Exact Solutions of the Generalized Richards Equation with Power-Law Nonlinearities
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IAL DIFFERENTIAL EQUATIONS
Exact Solutions of the Generalized Richards Equation with Power-Law Nonlinearities A. A. Kosov1∗ and E. I. Semenov1∗∗ 1
Matrosov Institute for System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences, Irkutsk, 664033 Russia e-mail: ∗ [email protected], ∗∗ [email protected] Received April 17, 2019; revised June 16, 2020; accepted June 26, 2020
Abstract—We consider a generalized Richards equation with power-law nonlinearities modeling filtration in porous media. Conditions are derived under which the problem can be reduced to the linear heat equation or to nonlinear equations with known solutions. The families of explicit exact solutions that can be expressed via elementary functions or Lambert’s W -function are found. Some examples illustrating the results are given. DOI: 10.1134/S0012266120090025
INTRODUCTION In this paper, we study the partial differential equation dθ ∂Ψ ∂ ∂Ψ = D(Ψ) − K(Ψ) , dΨ ∂t ∂x ∂x
4
Ψ = Ψ(x, t),
(1)
which will be called the generalized Richards equation. The classical Richards equation [1–3] is the special case of Eq. (1) with D(Ψ) ≡ K(Ψ) and describes moisture transfer in unsaturated soils. By analogy with the classical Richards equation, the variables occurring in Eq. (1) have the following physical meaning: Ψ is the volume content of water (a dimensionless quantity), θ(Ψ) is the water saturation function, K(Ψ) is the hydraulic conductivity, D(Ψ) is the moisture transfer (filtration) coefficient, t is time, and x is the vertical coordinate oriented in the direction of gravity. Equations of this kind are used when modeling various filtration processes (see, e.g., the paper [4] and the references therein). The Richards equation is nonlinear, and one can find its solutions only in special cases with coefficients of special form; therefore, research is being carried out in terms of both seeking such particular exact solutions [5, 6] and developing numerical methods for constructing approximate solutions of boundary value problems for this equation [7–10]. Here the exact solutions are of importance in that they are used to verify and adjust numerical methods [7]. The main goal of the present paper is to construct explicit exact solutions of Eq. (1) under the assumption that the water saturation function θ(Ψ), the hydraulic conductivity K(Ψ), and the moisture transfer coefficient D(Ψ) are power-law functions, θ(Ψ) = Ψµ , K(Ψ) = αΨλ , and D(Ψ) = Ψσ , where µ, λ, and σ are the nonlinearity parameters and α 6= 0 is an arbitrary constant. Thus, the object of study is the partial differential equation ∂ σ ∂Ψ λ µ−1 ∂Ψ µΨ = Ψ − αΨ , (2) ∂t ∂x ∂x where σ, λ 6= 0, µ 6= 0, and α 6= 0 are real parameters of the equation. The choice of the functions occurring in Eq. (1) in the class of power-law functions is motivated not only by the naturally encountered power-law dependences [5, 7] but also by the fact that, for certain parameter values, Eq. (2) is reduced to simpler (in particular, linear or well-known) equations, as will be shown below.
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