Neutral functional differential equations and systems of conservation laws

Periodic solutions for a class of neutral functional differential equations with infinite delay

An efficient IMEX method for nonlinear functional differential equations with state-dependent delay

Almost periodic solutions of neutral functional differential equations

Oscillation theorems for first-order nonlinear neutral functional differential equations

<p style='text-indent:20px;'>In this paper we study nonlinear autonomous retarded functional differential equations; that is, functional equations where the time derivative may depend on the past values of the variables. When the nonlinearities in such equations are comprised of elementary functions, we give a constructive proof of the existence of an embedding of the original coordinates yielding a polynomial differential equation. This embedding is a topological conjugacy between the semi-flow of the original differential equation and the semi-flow of the auxiliary polynomial differential equation. Further dynamical features are investigated; notably, for an equilibrium or a periodic orbit and its embedded counterpart, the stable and unstable eigenvalues have the same algebraic and geometric multiplicity.</p>

DOUGHERTY, E.L., CALIFORNIA RESEARCH CORP., LA HABRA, CALIF. JUNIOR MEMBER AIME Abstract A phenomenological theory for a one-dimensional unstable miscible displacement similar in type to the Buckley-Leverett model but including the effects of mixing is proposed An equation giving the fractional flow of pure solvent through an oil phase containing dissolved solvent is derived including the effects of gravity and nonhomogeneities. Numerical integration of the resulting pair of nonlinear hyperbolic partial differential equations gives volumes of dissolved and undissolved solvent as functions of space and time. Parameters in the model which characterize mixing due to dispersion were determined from experimental data; the parameters correlated with viscosity ratio. The results indicate that the rate of dispersive mixing is proportional to volumetric flow rate Incorporated into our equations were Koval's heterogeneity factor H, which characterizes mixing due to channeling. This allowed satisfactory predictions of displacement behavior observed in horizontal floods in nonhomogeneous cores; but in most cases the values of H which we used were considerably larger than those used by Koval. Introduction It has been shown that for favorable viscosity ratios, the diffusion equation with convection satisfactorily describes the behavior of a miscible displacement in a porous media. Numerous attempts to develop a satisfactory mathematical description for the case of unfavorable viscosity ratio have been reported, but they have met with only partial success. These attempts, which are reviewed by Koval, have taken two approaches:development of a flow model akin to the Buckley-Leverett approach for immiscible displacement neglecting the effects of mixing andsimultaneous application of Darcy's Law for flow and the diffusion equation with a convection term for mass transfer. We set out to construct a mathematical model of Type 1 including, though, the effects of mixing. Based upon a set of hypotheses, we derived a pair of non-linear hyperbolic partial differential equations which describe in a one-dimensional system the combined effects of flow and mixing. To work with these equations it was necessary to develop a fractional flow formula. The combined system was integrated numerically using the method of characteristics.In our equations the mixing process is characterized by four parameters. Three of these, labeled beta, p and p, account for dispersive type mixing. The fourth is the heterogeneity factor H, proposed by Koval to account for channeling due to nonhomogeneities in the porous media. Values of the parameters which would cause agreement between theory and experiment were determined by trial-and-error for miscible floods conducted in horizontal cores of both homogeneous and heterogeneous materials. Calculations were also performed for vertical floods in homogeneous cores.The purpose of this paper isto present the details of the mathematical analysis,to present the results of the calculations,to consider what light the results shed on the mixing process in an unstable miscible displacement, andto provide a firmer foundation for the correlation technique developed by Koval for predicting the behavior of unstable miscible floods. STATEMENT OF PROBLEM The problem is to construct a mathematical model which describes in one dimension the observed behavior in an unstable miscible displacement. The approach is phenomenological in that the equations are based on assumptions which violate certain physical precepts known to apply to the displacement phenomenon. However, the assumptions do allow us to account mathematically for the more essential phenomena. The work is of value if we can quantitatively predict experimental results which heretofore could not be predicted, even though the synthetic nature of the model belies complete explanation of the observations.We assume that the system is comprised of two contiguous flowing phases. One phase, which we call free solvent, has the properties of pure solvent. We call the fraction of the pore volume occupied by this phase the solvent saturation, designated s. SPEJ P. 155^

Stability in distribution of neutral stochastic functional differential equations with Markovian switching

The viscoelastic behavior of hydrated soft biological tissue is strongly influenced by the mechanical interaction of its interstitial fluid with the solid phase. The classical uniaxial steady biphasic theory for soft tissue with high moisture content under confined compression is extended by replacing the Darcy steady flow assumption with an unsteady internal fluid flow model that involves an unsteady flow coefficient based on non-equilibrium thermodynamics. The finite deformation, non-equilibrium, unsteady biphasic viscoelastic analysis derived from conservation of linear momentum, which includes the mass terms neglected in some steady models, produces a nonlinear hyperbolic partial differential equation for the solid phase displacements, with a finite propagation velocity of the disturbance from the plunger in contrast to the instantaneous propagation of the classical parabolic steady description for cartilage. For confined compression of a small cylinder of hydrated soft biological tissue, the new hyperbolic equation is solved for the solid phase local displacements during both loading and relaxation as a function of time and position by a finite difference scheme of Shampine. A parametric study shows the influence of the unsteady coefficient on the predicted behavior, for both a linear elastic and a linear viscoelastic solid phase and for a constant permeability, by a comparison of the respective solid phase displacement distributions at the end of a constant rate plunger displacement at 0.001/s, 1/s and 1,000/s. Models assuming the Darcy relation predict physically realistic responses only for slow rates. The concavity of the stress–time curve for the solid phase at the plunger during a constant displacement of the plunger even at slow rates may be adjusted from softening to hardening by increasing the unsteady coefficient to increase the drag force exerted by the fluid on the solid phase. Because of unsteady fluid flow, overshoot of the solid phase displacement may occur initially during relaxation and also again near the end of the approach to equilibrium. A Zener standard linear viscoelastic solid phase slows the relaxation to equilibrium in comparison with a linear elastic solid phase.

The work presents the analysis of the free boundary value problem related to the one-dimensional invasion model of new species in biofilm reactors. In the framework of continuum approach to mathematical modelling of biofilm growth, the problem consists of a system of non-linear hyperbolic partial differential equations governing the microbial species growth and a system of semi-linear elliptic partial differential equations describing the substrate trends. The model is completed with a system of elliptic partial differential equations governing the diffusion and reaction of planktonic cells, which are able to switch their mode of growth from planktonic to sessile when specific environmental conditions are found. Two systems of non-linear differential equations for the substrate and planktonic cells mass balance within the bulk liquid are also considered. The free boundary evolution is governed by a differential equation that accounts for detachment. The qualitative analysis is performed and a uniqueness and existence result is presented. Furthermore, two special models of biological and engineering interest are discussed numerically. The invasion of Anammox bacteria in a constituted biofilm inhabiting the deammonification units of the wastewater treatment plants is simulated. Numerical simulations are run to evaluate the influence of the colonization process on biofilm structure and activity.

This paper is devoted to the stability of neutral type functional differential equations whose principal terms are small in a certain sense. We derive the explicit conditions for the exponential and absolute stabilities, as well as for the L p -stability. Besides, solution estimates for the considered equations are established. They provide bounds for the regions of attraction of steady states. We also consider some classes of equations with neutral type linear parts and nonlinear causal mappings. These equations include differential, differential-delay, integro-differential, and other traditional equations. The main methodology presented in the paper is based on a combined usage of the recent norm estimates for matrix-valued functions with the generalized Bohl–Perron principle for neutral type functional differential equations. Our approach enables us to apply the well-known results of the theory of matrices to the stability analysis.

OSCILLATIONS OF NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS GENERATED BY RETARDED ACTIONS

Characteristics of lithium chloride in rotary heat and mass exchangers

We justify a method for reducing a wide class of nonlinear equations (including several partial differential equations) to ordinary differential equations in locally convex spaces. The possibilities of this method are demonstrated by an example of a class of nonlinear hyperbolic partial differential equations.

This book is devoted to the multiplicative differential calculus. Its seven pedagogically organized chapters summarize the most recent contributions in this area, concluding with a section of practical problems to be assigned or for self-study. Two operations, differentiation and integration, are basic in calculus and analysis. In fact, they are the infinitesimal versions of the subtraction and addition operations on numbers, respectively. From 1967 till 1970, Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, moving the roles of subtraction and addition to division and multiplication, and thus established a new calculus, called multiplicative calculus. It is also called an alternative or non-Newtonian calculus. Multiplicative calculus can especially be useful as a mathematical tool for economics, finance, biology, and engineering. Multiplicative Differential Calculus is written to be of interest to a wide audience of specialists such as mathematicians, physicists, engineers, and biologists. It is primarily a textbook at the senior undergraduate and beginning graduate level and may be used for a course on differential calculus. It is also for students studying engineering and science.   Authors Svetlin G. Georgiev is a mathematician who has worked in various areas of the study. He currently focuses on harmonic analysis, functional analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, integral equations, and dynamic calculus on time scales. He is also the author of Dynamic Geometry of Time Scales (CRC Press). He is a co-author of Conformable Dynamic Equations on Time Scales, with Douglas R. Anderson (CRC Press). Khaled Zennir earned his PhD in mathematics from Sidi Bel Abbès University, Algeria. He earned his highest diploma in Habilitation in Mathematics from Constantine University, Algeria. He is currently Assistant Professor at Qassim University in the Kingdom of Saudi Arabia. His research interests lie in the subjects of nonlinear hyperbolic partial differential equations: global existence, blowup, and long-time behavior. The authors have also published: Multiple Fixed-Point Theorems and Applications in the Theory of ODEs, FDEs and PDE; Boundary Value Problems on Time Scales, Volume 1 and Volume II, all with CRC Press.

The self-similar solutions for converging spherical and cylindrical strong shock waves in a non-ideal gas satisfying the equation of state of the Mie-Gruneisen type are investigated. The equations governing the flow, which are highly non-linear hyperbolic partial differential equations, are first reduced to a Poincare-type ordinary differential equation with suitable approximation. Such an approximation helps in obtaining the self-similar solutions and the similarity exponent numerically by phase-plane analysis.