Class Of Diffusion Equations Research Articles

A hybrid local energy formulation for plates mid-frequency flexural vibrations

Many models based on heat transfer equations have been developed in order to predict the vibroacoustic behavior of structures at medium and high frequencies. Propagation of vibrations in a plate or in coupled plates can be estimated using a diffusion equation. In this paper, a hybrid solution of this equation is developed to improve the predicted energy levels. The power injected inside the structure is thus calculated more precisely taking into account the interferences within the direct field. The reverberant field, due to wave scattering on plates boundaries, is assumed to be the summation of incoherent waves. This hybrid approach allows not only a better calculation of the power injected into the structure but also an accurate spatial description of vibrational fields when compared to the classical diffusion equation.

Description of dynamical properties of magnetisation using dedicated delayed functions

We propose a new approach to calculate the evolution of the induction B( t) in dynamically magnetized soft conducting magnetic samples. This method does not imply any assumption about the magnetization mechanisms. The quasi-static curve μ( H m) ( H m=maximum H and μ=average static permeability) and a set of cyclic characterizations at different frequencies f provide the basic information which allows the prediction of the response B ar( t) to an arbitrary monotonous—or cyclic—waveform H a( t). We firstly theoretically establish the validity of the method considering a sample of thickness e and conductivity σ, obeying the classical diffusion equation, excited with a Heaveside field. The method is then applied to a real iron–vobalt sample.

Kinetics for Supercritical CO<sub>2</sub> Debinding of Injection Molded ZrO<sub>2</sub>

Mercury porosimeter and scanning electron microscope (SEM) were used to analyze the pore structures evolution and distributions for supercritical CO2 debinding of injection molding ceramics. Classical diffusion equation was used to describe the mass transfer of supercritical CO2 debinding of the injection molded ZrO2 ceramics. The behavior and kinetics of the debinding were studied and analyzed. Results show that the solubility and diffusivity of soluble binder are the key factors in supercritical CO2 debinding, while the diffusivity is a dominant factor. The calculation data from the theoretical model are consistent with the experiment under the condition of enough long debinding time. It is shown that the diffusivity can be obtained by simple theoretical model combined with experimental data. The extraction rate and the extraction kinetics of the process can be predicted using the theoretical model.

Analytical solution for the space fractional diffusion equation by two-step Adomian Decomposition Method

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical superdiffusive problems in fluid flow, finance and other areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by two-step Adomian Decomposition Method (TSADM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form and then their solutions have been represented graphically. Two examples, the first one is one-dimensional and the second one is two-dimensional fractional diffusion equation, are presented to show the application of the present technique. The solutions obtained by the standard decomposition method have been numerically evaluated and presented in the form of tables and then compared with those obtained by TSADM. The present TSADM performs extremely well in terms of efficiency and simplicity.

A SPLITTING TYPE ALGORITHM FOR NUMERICAL SOLUTION OF PDES OF FRACTIONAL ORDER

Fractional order diffusion equations are generalizations of classical diffusion equations, treating super‐diffusive flow processes. In this paper, we examine a splitting type numerical methods to solve a class of two‐dimensional initial‐boundary value fractional diffusive equations. Stability, consistency and convergence of the methods are investigated. It is shown that both schemes are unconditionally stable. A numerical example is presented.

Monte Carlo simulations of drug release from matrices with periodic layers of high and low diffusivity

We have studied drug release from matrices with periodic layers of high and low diffusivity using Monte Carlo simulations. Despite the fact, that the differential equations relevant to this process have a form that is quite different from the classical diffusion equation with constant diffusion coefficient, we have found that the Weibull model continues to describe the release process as well as in the case of the “classical” diffusion controlled drug release. We examine the similarities and differences between release from matrices with periodic layers and matrices with random mixtures of high and low diffusivity area and show that the periodic geometrical arrangement of the low diffusivity areas has an influence in the release profile which is negligible for low diffusivity ratios, but becomes important in the case of high diffusivity ratios and for intermediate values of the periodic “length”. Such an arrangement in periodic layers leads to Weibull exponent a which are lower than those of the corresponding random arrangement and exponents b which are higher than those of the random case.

Improving dual-porosity-medium approaches to account for karstic flow in a fractured limestone: Application to the automatic inversion of hydraulic interference tests (Hydrogeological Experimental Site, HES – Poitiers – France)

A reassessment-improvement of a previous work by the authors is proposed on the inversion of hydraulic interference testing by means of dual-porosity-medium approaches. The tests have been performed over a fractured limestone aquifer (Poitiers, France) showing a preferential and rapid flow path along open karstic drains. All drawdown curves are merged together both in amplitude and time irrespective of their location relative to the single pumped well and classical dual-porosity-medium approaches are unable to mimic this behavior. A pressure wave propagation by advection is added to the classical diffusion equations of dual-porosity-medium in order to stress the medium almost evenly whatever the distance from the pumped well. The problem is set up to keep the continuous dual-porosity-medium framework with only two state variables, i.e. the heads in fractures and matrix. Even though the wave velocity is not well-conditioned because the ratio of the volume of drains to the total volume of the medium is unknown, automatic inversion remains feasible. The dual-porosity-medium (without the drains) can be homogeneous or fractal, having in the latter case hydraulic parameters ruled by scaling power laws. In both cases, it is shown that adding an advection term allows to invert accurately all experimental drawdown curves. Statistical fluctuations of the sought hydraulic parameters are quite small and do not show any trend with respect to the distance. Thus, modifying a dual-porosity-medium approach by adding an advection term allows for the homogenization of a reservoir that combines the general drainage of a fracture-matrix system and a rapid drainage through karstic conduits.

Application of thermomechanics equations in describing chloride extraction from concrete

During the extraction of chlorides from concrete there occur processes involving transfer of mass and electric charge via pore solution components. On the basis of the relations which exist between the concentrations of these components, the electric field intensity, the temperature, and the mechanical stress, thermomechanics equations of a multicomponent medium have been used for analytical description. On the basis of equations for the balance of mass, electric charge, momentum, energy and entropy inequality, equations of ion flows have been determined, which are justified in the general effect of the electric field action on the concrete covering. The occurrence of processes which are recognised as dominant ones during the chloride extraction has allowed us to significantly simplify the equations and reduce them to formulas permitting a practical use. By approximating the electrochemical potentials of the leading ions via the linear function of concentration and electric field potential, a flow equation has been determined which has a shape similar to the classical diffusion equation. In order to define the electrodiffusion coefficient occurring in this equation an inverse problem has been formulated and then numerical values of this coefficient have been evaluated using the results of experimental tests.

Application of modified decomposition method for the analytical solution of space fractional diffusion equation

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical super diffusive problems in fluid flow, finance and others areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by Modified decomposition method (MDM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form. The decomposition series analytic solution of the problem is quickly obtained by observing the existence of the self-cancelling “noise” phenomenon. Two examples, the first one is one-dimensional and the second one is two-dimensional fractional diffusion equation, are presented to show the application of the present technique. The present method performs extremely well in terms of efficiency and simplicity.

Generalized diffusion equation for anisotropic anomalous diffusion

Motivated by studies of comblike structures, we present a generalization of the classical diffusion equation to model anisotropic, anomalous diffusion. We assume that the diffusive flux is given by a diffusion tensor acting on the gradient of the probability density, where each component of the diffusion tensor can have its own scaling law. We also assume scaling laws that have an explicit power-law dependence on space and time. Solutions of the proposed generalized diffusion equation are consistent with previously derived asymptotic results for the probability density on comblike structures.

Quantum decoherence and El Naschie’s complex temporality

Abstract It is well-known that the Schrodinger equation reduces to a classical diffusion equation by means of Wick rotation (t → it), suggesting a correspondence between quantum and classical mechanics. Nonetheless, this result does not admit a clear conceptual interpretation. In the framework of his fractal E ( ∞ ) space-time theory, El Naschie showed that great conceptual advantage could be achieved by extending the imaginary time, it, to a perfectly symmetric, complex conjugate time 0 ± it. In this note we show through a simple analysis, involving formal analytic continuation (t → 0 ± it), that El Naschie’s time complexification provides the basis for a physical interpretation of the correspondence between quantum and classical mechanics in terms of quantum decoherence. We find that decoherent states inevitably arise due to time symmetry breaking as we go from the micro Cantorian space-time, where the two symmetric times, 0 + it and 0 − it, coexist to our 4-dimensional smooth space-time, where t is the only time.

A continuous time random walk approach to transient flow in heterogeneous porous media

We propose a new physical interpretation of the diffusion process for the piezometric head h(x, t) in heterogeneous aquifers based on the continuous time random walk (CTRW) theory. For the typical heterogeneities considered in this work, we find that a CTRW based diffusion equation for h(x, t) provides better fits to the transient flow simulations than the classical diffusion equation (DE). The DE is found to be a special case of the CTRW diffusion equation. The results of this work have clear implications for the interpretation of pumping tests and what information can be extracted from them.

Elliptic equation for random walks. Application to transport in microporous media

We consider a process of random walks with arbitrary residence time distribution. We show that in many cases this process may not be described by the classical (Fick) parabolic diffusion equation, but an elliptic equation. An additional term proportional to the second time derivative takes into account the distribution of the residence times of molecules in pores. The new elliptic diffusion equation is strictly derived by the operator approach. A criterion showing where the new equation should be applied instead of the standard diffusion equation is obtained. Boundary conditions are studied and a principle for selection of a unique bounded solution is formulated. Fundamental solutions are obtained and compared with the results of direct simulation of the random walks.

H1 Approximations of the Neutron Transport Equation and Associated Diffusion Equations

In this article, we give H1 approximations of the neutron transport equation based on variational formulations of the transport equation recently presented in Bourhrara (2004). These approximations are solutions of some variational problems on H1. We also show that the strong formulation of one of these approximate problems is no other than the classical neutron diffusion equation, taking into account the transport imposed incoming flux and the albedo boundary conditions. We also give an error estimate between the angular and the scalar flux of the transport equation and the H1 approximations.

Eliminating the mean-free-path inconsistency in classical phenomenological model of diffusion for fluids

Classical phenomenological model of diffusion in fluids is based on the concept of the mean-free-path, λ , and density distribution, n ( x , t ) , as a function of coordinate, x , and time, t . Under the assumption that (1) n ( x - λ , t ) - n ( x + λ , t ) ≈ - 2 λ ( ∂ n / ∂ x ) , this model results in the classical diffusion equation, (2) ∂ n ( x , t ) ∂ t = ∂ ∂ x D ∂ n ( x , t ) ∂ x , where (3) D = ( 1 / 3 ) V ¯ λ and V ¯ is the average velocity of molecules. However, Eq. (3) implies finite λ , but approximation (1) requires the limit of λ → 0 . Here we show that this (mean-free-path) inconsistency distorts the essential physics; in particular, it results in wrong solutions at finite and large λ (including an incorrect limit for the ideal gas). In this paper, the mean-free-path inconsistency is corrected by relaxing the unnecessary assumption (1). This gives a finite-difference equation for the classical diffusion model: (4) ∂ n ( x , t ) ∂ t = ∂ ∂ x D 2 λ [ n ( x + λ , t ) - n ( x - λ , t ) ] and its generalization taking into account statistical distribution of free-path lengths: (5) ∂ n ( x , t ) ∂ t = ∂ ∂ x D 2 λ 2 ∫ 0 ∞ [ n ( x + ξ , t ) - n ( x - ξ , t ) ] exp - ξ λ d ξ . We have solved and analyzed Eqs. (4) and (5) for capturing boundaries, for infinite system without boundaries, and for periodic initial conditions. Our analysis indicates that these new equations allow more accurate calculations for the classical model and predict new elements of diffusion mechanisms in fluids, such as diffusion correlations.

A second-order accurate numerical method for the two-dimensional fractional diffusion equation

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow, finance and others. In this paper, we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation. This numerical method combines the alternating directions implicit (ADI) approach with a Crank–Nicolson discretization and a Richardson extrapolation to obtain an unconditionally stable second-order accurate finite difference method. The stability and the consistency of the method are established. Numerical solutions for an example super-diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method.

Current‐induced spin dynamics in spin‐valve structures

AbstractThe Landau–Lifshitz–Gilbert equation for the magnetization dynamics, extended by including the torque due to spin‐transfer from conduction electrons, is used to study current‐induced magnetic switching and stationary precessional modes. The spin torque is described in terms of a macroscopic model based on the classical spin diffusion equations and relevant boundary conditions. The approach allows to investigate current‐driven switching, in particular the switching time and the frequency of the associated magnetoresistance signal in the precessional regime. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Current induced switching due to spin‐transfer in spin valves: macroscopic model

AbstractWe develop a macroscopic description of current‐induced torque due to spin transfer in layered systems consisting of ferromagnetic films separated by a nonmagnetic layer. The description is based on the classical spin diffusion equations for the distribution functions inside the films used in the theory of CPP‐GMR, and macroscopic boundary conditions for the longitudinal and transverse components of the spin current. Due to strong exchange field in ferromagnetic films, we assume that the perpendicular component of spin current is totally absorbed within the narrow interface region giving rise to the torque. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Some self‐similar solutions in river morphodynamics

Aggradation and degradation in one‐dimensional channels are often modeled with a simplified nonlinear diffusion equation. Different degrees of nonlinearity are obtained using the Chezy and Manning/Gauckler‐Strickler laws for the friction coefficient combined with a sediment transport equation having a generalized form of the Meyer‐Peter and Müller formula. Analytical self‐similar solutions for the “dam break” and the base‐level lowering are presented. While the linear case corresponds to the classic diffusion equation, the main effect of the nonlinearity appears to be the presence of singularities in the self‐similar solutions, related to the finite speed of propagation of perturbations.

Stochastic Schrodinger Equation for a Quantum Oscillator with Dissipation

In this paper, we construct an exact solution of the stochastic Schrodinger equation for a quantum oscillator with possible dissipation of energy taken into account. Using the explicit form of the solution, we calculate estimates for the characteristic damping time of free damped oscillations. In the case of forced oscillations, we obtain formulas for the Q-factor of the system and for the variance of the coordinate and momentum of a quantum oscillator with dissipation. We obtain the quantum analog of the classical diffusion equation and explicitly show that the equations of motion for the mean value of the momentum operator following from the solution of the stochastic Schrodinger equation play the role of the quantum Langevin equation describing Brownian motion under the action of a stochastic force.