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Differential Equations, 2023
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Shishkina, E. L., Yusupova, A. K.
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zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Shishkina, E. L., Yusupova, A. K.
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Morphisms of the Heat Equation
Annals of Global Analysis and Geometry, 1997The author extends the Fuglede-Ishihara characterization of harmonic morphisms to maps \(f:M\times \mathbb{R}^+\to N\times \mathbb{R}^+\) that pull back local solutions of the heat equation on \(N\) to those of \(M\), as well as to maps \(f:M\times M\times \mathbb{R}^+\to N\times N\times \mathbb{R}^+\) that pull back the heat kernel of \(N\) to that of
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On a Universality of the Heat Equation
Mathematische Nachrichten, 1997AbstractWe prove that there are solutions u(t,x) of the heat equation ut = uxx such that every continuous function f : [a, b] → ℝ can be uniformly approximated by a subsequence of u (n, ·), nϵ ℕ.
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On Symmetries of the Heat Equation
Acta Applicandae Mathematicae, 1989Two aims are pursued in this article. The first one is methodological and consists of demonstrating the symmetry calculation techniques based on the commutation relation on the example of the quasi-linear heat equation. The second one consists of investigating the following open question: is the algebra of higher symmetries of the linear heat equation ...
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2001
In this chapter we consider the Laplacian on spaces of continuous functions. If \(\Omega \subset \mathbb{R}^n\) is an open, bounded set with boundary \(\partial \Omega\) which is Dirichlet regular, we will show that the Laplacian generates a holomorphic semigroup on the space $$C_0 (\Omega ): = \{ u \in C(\bar \Omega ):u|_{\partial \Omega} =0\}.$$
Wolfgang Arendt +3 more
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In this chapter we consider the Laplacian on spaces of continuous functions. If \(\Omega \subset \mathbb{R}^n\) is an open, bounded set with boundary \(\partial \Omega\) which is Dirichlet regular, we will show that the Laplacian generates a holomorphic semigroup on the space $$C_0 (\Omega ): = \{ u \in C(\bar \Omega ):u|_{\partial \Omega} =0\}.$$
Wolfgang Arendt +3 more
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1986
The most frequently encountered equation of parabolic type is the heat or diffusion equation, which is the subject of this chapter. Our setting is again a region in En, but in addition a new variable t, which may be thought of as time, also appears. Consequently the type of problem to be studied is not a boundary value problem, as it was with Laplace’s
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The most frequently encountered equation of parabolic type is the heat or diffusion equation, which is the subject of this chapter. Our setting is again a region in En, but in addition a new variable t, which may be thought of as time, also appears. Consequently the type of problem to be studied is not a boundary value problem, as it was with Laplace’s
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1998
This chapter is devoted to the equation $$\frac{{\partial u}}{{\partial t}}\left( {x,t} \right) - {\omega ^2}\Delta u\left( {x,t} \right) = f\left( {x,t} \right),x \in \Omega \subset {R^n},t \in \left( {0,T} \right)$$ where Δ is the Laplace operator with respect to the spatial variable x and t is the time variable.
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This chapter is devoted to the equation $$\frac{{\partial u}}{{\partial t}}\left( {x,t} \right) - {\omega ^2}\Delta u\left( {x,t} \right) = f\left( {x,t} \right),x \in \Omega \subset {R^n},t \in \left( {0,T} \right)$$ where Δ is the Laplace operator with respect to the spatial variable x and t is the time variable.
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