On the uniqueness of the solution of a nonlinear filtration problem through a porous medium

The purpose of this work is to study a fluid flow through a porous medium governed by a nonlinear Darcy's law. We also impose a nonlinear semi-permeability condition on some part of the boundary of this medium. The main results are the continuity of the free boundary and the uniqueness of the solution. Received May 5, 1996 / In a revised form November 16, 1996 / Accepted December 17, 1996     

On a filtration problem through a porous medium

Summary We prove the existence of a unique solution for a free boundary problem relative to the stationary flow between two water reservoirs of different levels separated by a dam of a non-homogeneous porous medium. Entrata in Redazione il 5 maggio 1973.     

Two-dimensional filtration problem for solutions in a nonhomogeneous porous medium

We present a mathematical model of underground leaching by solutions filtering through a porous medium. The model describes the motion of solutions from injection to extraction boreholes, as well as dissolution and secondary deposition in reduced-pH regions. A numerical algorithm has been developed for solving the problem on a plane in the general case of a homogeneous medium that contains regions with various nonhomogeneities. The algorithm combines triangulation of the region with the finite element method. The model also allows slow variation over time of some of the process parameters, such as porosity and the filtration coefficient. Numerical results are reported for various cases. The model qualitatively describes the main regularities of underground leaching and can be used to study and understand the detailed dynamics of these processes. The model also fills gaps in geological prospecting data, and extraction curves constructed for different wells can be applied to determine the approximate location of a particular nonhomogeneity. Mathematical modeling can help to optimize mineral extraction by underground leaching. Translated from Prikladnaya Matematika i Informatika, No. 29, pp. 29–55, 2008.     

Numerical solution of an inverse electrocardiography problem for a medium with piecewise constant electrical conductivity

A numerical method is proposed for solving an inverse electrocardiography problem for a medium with a piecewise constant electrical conductivity. The method is based on the method of boundary integral equations and Tikhonov regularization.     

A filtration problem in a porous medium with variable permeability

Sunto Un problema di frontiera libera (moto di un fluido attraverso una diga in terra non omogenea) è ricondotto allo studio di una disequazione variazionale. Si danno condizioni sul coefficiente di permeabilità sufficienti ad assicurare che la parte bagnata della diga è un sottografico.

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Entrata in Redazione il 30 giugno 1976.

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This paper is partially supported by National Science Foundation Grant MPS 72-04959 A02, and by C.N.R. of Italy through L.A.N. of Pavia.     

Numerical solution of a linear bilevel problem

The linear bilevel programming problem in the optimistic formulation is studied. It is reduced to an optimization problem with a nonconvex constraint in the form of a d.c. function (that is, the difference of two convex functions). For this problem, local and global search methods are developed. Numerical experiments performed for numerous specially generated problems, including large-scale ones, demonstrate the efficiency of the proposed approach.     

Numerical solution of a Rayleigh-Taylor instability problem

The numerical solution of a Rayleigh-Taylor instability problem where an inviscid liquid of finite depth is accelerated into a gas of semi-infinite extent is obtained by transforming the irregular flow domain into a rectangular domain by a coordinate transformation. The free surface equation is solved by a Crank-Nicolson procedure. The boundary condition at the free surface for the velocity potential ø which contains the time derivative of ø is also treated by an implicit scheme. The numerical results agree well with those obtained by higher order perturbation analysis.     

Numerical solution of a quasilinear parabolic problem

A combined approach of linearisation techniques and finite difference method is presented for obtaining the numerical solution of a quasilinear parabolic problem. The given problem is reduced to a sequence of linear problems by using the Picard or Newton methods. Each problem of this sequence is approximated by Crank-Nicolson difference scheme. The solutions of the resulting system of algebraic equations are obtained by using Block-Gaussian elimination method. Two numerical examples are solved by using both linearisation procedures to illustrate the method. For these examples, the Newton method is found to be more effective, especially when the given nonlinear problem has steep gradients.     

Numerical solution to a stokes interface problem

A numerical algorithm is described for solving the generalized eigenvalue problem arising in the study of the spectrum of a preconditioned operator in the pressure equation derived from a Stokes interface problem. The algorithm is implemented for two finite element schemes. It is tested for a problem with an analytical solution and is applied to spectrum computations in the case of a piecewise constant viscosity. A large number of numerical experiments are analyzed, and recommendations are given for solving the Stokes interface problem in practice.     

Smooth solution for the porous medium equation in a bounded domain

In this paper, we show the short time existence of the smooth solution for the porous medium equations in a smooth bounded domain:

(0.1)     

Numerical solution of a multidimensional two-phase stefan problem

This paper treats a multidimensional two-phase Stefan problem with variable coefficients and mixed type boundary conditions. A numerical method for solving the problem is of fixed domain type, based on a variational inequality formulation of the problem. Numerical solutions are obtained by using piecewise linear finite elements in space and finite difference in time, and by solving a strictly convex minimization problem at each time step. Some computational results are presented.     

Numerical solution of a nonlinear time-optimal control problem

Nonlinear systems with a stationary (i.e., explicitly time independent) right-hand side are considered. For time-optimal control problems with such systems, an iterative method is proposed that is a generalization of one used to solve nonlinear time-optimal control problems for systems divided by phase states and controls. The method is based on constructing finite sequences of simplices with their vertices lying on the boundaries of attainability domains. Assuming that the system is controllable, it is proved that the minimizing sequence converges to an ɛ-optimal solution after a finite number of iterations. A pair {T, u(·)} is called an ɛ-optimal solution if |T − T _opt| − ɛ, where T _opt is the optimal time required for moving the system from the initial state to the origin and u is an admissible control that moves the system to an ɛ-neighborhood of the origin over the time T.     

Numerical solution of a two-dimensional hyperbolic transmission problem

An initial boundary value problem for a two-dimensional hyperbolic equation in two disjoint rectangles is investigated. The existence and uniqueness and a priori estimates for weak solutions in appropriate Sobolev-like spaces are proved. Few finite difference schemes approximating this problem are proposed and analyzed.     

Helical flows of a heated generalized Oldroyd-B fluid subject to a time-dependent shear stress in porous medium

This paper presents an analysis for helical flows of a heated generalized Oldroyd-B fluid subject to a time-dependent shear stress in porous medium, where the motion is due to the longitudinal time-dependent shear stress and the oscillating velocity in boundary. The exact solutions are established by using the sequential fractional derivatives Laplace transform coupled with finite Hankel transforms in terms of generalized G function. Moreover, the effects of various parameters (relaxation time, fractional parameter, permeability and porosity) on the flow and heat transfer are analyzed in detail by graphical illustrations.     

Numerical solution of a problem in the theory of epidemics

We propose a numerical algorithm for a model of the spread of infection in the theory of epidemics. This model involves a threshold for becoming infective and leads to a system of delay-differential equations with the delay function which depends on the values of the unknowns over an interval. The numerical algorithm is based on diagonally implicit multistage integration formulas with stage order equal to the order of the method. This makes possible the efficient evaluation of past values to the accuracy compatible with the requested error tolerance. This algorithm is appicable to the general model without any simplifying assumptions on the parameters of the delay-differential system. Numerical results illustrate the effect of varying various parameters of the model on the spread of infection in a constant population.     

A free boundary problem for a fluid flow in a heterogeneous porous medium

In this paper we study the problem of seepage of a fluid through a porous medium, assuming the flow governed by a nonlinear Darcy law and nonlinear leaky boundary conditions. We prove the continuity of the free boundary and the existence and uniqueness of minimal and maximal solutions. We also prove the uniqueness of theS ₃-connected solution in various situations.     

Exact analytical solution for a thermal boundary layer in a saturated porous medium

The forced convection thermal boundary layer in a porous medium as an analytically tractable special case of a mixed convection problem is considered. It is shown that some general features of the mixed convection solutions reported recently by other authors [B. Brighi, J.-D. Hoernel, On the concave and convex solutions of mixed convection boundary layer approximation in a porous medium, Appl. Math. Lett. (published online, 2005); M. Guedda, Multiple solutions of mixed convection boundary layer approximations in a porous medium, Appl. Math. Lett. (published online, 2005)] can already be recovered from this exactly solvable case.