Abstract
The time-fractional diffusion equation with mass absorption is studied in a half-line domain under the Dirichlet boundary condition varying harmonically in time. The Caputo derivative is employed. The solution is obtained using the Laplace transform with respect to time and the sin-Fourier transform with respect to the spatial coordinate. The results of numerical calculations are illustrated graphically.
Highlights
From a mathematical point of view, diffusion and heat conduction are described by the same equation of the parabolic type ∂u = a∆u (1)∂t where a is the diffusivity coefficient, t denotes time, ∆ is the Laplace operator, and u stands for concentration in the case of diffusion and for temperature in the case of heat conduction.Ångström [1] was the first to consider Equation (1) under harmonic impact
Introducing of oscillations into the diffusion equation can be done by two ways
Another possibility to introduce oscillations in the diffusion equation consists in imposing the harmonic boundary condition
Summary
From a mathematical point of view, diffusion and heat conduction are described by the same equation of the parabolic type. The first possibility consists in considering the harmonic source term. In the domain −∞ < x < ∞ with δ( x ) being the Dirac delta function under assumption u( x, t) = U ( x ) eiωt. Another possibility to introduce oscillations in the diffusion equation consists in imposing the harmonic boundary condition. Is considered in the domain 0 < x < ∞ under condition x = 0 : u( x, t) = u0 eiωt (8). In a medium with a chemical reaction or with heat absorption/release, in Equation (1) , there appears an additional linear term [11,12]. We study Equation (19) in a half-line domain under the Dirichlet boundary condition varying harmonically in time; the paper develops the results of [44]
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