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
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 ).