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In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations.
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity (with units). In a physical context, scalar fields are required to be independent of the choice of reference frame. That is, any two observers using the same units will agree on the value of the scalar field at the same absolute point in space (or spacetime) regardless of their respective points of origin.
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
We prove the semi-global controllability and stabilization of the (1+1)−dimensional wave maps equation with spatial domain 𝕊1 and target Sk. First we show that damping stabilizes the system when t
We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. This classical framework allows us to present the backste
We revisit the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. We prove that, under some assumption on the control ope