existence of an isolated point of the continuous spectrum for a nonself-adjoint operator with involution.
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a criterion of solvability of the elliptic cauchy problem in a multi dimensional Cylindrical Domain
Complex Variables and Elliptic Equations, 2019Co-Authors: Tynysbek Sh Kalmenov, Makhmud A Sadybekov, Berikbol T TorebekAbstract:In this paper, we consider the Cauchy problem for multidimensional elliptic equations in a Cylindrical Domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations...
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a criterion of solvability of the elliptic cauchy problem in a multi dimensional Cylindrical Domain
arXiv: Analysis of PDEs, 2016Co-Authors: Tynysbek Sh Kalmenov, Makhmud A Sadybekov, Berikbol T TorebekAbstract:In this paper we consider the Cauchy problem for multidimensional elliptic equations in a Cylindrical Domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations with deviating argument establishes a criterion of the strong solvability of the considered elliptic Cauchy problem. It is shown that the ill-posedness of the elliptic Cauchy problem is equivalent to the existence of an isolated point of the continuous spectrum for a self-adjoint operator with deviating argument.
Jorge Ferreira - One of the best experts on this subject based on the ideXlab platform.
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a reaction diffusion model for a class of nonlinear parabolic equations with moving boundaries existence uniqueness exponential decay and simulation
Applied Mathematical Modelling, 2014Co-Authors: Rui Robalo, Rui M P Almeida, Maria Do Carmo Coimbra, Jorge FerreiraAbstract:Abstract The aim of this paper is to establish the existence, uniqueness and asymptotic behaviour of a strong regular solution for a class of nonlinear equations of reaction–diffusion nonlocal type with moving boundaries: u t - a ∫ Ω t u ( x , t ) dx u xx = f ( x , t ) , ( x , t ) ∈ Q t , u ( α ( t ) , t ) = u ( β ( t ) , t ) = 0 , t > 0 , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω 0 = ] α ( 0 ) , β ( 0 ) [ , where Q t is a bounded non-Cylindrical Domain defined by Q t = ( x , t ) ∈ R 2 : α ( t ) x β ( t ) , for all 0 t T . Moreover, we study the properties of the solution and implement a numerical algorithm based on the Moving Finite Element Method (MFEM) with polynomial approximations of any degree, to solve this class of problems. Some numerical tests are investigated to evaluate the performance of our Matlab code based on the MFEM and illustrate the exponential decay of the solution.
Tomasz Piasecki - One of the best experts on this subject based on the ideXlab platform.
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on an inhomogeneous slip inflow boundary value problem for a steady flow of a viscous compressible fluid in a Cylindrical Domain
Journal of Differential Equations, 2010Co-Authors: Tomasz PiaseckiAbstract:Abstract We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a Cylindrical Domain Ω = Ω 0 × ( 0 , L ) ∈ R 3 . We show existence of a solution ( v , ρ ) ∈ W p 2 ( Ω ) × W p 1 ( Ω ) , p > 3 , where v is the velocity of the fluid and ρ is the density, that is a small perturbation of a constant flow ( v ¯ ≡ [ 1 , 0 , 0 ] , ρ ¯ ≡ 1 ). We also show that this solution is unique in a class of small perturbations of ( v ¯ , ρ ¯ ) . The term u ⋅ ∇ w in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence ( v n , ρ n ) that is bounded in W p 2 ( Ω ) × W p 1 ( Ω ) and satisfies the Cauchy condition in a larger space L ∞ ( 0 , L ; L 2 ( Ω 0 ) ) what enables us to deduce that the weak limit of a subsequence of ( v n , ρ n ) is in fact a strong solution to our problem.
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on an inhomogeneous slip inflow boundary value problem for a steady flow of a viscous compressible fluid in a Cylindrical Domain
arXiv: Mathematical Physics, 2009Co-Authors: Tomasz PiaseckiAbstract:We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a Cylindrical Domain $\Omega = \Omega_0 \times (0,L) \in \mathbb{R}^3$. We show existence of a solution $(v,\rho) \in W^2_p(\Omega) \times W^1_p(\Omega)$, where $v$ is the velocity of the fluid and $\rho$ is the density, that is a small perturbation of a constant flow $(\bar v \equiv [1,0,0], \bar \rho \equiv 1)$. We also show that this solution is unique in a class of small perturbations of $(\bar v,\bar \rho)$. The term $u \cdot \nabla w$ in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence $(v^n,\rho^n)$ that is bounded in $W^2_p(\Omega) \times W^1_p(\Omega)$ and satisfies the Cauchy condition in a larger space $L_{\infty}(0,L;L_2(\Omega_0))$ what enables us to deduce that the weak limit of a subsequence of $(v^n,\rho^n)$ is in fact a strong solution to our problem.
Danillo Silva De ,oliveira - One of the best experts on this subject based on the ideXlab platform.
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Heat conduction equation solution in the presence of a change of state in a bounded axisymmetric Cylindrical Domain
'Universidade de Sao Paulo Agencia USP de Gestao da Informacao Academica (AGUIA)', 2019Co-Authors: Danillo Silva De ,oliveiraAbstract:O problema da condução de calor, envolvendo mudança de fase, foi resolvido para o caso de uma cavidade limitada por duas superfícies cilíndricas indefinidamente longas. As condições de contorno impostas consistem em manter a temperatura da superfície interna fixa e abaixo da temperatura de fusão do material que preenche a cavidade, enquanto que a temperatura da superfície externa é mantida fixa e acima da temperatura de fusão. Como condição inicial se fixou a temperatura de todo o material que preenche a cavidade no valor da temperatura da superfície externa. A solução obtida consiste em duas soluções da equação de condução de calor, uma escrita para o material solidificado e outra escrita para o material em estado líquido. As duas soluções são formalmente escritas em termos da posição da frente de mudança de fase, que é representada por uma superfície cilíndrica com raio em expansão dentro da cavidade. A posição dessa superfície é, a princípio, desconhecida e é calculada impondo o balanço de energia através da frente da mudança de fase. O balanço de energia é expresso por uma equação diferencial de primeira ordem, cuja solução numérica fornece a posição da frente como função do tempo. A substituição da posição da frente de mudança de fase em um instante particular, nas soluções da equação de condução de calor, fornece a temperatura nas duas fases naquele instante. A solução obtida é ilustrada através de exemplos numéricos.The heat conduction problem, in the presence of a change of state, was solved for the case of an indefinitely long Cylindrical layer cavity. As boundary conditions it is imposed that the internal surface of the cavity is maintained below the fusion temperature of the infilling substance and the external surface is kept above it. The solution, obtained in non-dimensional variables, consists in two closed form heat conduction equation solutions for the solidified and liquid regions, which formally depend of the, at first, unknown position of the phase change front. The energy balance through the phase change front furnishes the equation for time dependence of the front position, which is numerically solved. Substitution of the front position for a particular instant in the heat conduction equation solutions gives the temperature distribution inside the cavity at that moment. The solution is illustrated with numerical examples
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heat conduction equation solution in the presence of a change of state in a bounded axisymmetric Cylindrical Domain
Journal of Heat Transfer-transactions of The Asme, 2011Co-Authors: Danillo Silva De ,oliveira, Fernando Brenha RibeiroAbstract:The heat conduction problem, in the presence of a change of state, was solved for the case of an indefinitely long Cylindrical layer cavity. As boundary conditions, it is imposed that the internal surface of the cavity is maintained below the fusion temperature of the infilling substance and the external surface is kept above it. The solution, obtained in nondimensional variables, consists in two closed form heat conduction equation solutions for the solidified and liquid regions, which formally depend of the, at first, unknown position of the phase change front. The energy balance through the phase change front furnishes the equation for time dependence of the front position, which is numerically solved. Substitution of the front position for a particular instant in the heat conduction equation solutions gives the temperature distribution inside the cavity at that moment. The solution is illustrated with numerical examples.
Enrico Valdinoci - One of the best experts on this subject based on the ideXlab platform.
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the effect on fisher kpp propagation in a cylinder with fast diffusion on the boundary
Siam Journal on Mathematical Analysis, 2017Co-Authors: Luca Rossi, Andrea Tellini, Enrico ValdinociAbstract:In this paper we consider a reaction-diffusion equation of Fisher-KPP type inside an infinite Cylindrical Domain in $\mathbb{R}^{N+1}$, coupled with a reaction-diffusion equation on the boundary of the Domain, where potentially fast diffusion is allowed. We will study the existence of an asymptotic speed of propagation for solutions of the Cauchy problem associated with such a system, as well as the dependence of this speed on the diffusivity at the boundary and the amplitude of the cylinder. When $N=1$ the Domain reduces to a strip between two straight lines. This models the effect of two roads with fast diffusion on a strip-shaped field bounded by them.
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the effect on fisher kpp propagation in a cylinder with fast diffusion on the boundary
Weierstrass Institute for Applied Analysis and Stochastics: Preprint 2103, 2015Co-Authors: Luca Rossi, Andrea Tellini, Enrico ValdinociAbstract:In this paper we consider a reaction-diusion equation of Fisher-KPP type inside an innite Cylindrical Domain in
