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Laplace Transforms
Published in Steven G. Krantz, Differential Equations, 2022
Here ν is a physical constant known as the kinematic viscosity. The constant U is determined by the initial state of the system. The partial differential equation is a version of the classical heat equation. It is parabolic in form. It can also be used to model other diffusive systems, such as a semi-infinite bar of metal, insulated along its sides, suddenly heated up at one end. The system we are considering is known as Rayleigh's problem. This mathematical model shows that the only process involved in the flow is the diffusion of x-momentum into the bulk of the fluid (since u represents unidirectional flow in the x-direction).
Solutions of Partial Differential Equations
Published in Mayer Humi, Introduction to Mathematical Modeling, 2017
The algorithm defined by Equation (8.110) or Equation (8.111) is called the explicit method for the numerical solution of the heat equation. Its drawback is that in order to ensure its stability, that is, to prevent the accumulated numerical error from becoming too large and therefore rendering the numerical solution meaningless, we must choose r≤12 $ r \le \frac{1}{2} $ . This condition places a restriction on ▵ t that must satisfy ▵ t ≤ ( ▵ x2)/2k and, therefore, increases the computational effort needed to obtain the required solution.
Laplace Transforms
Published in Steven G. Krantz, Differential Equations, 2015
Here ν is a physical constant known as the kinematic viscosity. The constant is determined by the initial state of the system. The partial differential equation is a version of the classical heat equation. It is parabolic in form. It can also be used to model other diffusive systems, such as a semi-infinite bar of metal, insulated along its sides, suddenly heated up at one end. The system we are considering is known as Rayleigh’s problem. This mathematical model shows that the only process involved in the flow is the diffusion of x-momentum into the bulk of the fluid (since u represents unidirectional flow in the x-direction).
Thermal stresses for a generalized magneto-thermoelasticity on non-homogeneous orthotropic continuum solid with a spherical cavity
Published in Mechanics Based Design of Structures and Machines, 2022
S. M. Abo-Dahab, Nahed S. Hussein, A. M. Abd-Alla, H. A. Alshehri
More attentions in the last decades have been considered to thermoelasticity theories that release the paradox for admitting a finite speed for the propagation of thermal signals due to the effect of thermal field. In contrast to the conventional theories based on Fourier’s heat equation in the parabolic-type partial differential equation, these theories are referred to generalized theories modifying temperature equation to parabolic partial differential equation that admits finite speed for the propagation of waves. These theories are more realistic than conventional thermoelasticity theories in dealing with experimental evidence and practical problems involving very short time intervals and high heat fluxes such as those occurring in laser units, nuclear reactors, energy channels, etc. Since the 19th century, the phenomenon of coupling between the thermal field behavior of materials and magnetic behavior of materials has been studied and has more applications in diverse fields as geophysics, biology, geology, and engineering.
Effect of Atangana–Baleanu fractional derivative on a two-dimensional thermoviscoelastic problem for solid sphere under axisymmetric distribution
Published in Mechanics Based Design of Structures and Machines, 2023
Polymers are considered a significantly important industry material due to their physical properties and ease of deformation. So, this importance prompted Gross (1953) to find a mathematical model that describes this phenomenon. This model has been developed by many authors like Ferry (1961), Gurtin and Sternberg (1962), Stratonva (1971), and Pobedrya (1979). Il’yushin (1968) and Pobedrya (1969) introduced the coupled theory of thermoviscoelasticity. This theory has been discussed by Kovalenko and Karnaukhov (1972) and Medri (1988). Contrary to physical observations, the heat equation of this theory predicts an infinite speed of propagation for heatwaves because it is a parabolic partial differential equation.
Wine and maths: mathematical solutions to wine–inspired problems
Published in International Journal of Mathematical Education in Science and Technology, 2018
The mathematical problem is described by a partial differential equation of second-order which falls in the class of equations known as the heat equation. The objective is to solve this equation, to derive the optimal depth of a wine cellar in order to minimize temperature fluctuations which disturb proper wine ageing.