Nonlinear Diffusion Coefficient Research Articles

On the uniqueness of nonlinear diffusion coefficients in the presence of lower order terms

We consider the identification of nonlinear diffusion coefficients of the form or in quasi-linear parabolic and elliptic equations. Uniqueness for this inverse problem is established under very general assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof of our main result relies on the construction of a series of appropriate Dirichlet data and test functions with a particular singular behavior at the boundary. This allows us to localize the analysis and to separate the principal part of the equation from the remaining terms. We therefore do not require specific knowledge of lower order terms or initial data which allows to apply our results to a variety of applications. This is illustrated by discussing some typical examples in detail.

A family of analytical solutions of a nonlinear diffusion–convection equation

Despite its popularity in many engineering fields, the nonlinear diffusion–convection equation has no general analytical solutions. This work presents a family of closed-form analytical traveling wave solutions for the nonlinear diffusion–convection equation with power law nonlinearities. This kind of equations typically appears in nonlinear problems of flow and transport in porous media. The solutions that are addressed are simple and fully analytical. Three classes of analytical solutions are presented depending on the type of the nonlinear diffusion coefficient (increasing, decreasing or constant). It has shown that the structure of the traveling wave solution is strongly related to the diffusion term. The main advantage of the proposed solutions is that they are presented in a unified form contrary to existing solutions in the literature where the derivation of each solution depends on the specific values of the diffusion and convection parameters. The proposed closed-form solutions are simple to use, do not require any numerical implementation, and may be implemented in a simple spreadsheet. The analytical expressions are also useful to mathematically analyze the structure and properties of the solutions.

On the differentiability of solutions of stochastic evolution equations with respect to their initial values

In this article we study the differentiability of solutions of parabolic semilinear stochastic evolution equations (SEEs) with respect to their initial values. We prove that if the nonlinear drift coefficients and the nonlinear diffusion coefficients of the considered SEEs are n-times continuously Fréchet differentiable, then the solutions of the considered SEEs are also n-times continuously Fréchet differentiable with respect to their initial values. Moreover, a key contribution of this work is to establish suitable enhanced regularity properties of the derivative processes of the considered SEE in the sense that the dominating linear operator appearing in the SEE smoothes the higher order derivative processes.

Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations

In this work, we consider a one-dimensional Itô diffusion process Xt with possibly nonlinear drift and diffusion coefficients. We show that, when the diffusion coefficient is known, the drift coefficient is uniquely determined by the observation of the expectation of the process during a small time interval, and starting from any value X0 in a given subset of . With the same type of observation, and given the drift coefficient, we also show that the diffusion coefficient is uniquely determined. When both coefficients are unknown, we show that they are simultaneously uniquely determined by the observation of the expectation and variance of the process, during a small time interval, and starting again from any value X0 in a given subset of . To derive these results, we apply the Feynman-Kac theorem which leads to a linear parabolic equation with unknown coefficients in front of the first and second order terms. We then solve the corresponding inverse problem with PDE technics which are mainly based on the strong parabolic maximum principle.

Perpendicular Diffusion Coefficient of Comic Rays: The Presence of Weak Adiabatic Focusing

Abstract The influence of adiabatic focusing on particle diffusion is an important topic in astrophysics and plasma physics. In the past, several authors have explored the influence of along-field adiabatic focusing on the parallel diffusion of charged energetic particles. In this paper, using the unified nonlinear transport theory developed by Shalchi and the method of He and Schlickeiser, we derive a new nonlinear perpendicular diffusion coefficient for a non-uniform background magnetic field. This formula demonstrates that the particle perpendicular diffusion coefficient is modified by along-field adiabatic focusing. For isotropic pitch-angle scattering and the weak adiabatic focusing limit, the derived perpendicular diffusion coefficient is independent of the sign of adiabatic focusing characteristic length. For the two-component model, we simplify the perpendicular diffusion coefficient up to the second order of the power series of the adiabatic focusing characteristic quantity. We find that the first-order modifying factor is equal to zero and that the sign of the second order is determined by the energy of the particles.

A closed form approximation and error quantification for the response transition probability density function of a class of stochastic differential equations

A closed-form analytical approximation is derived for the response transition probability density function (PDF) of a certain class of stochastic differential equations with constant drift and nonlinear diffusion coefficients. This is done by resorting to a recently developed Wiener path integral based technique (WPI) in conjunction with a Cauchy–Schwarz inequality treatment of the problem. The derived approximation can be used, due to its analytical nature, as a direct SDE response PDF estimate that requires zero computational effort for its determination. Further, it facilitates an error quantification analysis, which yields an a priori estimate of the anticipated accuracy obtained by applying the approximate methodology. The reliability of the approximation is demonstrated via several engineering mechanics/dynamics related numerical examples pertaining to the stochastic beam bending problem, as well as to the response determination of stochastically excited nonlinear oscillators.

Identification of the initial condition in backward problem with nonlinear diffusion and reaction

In this paper, we study the backward in time problem for nonlinear parabolic equations associated with nonlinear diffusion coefficient. The problem is ill-posed in the sense of Hadamard. Under an a priori assumption on the exact solution in Gevrey space, we propose a new regularization method for stabilising the ill-posed problem. Our results extend some earlier works on backward problems for nonlinear parabolic equations.

Estimation of the non-linear diffusion coefficient with Marcov Chain Monte Carlo method based on the integral information

PurposeThis paper aims to present development and application of the Bayesian inverse approach for retrieving parameters of non-linear diffusion coefficient based on the integral information.Design/methodology/approachThe Bayes formula was used to construct posterior distribution of the unknown parameters of non-linear diffusion coefficient. The resulting aposteriori distribution of sought parameters was integrated using Markov Chain Monte Carlo method to obtain expected values of estimated diffusivity parameters as well as their confidence intervals. Unsteady non-linear diffusion equation was discretised with the Global Radial Basis Function Collocation method and solved in time using Crank–Nicholson technique.FindingsA number of manufactured analytical solutions of the non-linear diffusion problem was used to verify accuracy of the developed inverse approach. Reasonably good agreement, even for highly correlated parameters, was obtained. Therefore, the technique was used to compute concentration dependent diffusion coefficient of water in paper.Originality/valueAn original inverse technique, which couples efficiently meshless solution of the diffusion problem with the Bayesian inverse methodology, is presented in the paper. This methodology was extensively verified and applied to the real-life problem.

SUNSPOT AND STARSPOT LIFETIMES IN A TURBULENT EROSION MODEL

ABSTRACT Quantitative models of sunspot and starspot decay predict the timescale of magnetic diffusion and may yield important constraints in stellar dynamo models. Motivated by recent measurements of starspot lifetimes, we investigate the disintegration of a magnetic flux tube by nonlinear diffusion. Previous theoretical studies are extended by considering two physically motivated functional forms for the nonlinear diffusion coefficient D: an inverse power-law dependence D ∝ B −ν and a step-function dependence of D on the magnetic field magnitude B. Analytical self-similar solutions are presented for the power-law case, including solutions exhibiting “superfast” diffusion. For the step-function case, the heat-balance integral method yields approximate solutions, valid for moderately suppressed diffusion in the spot. The accuracy of the resulting solutions is confirmed numerically, using a method which provides an accurate description of long-time evolution by imposing boundary conditions at infinite distance from the spot. The new models may allow insight into the differences and similarities between sunspots and starspots.

Solvable Diffusion Models with Linear and Mean-Reverting Nonlinear Drifts

We present new extensions to a method for constructing families of solvable one-dimensional time-homogeneous diffusions whose transition densities and other related quantities are obtainable in analytically closed form. Our approach is based on a dual application of the so-called diffusion canonical transformation method that combines a smooth monotonic differentiable map with nonzero derivative (diffeomorphism) and a measure change via a Doob-$h$ transform. This gives rise to new multiparameter solvable diffusions that are divided into two main classes; the first is specified by having affine (linear) drift with various resulting nonlinear diffusion coefficient functions, while the second class allows for several specifications of a (generally nonlinear) diffusion coefficient with resulting nonlinear drift function. The first class of models, having linear drift and nonlinear (state-dependent) volatility functions, is useful for pricing equity and foreign exchange (FX) options in finance, while the second class of diffusions contains new models that are mean-reverting and are applicable to pricing interest-rate and other path-dependent derivatives such as volatility index (VIX) options. As specific examples of the first class of affine drift models, we present explicit results for two new families of models that arise from the squared Bessel process (the Bessel family) and the Ornstein--Uhlenbeck diffusion (the OU family). For the second class of nonlinear drift models, we give examples of solvable subfamilies called the Bessel family of mean-reverting diffusions and derive closed-form integral formulas for conditional expectations of certain functionals. In particular, we derive a new closed-form analytical formula for the Laplace transform with respect to the strike price of a standard call VIX option. We then succeed in Laplace inverting the expression to obtain numerically exact VIX call option prices with realistic implied volatilities with respect to strike and maturity. Moreover, we accurately calibrate the OU family of models to FX option market data, exhibiting pronounced implied volatility smiles across several strikes and maturities.

Linearly implicit BDF methods for nonlinear parabolic interface problems

We analyze linearly implicit BDF methods for the time discretization of a nonlinear parabolic interface problem, where the computational domain is divided into two subdomains by an interface, and the nonlinear diffusion coefficient is discontinuous across the interface. We prove optimal-order error estimates without assuming any growth conditions on the nonlinear diffusion coefficient and without restriction on the stepsize. Due to the existence of the interface and the lack of global Lipschitz continuity of the diffusion coefficient, we use a special type of test functions to analyze high-order \(A(\alpha )\)-stable BDF methods. Such test functions avoid any interface terms upon integration by parts and are used to derive error estimates in the piecewise \(H^2\) norm.

Heat transfer analysis for a free boundary problem arising in n-diffusion equation

This paper presents research on a free boundary value problem arising in a nonlinear n-diffusion equation by using a homotopy analysis method (HAM). Approximate analytical solutions are obtained for special nonlinear diffusion functional coefficient (variable thermal conduction) k(s)=s^{i} for i=1, 3, 5 and convection functional coefficient h(s)=s^{j} for j=1, 4 and power law parameter of n=0.2, 0.5, 1.0, 2.5. Reliability and efficiency of the approximate solutions are verified by numerical ones showing good agreement. The effects of the power law exponent, the nonlinear diffusion or convection functional coefficients, and the free boundary parameter on the flux transport characteristics are presented graphically and analyzed in detail. The mathematical techniques employed in this paper have the significance in studying some other problems of engineering.

Vertex-Centered Linearity-Preserving Schemes for Nonlinear Parabolic Problems on Polygonal Grids

On arbitrary polygonal grids, a family of vertex-centered finite volume schemes are suggested for the numerical solution of the strongly nonlinear parabolic equations arising in radiation hydrodynamics and magnetohydrodynamics. We define the primary unknowns at the cell vertices and derive the schemes along the linearity-preserving approach. Since we adopt the same cell-centered diffusion coefficients as those in most existing finite volume schemes, it is required to introduce some auxiliary unknowns at the cell centers in the case of nonlinear diffusion coefficients. A second-order positivity-preserving algorithm is then suggested to interpolate these auxiliary unknowns via the primary ones. All the schemes lead to symmetric and positive definite linear systems and their stability can be rigorously analyzed under some standard and weak geometry assumptions. More interesting is that these vertex-centered schemes do not have the so-called numerical heat-barrier issue suffered by many existing cell-centered or hybrid schemes (Lipnikov et al. in J Comput Phys 305:111---126, 2016). Numerical experiments are also presented to show the efficiency and robustness of the schemes in simulating nonlinear parabolic problems.

Numerical methods for a nonlinear reaction–diffusion system modelling a batch culture of biofilm

A biofilm is usually defined as a layer of bacterial cells anchored to a surface. These cells are embedded into a polymer matrix that keeps them attached to each other and to a solid surface. Among a large variety of biofilms, in this paper we consider batch cultures. The mathematical model is formulated in terms of a quasilinear system of diffusion–reaction equations for biomass and nutrients concentrations, which exhibits possible degeneracy and singularities in the nonlinear diffusion coefficient. In the present paper, we propose a set of efficient numerical methods that speeds up the solution of the model. Mainly, Crank–Nicolson finite differences techniques for discretisation are combined with a Newton algorithm for the nonlinearities. Moreover, some numerical examples show the expected behaviour of the biomass and nutrients concentrations and also clearly illustrate some theoretically proved qualitative properties related to exponential decays or convergence to a critical biomass concentration depending on the values of the model parameters.

Exact solutions for logistic reaction–diffusion equations in biology

Reaction-diffusion equations with a nonlinear source have been widely used to model various systems, with particular application to biology. Here, we provide a solution technique for these types of equations in $N$-dimensions. The nonclassical symmetry method leads to a single relationship between the nonlinear diffusion coefficient and the nonlinear reaction term; the subsequent solutions for the Kirchhoff variable are exponential in time (either growth or decay) and satisfy the linear Helmholtz equation in space. Example solutions are given in two dimensions for particular parameter sets for both quadratic and cubic reaction terms.

Burgers equation with no-flux boundary conditions and its application for complete fluid separation

Burgers equation in a one-dimensional bounded domain with no-flux boundary conditions at both ends is proven to be exactly solvable. Cole–Hopf transformation converts not only the governing equation to the heat equation with an extra damping but also the nonlinear mixed boundary conditions to Dirichlet boundary conditions. The average of the solution v¯ is conserved. Consequently, from an arbitrary initial condition, solutions converge to the equilibrium solution which is unique for the given v¯. The problem arises naturally as a continuum limit of a network of certain micro-devices. Each micro-device imperfectly separates a target fluid component from a mixture of more than one component, and its input–output concentration relationships are modeled by a pair of quadratic maps. The solvability of the initial boundary value problem is used to demonstrate that such a network acts as an ideal macro-separator, separating out the target component almost completely. Another network is also proposed which leads to a modified Burgers equation with a nonlinear diffusion coefficient.

Drug release kinetics from carboxymethylcellulose-bacterial cellulose composite films

Composite films of sodium carboxymethyl cellulose and bacterial cellulose (NaCMC–BC) cross-linked with citric acid (CA) were prepared by solution casting method. Ibuprofen sodium salt (IbuNa) has been used to study the mechanism of drug release from composite films. Surface morphology was investigated by scanning electron microscopy (SEM) and proved that the BC content influences the aspect of the films. Fourier transformed infrared spectroscopy (FTIR) revealed specific peaks in IR spectra of composite films which sustain that NaCMC was cross-linked with CA. Starting from swelling observations, the release kinetic of IbuNa was described using a model which neglects the volume expansion due to polymer swelling and which considers non-linear diffusion coefficients for drug and solvent. The IbuNa release is also influenced by BC content, the drug release rate was decreasing with the increase of BC content.

Does communication enhance pedestrians transport in the dark?

We study the motion of pedestrians through an obscure tunnel where the lack of visibility hides the exits. Using a lattice model, we explore the effects of communication on the effective transport properties of the crowd of pedestrians. More precisely, we study the effect of two thresholds on the structure of the effective nonlinear diffusion coefficient. One threshold models pedestrian communication efficiency in the dark, while the other one describes the tunnel capacity. Essentially, we note that if the evacuees show a maximum trust (leading to a fast communication), they tend to quickly find the exit and hence the collective action tends to prevent the occurrence of disasters. Nous étudions la dynamique des mouvements de foules dans un tunnel dans lequel la visibilité est très réduite. Nous nous intéressons tout particulièrement à des tunnels dont les sorties ne sont pas visibles. À l'aide de notre modèle – un automate cellullaire – nous exploitons les effets de la communication interpersonnelle parmi les piétons sur la structure de la non-linearité du coefficient de diffusion. Nous modélisons l'efficacité de la communication interpersonnelle ainsi que la capacité des sorties à l'aide de deux barrières.

Analysis of degenerate cross-diffusion population models with volume filling

A class of parabolic cross-diffusion systems modeling the interaction of an arbitrary number of population species is analyzed in a bounded domain with no-flux boundary conditions. The equations are formally derived from a random-walk lattice model in the diffusion limit. Compared to previous results in the literature, the novelty is the combination of general degenerate diffusion and volume-filling effects. Conditions on the nonlinear diffusion coefficients are identified, which yield a formal gradient-flow or entropy structure. This structure allows for the proof of global-in-time existence of bounded weak solutions and the exponential convergence of the solutions to the constant steady state. The existence proof is based on an approximation argument, the entropy inequality, and new nonlinear Aubin–Lions compactness lemmas. The proof of the large-time behavior employs the entropy estimate and convex Sobolev inequalities. Moreover, under simplifying assumptions on the nonlinearities, the uniqueness of weak solutions is shown by using the H−1 method, the E-monotonicity technique of Gajewski, and the subadditivity of the Fisher information.

Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2

In this article we are concerned with the study of the existence and uniqueness of pathwise mild solutions to evolutions equations driven by a Holder continuous function with Holder exponent in $(1/3,1/2)$. Our stochastic integral is a generalization of the well-known Young integral. To be more precise, the integral is defined by using a fractional integration by parts formula and it involves a tensor for which we need to formulate a new equation. From this it turns out that we have to solve a system consisting of a path and an area equations. In this paper we prove the existence of a unique local solution of the system of equations. The results can be applied to stochastic evolution equations with a non-linear diffusion coefficient driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$, which in particular includes white noise.