Nonlinear Filtration Equation Research Articles

Multidimensional nonlinear pseudo-differential evolution equation with p-adic spatial variables

We study the Cauchy problem for p-adic non-linear evolutionary pseudo-differential equations for complex-valued functions of a real positive time variable and p-adic spatial variables. Among the equations under consideration there is the p-adic analog of the porous medium equation (or more generally, the nonlinear filtration equation) which arise in numerous application in mathematical physics and mathematical biology. Our approach is based on the construction of a linear Markov semigroup on a p-adic ball and the proof of m-accretivity of the appropriate nonlinear operator. The latter result is equivalent to the existence and uniqueness of a mild solution of the Cauchy problem of a nonlinear equation of the porous medium type.

Construction and Study of Exact Solutions to A Nonlinear Heat Equation

We construct and study exact solutions to a nonlinear second order parabolic equation which is usually called the “nonlinear heat equation” or “nonlinear filtration equation” in the Russian literature and the “porous medium equation” in other countries. Under examination is the special class of solutions having the form of a heat wave that propagates through cold (zero) background with finite velocity. The equation degenerates on the boundary of a heat wave (called the heat front) and its order decreases. The construction of these solutions by passing to an overdetermined system and analyzing its solvability reduces to integration of nonlinear ordinary differential equations of the second order with an initial condition such that the equations are not solvable with respect to the higher derivative. Some admissible families of heat fronts and the corresponding exact solutions to the problems in question are obtained. A detailed study of the global properties of solutions is carried out by the methods of the qualitative theory of differential equations and power geometry which are adapted for degenerate equations. The results are interpreted from the point of view of the behavior and properties of heat waves with a logarithmic front.

Approximate symmetry group classification for a nonlinear fractional filtration equation of diffusion-wave type

A one-dimensional nonlinear fractional filtration equation with the Riemann–Liouville time-fractional derivative is proposed for modeling fluid flow through a porous medium. This equation is derived under an assumption that the fluid has a fractional equation of state in which the fluid density depends on the time-fractional derivative of pressure. The obtained equation belongs to the diffusion-wave type of equations. A case when the order of fractional differentiation is close to an integer number is considered, and a small parameter is introduced into the fractional filtration equation under consideration. An expansion of the Riemann–Liouville time-fractional derivative into the series with respect to this small parameter is obtained. Using this expansion, a first-order approximation of the derived fractional filtration equation is performed. Next, the problem of approximate Lie point symmetry group classification for this approximate nonlinear filtration equation with a small parameter is studied. It is shown that approximate symmetry groups admitted by different realizations of the approximate filtration equation have much more dimensions than the corresponding exact Lie point symmetry groups admitted by unperturbed fractional diffusion-wave equations. Obtained classification results permit to construct approximate invariant solutions for the considered nonlinear time-fractional filtration equations.

Finite dimensional dynamics for nonlinear filtration equation

We construct new finite dimensional submanifolds in the solution space of nonlinear differential filtration equations and describe the corresponding evolutionary dynamics. This method is implemented in a computer program of symbolic computations Maple.

Approximate generalized conditional symmetries and solutions for nonlinear filtration equation with a small parameter

The approximate generalized conditional symmetries of nonlinear filtration equation with a small parameter are studied. The initial-value problem of the equation can be transformed to perturbed Cauchy problem of perturbed first-order ordinary differential equations systems and approximate solutions are obtained by using these symmetries.

Group classification and conservation laws of nonlinear filtration equation with a small parameter

Group classification of the perturbed nonlinear filtration equation is performed assuming that the perturbation is an arbitrary function of the dependent variable. The nonlinear self-adjointness of the equation under consideration is investigated. Using these results, the approximate conservation laws are constructed.

Optimal control problem for the Oskolkov nonlinear filtration equation

Optimal control problem for the Oskolkov nonlinear filtration equation

Intermediate Asymptotics Beyond Homogeneity and Self-Similarity: Long Time Behavior for u t = Δϕ(u)

We investigate the long time asymptotics in L1+(R) for solutions of general nonlinear diffusion equations u t = Δϕ(u). We describe, for the first time, the intermediate asymptotics for a very large class of non-homogeneous nonlinearities ϕ for which long time asymptotics cannot be characterized by self-similar solutions. Scaling the solutions by their own second moment (temperature in the kinetic theory language) we obtain a universal asymptotic profile characterized by fixed points of certain maps in probability measures spaces endowed with the Euclidean Wasserstein distance d2. In the particular case of ϕ(u) ~ u m at first order when u ~ 0, we also obtain an optimal rate of convergence in L1 towards the asymptotic profile identified, in this case, as the Barenblatt self-similar solution corresponding to the exponent m. This second result holds for a larger class of nonlinearities compared to results in the existing literature and is achieved by a variation of the entropy dissipation method in which the nonlinear filtration equation is considered as a perturbation of the porous medium equation.

Asymptotic non-stability and blow-up at boundary for solutions of a filtration equation

For a class of nonlinear filtration equation with nonlinear second-third boundary value condition, it is shown that a priori boundary of the solution can be estimated and controlled by initial data and integral on the boundary of the region. The priori estimate of the solutions was established by iterative method. By using this estimate the solutions may blow-up on the boundary of the region and thus it may have asymptotic non-stability.

Group Classification and Some Similarity Solutions for a Nonlinear Filtration Equation

A nonlinear filtration equation in which the filter coefficient is an arbitrary function of the specific deposit is considered. Lie Group theory is applied to the coupled system of partial differential equations. Group classification is performed with respect to the arbitrary filter coefficient. Some similarity solutions are constructed using the symmetries.

On the behaviour of blow-up interfaces for an inhomogeneous filtration equation

We study the asymptotic behaviour of blow-up interfaces of the solutions to the one-dimensional nonlinear filtration equation in inhomogeneous media p(x)u 1 = (u m ) xx in Q= R x R + , where m > 1 is a constant and p(x) = |x| -α (for |x| ≥ 1, with α > 2) is a bounded, positive, smooth, and symmetric function. The initial data are assumed to be smooth, bounded, compactly supported, symmetric, and monotone. It is known that due to the fast decay of the density p(x) as |x| → ∞ the support of the solution increases unboundedly in a finite time T. We prove that as t → T - the interface behaves like O((T - t) -b ), where the exponent b > 0 (which depends on m and α only) is given by a unique self-similar solution of the second kind satisfying the equation |x| -α u t = (u m ) xx . The corresponding rescaled profiles also converge. We establish the stability of the self-similar solution of the second kind for the exponential density p(x) = e -|x| for |x| ≥ 1. We give a formal asymptotic analysis of the blow-up behaviour for the non-self-similar density p(x) = e -|x|2 Several exact self-similar solutions and their corresponding asymptotics are constructed.

On self-similar solutions of Barenblatt's nonlinear filtration equation

We derive the asymptotic form of the self-similar solutions of the second kind of the Cauchy problem for Barenblatt's nonlinear filtration equation by perturbing the Lie group of the underlying linear problem. We also show that the decay rate, appearing in the similarity solutions, can be found by a simple inspection of the corresponding energy dissipation law.

The filtration equation in a class of functions decreasing at infinity

We deal with the Cauchy and external boundary problems for the nonlinear filtration equation with variable density. For each density we define a class ϕ \phi of initial functions φ \varphi , such that for any φ ∈ ϕ \varphi \in \phi the problem is uniquely solvable in some set of functions decreasing at infinity with respect to space variables.

The Cauchy problem for the non-linear filtration equation in an inhomogeneous medium

For the filtration equation with a density which is rapidly decreasing at infinity, we have found a class of unique solvability of the Cauchy problem. Furthermore, we have studied the behaviours of solutions as t → ∞.

On the porous medium equations with lower order singular nonlinear terms

where rn_~l, 2>0, p>0 , b>0, c > 0 are constants. The function uo(x) is supposed to be bounded, continuous and nonnegative, satisfying uo(x) >0 on land uo(x) =-0 in R \ I for a bounded interval I = (al, a2)c R. Equation (1) is parabolic when u>0 and degenerates when u=0. In the case b = c = 0 (1) is called the equation of nonlinear heat conductivity or filtration equation. The terms cu p and b(uX)~ in (1) correspond to the absorption and to the transport of heat or matter, respectively. The Cauchy problem (1), (2) usually has no classical solution having continuous derivatives which appear in (1) even in the case b = c = 0 (see Zeldovich and Komponeets [7]). Oleinik, Kalastmikov and YuMin [6] proved that the Cauchy problem (1), (2) has a unique generalized solution in the case b = c = 0 .