Evolutionary Differential Equations Research Articles

Mathematical Models of Diffusion in Physiology.

Diffusion is a mass transport phenomenon caused by chaotic thermal movements of molecules. Studying the transport in specific domain is simplified by using evolutionary differential equations for local concentration of the molecules instead of complete information on molecular paths [1]. Compounds in a fluid mixture tend to smooth out its spatial concentration inhomogeneities by diffusion. Rate of the transport is proportional to the concentration gradient and coefficient of diffusion of the compound in ordinary diffusion. The evolving concentration profile c(x,t) is then solution of evolutionary partial differential equation deltac/deltat=DDeltac where D is diffusion coefficient and Delta is Laplacian operator. Domain of the equation may be a region in space, plane or line, a manifold, such as surface embedded in space, or a graph. The Laplacian operates on smooth functions defined on given domain. We can use models of diffusion for such diverse tasks as: a) design of method for precise measurement of receptors mobility in plasmatic membrane by confocal microscopy [2], b) evaluation of complex geometry of trabeculae in developing heart [3] to show that the conduction pathway within the embryonic ventricle is determined by geometry of the trabeculae.

Some New Aspects of Quantum Gravity

We have proposed the quantization of the gravitational field in a synchronous reference frame taking as independent position fields, the six spatial components of the metric tensor. The Einstein-Hilbert Lagrangian is quadratic in the space time derivatives of these metric tensor components and hence in particular, the momentum fields become linear functions of the space-time derivatives of the position fields. It is this fact that gives a simple form to the Hamiltonian density of the gravitational field in a synchronous frame, this simple form of the Hamiltonian being a quadratic function of the momentum fields with a shift that is linear in the spatial derivatives of the metric, very much like the Hamiltonian of a non-relativistic particle moving in a vector potential. Differential equations for the gravitational field propagator are derived and we explain how approximations to this propagator can be derived and used to deduce the graviton propagator corrections caused by nonlinear interactions of the graviton field with itself. We explain how this corrected graviton propagator can be used to deduce how much mass the graviton acquires due to these self-interactions of cubic and higher order. We then consider the important problem involving the coupling of a nonlinear field theory described by its Lagrangian density to a quantum noisy bath and explain how the resulting Hamiltonian of the field plus bath can be used to derive the Hudson-Parthasarathy noisy Schrodinger equation (HPS) which is a quantum stochastic differential equation for the joint unitary evolution of the field interacting with the noisy bath. We explain this in the context of gravity coupled to a noisy bath like a noisy electromagnetic field. The HPS equation contains linear as well as quadratic terms in the white bath noise with the linear terms representing quantum annihilation and creation/quantum Brownian motion process differentials and the quadratic terms representing quantum conservation/Poisson processes differentials. Finally we explain how using Feynman path integrals for fields for evaluating the quantum effective action produced by higher order cumulants of the current field, we can calculate corrections to the quantum effective action produced by higher order cumulants of the current field and hence demonstrate how gauge symmetries of the classical action get broken when we pass over to the quantum effective action with additional symmetry breaking terms produced by the presence of higher order cumulants of the current. This kind of approximate symmetry breaking is known to give masses to massless particles or more generally, corrections to the masses of already massive particles and we illustrate this idea in the context of interactions of the gravitational field with a random electromagnetic field being regarded as the current. This interaction is the standard Maxwell action used in general relativity. The drawback of our approach to quantum gravity is that is its not diffeomorphic invariant since we have chosen our frame to be always synchronous. Further work on how one can incorporate interactions of the gravitational field with a random non-Abelian gauge field is in progress which becomes important because it generates non only quadratic but also cubic and fourth degree terms in the gauge field when it interacts with gravity.

Accelerated particle beams in a 3D simulation of the quiet Sun

Context. Charged particles are constantly accelerated to non-thermal energies by the reconnecting magnetic field in the solar atmosphere. Our understanding of the interactions between the accelerated particles and their environment can benefit considerably from three-dimensional atmospheric simulations that account for non-thermal particle beam generation and propagation. In a previous publication, we presented the first results from such a simulation, which considers quiet Sun conditions. However, the original treatment of beam propagation ignores potentially important phenomena such as the magnetic gradient forces associated with a converging or diverging magnetic field. Aims. Here we present a more general beam propagation model incorporating magnetic gradient forces, the return current, acceleration by the ambient electric field, corrected collision rates due to the ambient temperature, and collisions with heavier elements than hydrogen and the free electrons they contribute. Neglecting collisional velocity randomisation makes the model sufficiently lightweight to simulate millions of beams. We investigate how each new physical effect in the model changes the non-thermal energy transport in a realistic three-dimensional atmosphere. Methods. We applied the method of characteristics to the steady-state continuity equation for electron flux to derive ordinary differential equations for the mean evolution of energy, pitch angle, and flux with distance. For each beam, we solved these numerically for a range of initial energies to obtain the evolving flux spectrum, from which we computed the energy deposited into the ambient plasma. Results. Magnetic gradient forces significantly influence the spatial distribution of deposited beam energy. The magnetic field converges strongly with depth in the corona above loop footpoints. This convergence leads to a small coronal peak in deposited energy followed by a heavy dip caused by the onset of magnetic mirroring. Magnetically reflected electrons carry away 5 to 10% of the injected beam energy on average. The remaining electrons are relatively energetic and produce a peak in deposited energy below the transition region a few hundred kilometres deeper than they would in a uniform magnetic field. A diverging magnetic field at the beginning of the trajectory, which is common in the simulation, enhances the subsequent impact of magnetic mirroring. The other new physical effects do not qualitatively alter the picture of non-thermal energy transport for the atmospheric conditions under consideration.

Evolution Equations with Liouville Derivative on R without Initial Conditions

New classes of evolution differential equations with the Liouville derivative in Banach spaces are studied. Equations are considered on the whole real line and are not endowed by the initial conditions. Using the methods of the Fourier transform theory, we prove the unique solvability in the sense of classical solutions for the equation solved with respect to the Liouville fractional derivative with a bounded operator at the unknown function. This allows us to obtain the analogous result for the equation with a linear degenerate operator at the fractional derivative and with a spectrally bounded pair of operators. Abstract results are applied to obtain new results on the unique solvability of systems of ordinary differential equations, boundary problems to partial differential equations, and systems of equations.

Singular Stochastic Differential Equations for Time Evolution of Stocks Within Non-white Noise Approach

Singular Stochastic Differential Equations for Time Evolution of Stocks Within Non-white Noise Approach

Influence of pore-crack environment on heat propagation of underground coalfield fire: A case study in Daquanhu, Xinjiang, China

The development of pores and cracks in coalfield fire area promote the abnormal temperature, the greenhouse effect, and the release of toxic gases. It is of great significance to study the dynamic distribution characteristics of temperature field and concentration field of typical gas products in a coalfield fire area under different pore-crack environments for revealing the spreading law of underground coalfield fire. In this paper, the combined effect of coal combustion chemical reaction, heat energy transmission, flue gas seepage and pore-crack environment were considered. Taking Daquanhu coalfield fire area as an example, an optimized multi-field coupling model, which included the chemical reaction of coal combustion, convection heat transfer in cracks, heat transfer in porous media, gas seepage and improved partial differential equation of pore-crack evolution was constructed. The coalfield fires area was divided into combustion zone, caving zone, sagging subsidence zone, crack zone. The evolution law of temperature field, CO2 concentration field in the coalfield fire area under different air leakage and geological crack conditions was analysed. The results showed that the temperature development of coalfield fire is divided into rapid heating stage and stable development stage. At the rapid heating stage, the greater the air leakage flow was, the faster the maximum temperature rise in the combustion zone. At the stable development stage, with the air leakage decreased, the thermal buoyancy increased. The greater the number of cracks was, the greater the area of the drying zone (673.15 K < T < 1073.15 K) closed to the crack zone. The influence of crack environment on heat flux in the sagging subsidence zone and crack zone was obvious. The total heat flow in the crack zone was about 25 times that in the caving zone. The relationship between the maximum temperature in the combustion zone and the CO2 concentration at the crack outlet showed an exponential model (ExpGro1 exponential model). The area spread by coalfield fire increased linearly with the rang of surface temperature abnormal. The research results provided a basis for judging the spread trend of coalfield fire and extracting coalfield fire of heat energy.

Lie symmetry analysis, numerical solutions by spectral method and travelling wave solutions of a (5+1)-dimensional partial differential equation

ABSTRACT The purpose of this research is to investigate the Lie symmetries and the numerical solution of the (5 + 1)-dimensional evolutionary differential equation using the Chebyshev spectral method. Also, the exact solution of this equation is presented by the traveling wave method. These equations play a very important role in mathematical physics and engineering sciences. The implemented algorithm is quite efficient and practically suitable for these problems. The use of Maple and MATLAB software allows us to perform complex and boring calculations. The most distinctive aspect of this method is that no prescribed assumptions are required, and enormous computational effort is reduced and rounding errors are avoided. Our proposed methods for checking and solving the considered equation show that this method is very accurate and practical.

On Local Unique Solvability for a Class of Nonlinear Identification Problems

Nonlinear identification problems for evolution differential equations, solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are studied. An equation of the considered class contains a linear unbounded operator, which generates analytic resolving families for the corresponding linear homogeneous equation, and a continuous nonlinear operator, which depends on lower-order Dzhrbashyan–Nersesyan derivatives and a depending on time unknown element. The identification problem consists of the equation, Dzhrbashyan–Nersesyan initial value conditions and an abstract overdetermination condition, which is defined by a linear continuous operator. Using the contraction mappings theorem, we prove the unique local solvability of the identification problem. The cases of mild and classical solutions are studied. The obtained abstract results are applied to an investigation of a nonlinear identification problem to a linearized phase field system with time dependent unknown coefficients at Dzhrbashyan–Nersesyan time-derivatives of lower orders.

Application of the Constrained Formulation to the Nonlinear Sloshing Problem Based on the Arbitrary Lagrangian–Eulerian Method

Abstract This study deals with an application of constrained formulation to a nonlinear sloshing problem based on the arbitrary Lagrangian–Eulerian finite element method (ALE). The ALE method incorporates a discretized form of equations of motion with mesh updating algorithms in order to prevent a problem of mesh distortion. This paper focuses on an analytical aspect of such treatments as constrained systems in the formulation of the ALE method. Since the mesh updating algorithms give algebraic relations for nodal coordinates, this study treats these relations as constraints. Then, we introduce formulation for constrained systems based on the method of Lagrange multipliers. As a result of this formulation, equations of motion are given by differential algebraic equations (DAEs) consisting of differential equations for time evolution of physical quantities and algebraic equations (constraints). The present method can be classified into a kind of augmented formulation. Moreover, we present a size-reduction technique used in the Newton–Raphson method in order to remove a part of the redundant degrees-of-freedom in the iterative procedures, because the resulting set of DAEs involves a larger number of unknowns than the minimal number of degrees-of-freedom due to the introduction of the constrained formulation. The derived reduced system still holds the physically essential part of equations of motion for the sloshing problem. Even though it is not reduced to the minimal degrees-of-freedom, it does not involve algebraically complicated and inefficient procedures in computation. In addition, this study presents a method to introduce damping effects defined in the modal space into the FEM models. The proposed approach is validated by comparisons with experimental data in the time domain analysis.

Rimming flow in a rotating horizontal cylinder with phase change at the interface

The two-dimensional solutions and stability analysis are presented for an evaporating thin viscous liquid film flowing inside a uniformly heated rotating horizontal cylinder. A non-linear, fourth-order, partial differential evolution equation is obtained by simplifying mass, momentum, and energy conservation equations within the lubrication approximation. The effect of evaporation, gravity, viscous drag, surface tension, thermocapillary stress, and intermolecular forces has been taken into account. The numerical solutions of the model are validated against the existing experimental as well as the numerical results, along with the analytical result for the limiting cases of the present model. The film thickness model is solved to elucidate two-dimensional spatiotemporal solutions and their stability for a wide range of thermal and other parameters. The evaporative mass flux at the liquid–air interface results in unsteady solutions which are oscillatory in nature, and the amplitude of the oscillations increases with an increase in the evaporative flux. The film ruptures after some time and the rupture time is found to be inversely proportional to the evaporation number, a non-dimensional number quantifying the rate of evaporation. The linear stability analysis shows that the thermocapillary stresses as well as the long-range molecular forces destabilize the film. A negative disjoining pressure is shown to reduce the rupture time and vice versa. Evaporation (condensation) plays a destabilizing (stabilizing) role in the thin film flow. Non-linear computations are carried out for the steady profiles, validating the growth rates obtained from the linear stability analysis.

Joint Distribution of Ages of Information in Networks

We study a general setting of status updating systems in which a set of source nodes provide status updates about some physical process(es) to a set of monitors. The freshness of information available at each monitor is quantified in terms of the Age of Information (AoI), and the vector of AoI processes at the monitors (or equivalently the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">age vector</i> ) models the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">continuous</i> state of the system. While the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">marginal</i> distributional properties of each AoI process have been studied for a variety of settings using the stochastic hybrid system (SHS) approach, we lack a counterpart of this approach to systematically study their <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">joint</i> distributional properties. Developing such a framework is the main contribution of this paper. In particular, we model the discrete state of the system as a finite-state continuous-time Markov chain, and describe the coupled evolution of the continuous and discrete states of the system by a piecewise linear SHS with linear reset maps. Using the notion of tensors, we first derive first-order linear differential equations for the temporal evolution of both the joint moments and the joint moment generating function (MGF) for an arbitrary set of age processes. We then characterize the conditions under which the derived differential equations are asymptotically stable. The generality of our framework is demonstrated by recovering several existing results as special cases. Finally, we apply our framework to derive closed-form expressions of the stationary joint MGF in a multi-source updating system under non-preemptive and source-agnostic/ source-aware preemptive in service queueing disciplines.

Enthalpy effect on the kinetics of concurrent nucleation and chemical aging of aqueous organic aerosols: The stage of thermal relaxation

The size and composition distribution of an ensemble of aqueous organic droplets, evolving via nucleation and concomitant chemical aging, may be affected by the latent heat of condensation and enthalpy of heterogeneous chemical reactions, so the temperature of the droplet may deviate from the air temperature and thus become an independent variable of its state (additional to its size and composition variables). Using the formalism of the classical nucleation theory, we derive a partial differential equation for the temporal evolution of the distribution of an ensemble of such droplets with respect to all their variables of state via Taylor series expansions of the corresponding multidimensional discrete equation of balance, describing the material and heat exchange between droplets and air. The resulting kinetic equation goes beyond the framework of the Fokker-Planck approximation with respect to the temperature variable. A hierarchy of time scales of nonisothermal nucleation and concomitant chemical aging of aqueous organic aerosols is established and an analytical description of their thermal relaxation stage is developed, allowing one to estimate the characteristic time of the establishment of the equilibrium distribution of aerosol particles with respect to their temperatures. Theoretical results are illustrated with numerical calculations for the concurrent nucleation and chemical aging of model aqueous hydrophilic-hydrophobic organic aerosols in air.

On the Existence of Optimal Control in the Problem of Optimizing the Lowest Coefficient of a Semilinear Evolutionary Equation

The paper studies the problem of optimizing the lowest coefficient understood as a function with values in a Banach space, which enters linearly into an abstract semilinear pseudoparabolic evolutionary differential equation in a Banach space. For this problem, an existence theorem for an optimal control is proved. Due to the nonlinearity of the equation under study, the author uses his previous results on the total preservation of the unique global solvability (on the totally global solvability) and on the estimation of solutions for similar equations. This estimate turns out to be significant in the course of the study. As an example, the Oskolkov’s hydrodynamic system of equations is considered.

Lifetime Prediction with Combined Hygro‐Thermo‐Mechanical Influences on Thick Layer Adhesives

AbstractSemi‐structural bonded joints of steel components are exposed to long‐term stresses in addition to water diffusion, temperature changes and varying external forces during operation. These hygro‐thermomechanical loadings lead to successive degradation of stiffness and strength due to damage resulting from chemical aging and fatigue processes, causing failure of the bonded joint when its service life is reached. In the following contribution, a methodology is presented to predict the service life of hygro‐thermo‐mechanically loaded semistructural adhesive joints with transient FE simulation. The lifetime prediction is based on a constitutive model with a viscoelastic and a damage part. The first part for viscoelasticity consists of the generalized Maxwell model, in which the effects of temperature changes and varying humidity on the viscoelastic properties of the adhesive bond are captured by the time‐temperature and time‐water concentration shifts. The second part for the material damage is based on an ordinary differential equation for the damage evolution. It consists of a creep and a moisture damage part for the void developments caused by the local water concentration due to mechanical stress and chemical aging. Both damage parts are multiplied by an Arrhenius‐type function to account for the effect of temperature change on the defect growth. The interaction of the creep and the moisture damage part is studied by applying the analytical and numerical solution of the damage differential equation. All parameters of the damage model are determined by using fracture times from creep tests under different climatic conditions and loading levels. The local water concentration in the adhesive is calculated by Fick's model and a concentration boundary condition. The diffusion parameters are determined by gravimetric measurements. A numerical example is setup to demonstrate the lifetime prediction methodology and to show the influence of viscoelasticity on the predicted lifetimes for different hygro‐thermo‐mechanical loading cases.

A third transition in science?

Since Newton, classical and quantum physics depend upon the 'Newtonian paradigm'. The relevant variables of the system are identified. For example, we identify the position and momentum of classical particles. Laws of motion in differential form connecting the variables are formulated. An example is Newton's three laws of motion. The boundary conditions creating the phase space of all possible values of the variables are defined. Then, given any initial condition, the differential equations of motion are integrated to yield an entailed trajectory in the prestated phase space. It is fundamental to the Newtonian paradigm that the set of possibilities that constitute the phase space is always definable and fixed ahead of time. This fails for the diachronic evolution of ever-new adaptations in any biosphere. Living cells achieve constraint closure and construct themselves. Thus, living cells, evolving via heritable variation and natural selection, adaptively construct new-in-the-universe possibilities. We can neither define nor deduce the evolving phase space: we can use no mathematics based on set theory to do so. We cannot write or solve differential equations for the diachronic evolution of ever-new adaptations in a biosphere. Evolving biospheres are outside the Newtonian paradigm. There can be no theory of everything that entails all that comes to exist. We face a third major transition in science beyond the Pythagorean dream that 'all is number' echoed by Newtonian physics. However, we begin to understand the emergent creativity of an evolving biosphere: emergence is not engineering.

Asymptotic behavior of stochastic functional differential evolution equation

In this work we study the long time behavior of nonlinear stochastic functional-differential equations in Hilbert spaces. In particular, we start with establishing the existence and uniqueness of mild solutions. We proceed with deriving a priory uniform in time bounds for the solutions in the appropriate Hilbert spaces. These bounds enable us to establish the existence of invariant measure based on Krylov-Bogoliubov theorem on the tightness of the family of measures. Finally, under certain assumptions on nonlinearities, we establish the uniqueness of invariant measures.

On Complete Group Classification of Time Fractional Systems Evolution Differential Equation with a Constant Delay

A fractional order system of evolution partial differential equations with a constant delay is considered. By exploiting the Lie symmetry method, we give a complete group classification of the system. Furthermore, we establish the corresponding symmetry reductions and construct some analytical solutions to the system.

Numerical solution for a class of evolution differential equations with [formula omitted]-Laplacian and memory

In this paper, we study a partial integral differential equation with p-Laplacian using a mixed finite element method. Two stable and convergent fixed point schemes are proposed to solve the nonlinear algebraic system. Using the implementation of the method in Matlab environment, we numerically analyse the convergence with an example. Some other examples, which illustrate several asymptotic behaviours and some localization effects of the solutions, are presented.

On solutions of a class of neutral evolution equations with discrete nonlocal conditions

This paper studies the existence, regularity, and asymptotic properties of solutions for a class of neutral differential evolution equations with nonlocal initial conditions on an infinite interval. The existence and regularity of solutions of the considered equation are, respectively, investigated by the theory of fractional power operators and fixed point theorems under some assumptions for nonlinear functions. Then, under suitable conditions, asymptotic properties, including stability and existence of global attracting sets and quasi-invariant sets of mild solutions, are also discussed in the context. Finally, an example is presented to illustrate the applications of the obtained abstract results.

Dynamics of Evolutionary Differential Equations with Several Spatial Variables

The article is devoted to a method for constructing exact and approximate solutions of evolutionary partial differential equations with several spatial variables. The method is based on the theory of completely integrable distributions. Examples of applying this method to calculating exact solutions of the generalized Kolmogorov–Petrovskii–Piskunov–Fishev equations with two space variables are given.