Functional approach to nonlinear models of water flow in by G. Marinoschi

By G. Marinoschi

This paintings of utilized arithmetic specializes in the sensible learn of the nonlinear boundary worth difficulties on the subject of water movement in porous media, a subject matter which has less than now been explored in e-book shape. the writer indicates that summary thought can be occasionally more uncomplicated and richer in effects for functions than regular classical ways are. the amount bargains with diffusion variety versions, emphasizing the mathematical remedy in their nonlinear aspects.

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Extra info for Functional approach to nonlinear models of water flow in soils

Example text

There is no fixed procedure to extend the functions but, in general, this is done by continuity. The most important thing is to succeed to prove, at the end, that the extension procedure did not introduce artificial solutions that may change the feature of the original problem. In this work we shall extend the functions having in view two things: the first is to preserve some basic properties of the original functions, and the second to endow the extensions with the properties specified below. 1) β(θ) = ρ, K(θ) = Kr , for θ < θr , and as we can see, the properties of continuity and monotonicity of all functions β ∗ and K are still satisfied.

As the scope of this book is to extend the mathematical approach a little outside the framework of water infiltration models, to the classes of diffusion processes specified before, we have tried to reveal properties of the hydraulic models that include them in a diffusion category or another. Thus, the mathematical results which will be obtained in a general abstract framework for the various types of diffusion processes will apply in particular to the specific hydraulic models, but also to other models like those just enumerated.

19) 26 2 Settlement of the mathematical models of nonhysteretic infiltration (k C − kC )C − 3C (k C − kC ) > 0, on [θr , θs ). 21) θr ⎪ ⎩ [K ∗ , +∞) for θ = θs . 17) we deduce that the function β ∗ satisfies the inequality (β ∗ (θ) − β ∗ (θ))(θ − θ) ≥ ρ(θ − θ)2 , ∀θ, θ ∈ [θr , θs ]. 22) This can be very easily checked for θ, θ < θs , or θ = θ = θs . 14) we have (β ∗ (θs ) − β ∗ (θ))(θs − θ) ≥ (Ks∗ − β ∗ (θ))(θs − θ) lim K ∗ ((C ∗ )−1 (θ)) − β ∗ (θ) (θs − θ) = θ θs = lim θ θs K ∗ ((C ∗ )−1 (θ)) − K ∗ ((C ∗ )−1 (θ)) (θ − θ) ≥ lim (β ∗ (θ) − β ∗ (θ))(θ − θ) ≥ ρ(θs − θ)2 , θ θs since K ∗ ◦ (C ∗ )−1 is a monotonically increasing function on [θr , θs ).

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