Nonlinear Diffusion Phenomena Research Articles

Asymptotic behavior of solutions to bipolar Euler–Poisson equations with time-dependent damping

In this paper, we study the one-dimensional Euler–Poisson equations of bipolar hydrodynamic model for semiconductor device with time-dependent damping effect − J ( 1 + t ) λ for − 1 < λ < 1 , where the damping effect is time-gradually-degenerate for λ > 0 , and time-gradually-enhancing for λ < 0 . Such a damping effect makes the hydrodynamic system possess the nonlinear diffusion phenomena time-asymptotic-weakly or strongly. Based on technical observation, and by using the time-weighted energy method, where the weights are artfully chosen, we prove that the system admits a unique global smooth solution, which time-asymptotically converges to the corresponding diffusion wave, when the initial perturbation around the diffusion wave is small enough. The convergence rates are specified in the algebraic forms O ( t − 3 4 ( 1 + λ ) ) and O ( t − ( 1 − λ ) ) according to different values of λ in ( − 1 , 1 7 ) and ( 1 7 , 1 ) , respectively, where λ = 1 7 is the critical point, and the convergence rate at the critical point is O ( t − 6 7 ln ⁡ t ) . All these convergence rates obtained in different cases are optimal in the sense when the initial perturbations are L 2 -integrable. Particularly, when λ = 1 7 , the convergence rate is the fastest, namely, the asymptotic profile of the original system at the critical point is the best.

A numerical study of nonlinear diffusion phenomena in heterogeneous media: energy transfer at diverse blow-up modes and self-organisation processes

A detailed analysis of a new method for numerical simulation of nonlinear diffusion phenomena is carried out. The method is based on operator splitting performed in time and space, and yields highly accurate solutions in complex 2D and 3D computational domains. After providing a circumstantial mathematical description of the developed method, we test it in several numerical experiments aimed, firstly, to model energy transfer at diverse modes of evolution of the dynamical system, and, secondly, to simulate self-organisation processes typical for real-world applications. A discussion of the outcomes of the numerical experiments is given. This is a follow-up paper of our recent original results presented at the 19th European conference on mathematics for industry.

A novel analytic approximation method with a convergence acceleration parameter for solving nonlinear problems

In this paper, a new analytic approximation method with a convergence acceleration parameter c is first proposed. The parameter c is used to adjust and control the convergence region and rate of the resulting series solution. It turns out that the convergence region and rate can be greatly enlarged by choosing a proper value of c. Furthermore, a numerical approach for finding the optimal value of the convergence acceleration parameter is given. At the same time, it is found that the traditional Adomian decomposition method is only a special case of the new method. The effectiveness and applicability of the new technique are demonstrated by several physical models including nonlinear heat transfer problems, nano-electromechanical systems, diffusion and dissipation phenomena, and dispersive waves.

Modeling of Magnetization Dynamics Using Saturation Wavefronts

This paper presents a detailed theoretical background of the saturation wave model (SWM), which offers a simplified homogenized solution of the nonlinear diffusion phenomena inside soft magnetic steel sheets (SMSSs). The SWM is capable of predicting the complicated dynamic magnetization of SMSS without discretization of the cross section of the SMSSs. Despite its simplicity, the SWM predicts exact solutions of the discussed problems when step-like magnetization curves are assumed and reasonably accurate solutions for real magnetization curves. In this paper, the SWM is analyzed in comparison to the numerical solution of the diffusion phenomena using the parametric magnetodynamic model, where interesting properties of the SWM are discussed and pointed out.

Nonlinear thermal system identification using fractional Volterra series

Linear fractional differentiation models have already proven their efficacy in modeling thermal diffusive phenomena for small temperature variations involving constant thermal parameters such as thermal diffusivity and thermal conductivity. However, for large temperature variations, encountered in plasma torch or in machining in severe conditions, the thermal parameters are no longer constant, but vary along with the temperature. In such a context, thermal diffusive phenomena can no longer be modeled by linear fractional models. In this paper, a new class of nonlinear fractional models based on the Volterra series is proposed for modeling such nonlinear diffusive phenomena. More specifically, Volterra series are extended to fractional derivatives, and fractional orthogonal generating functions are used as Volterra kernels. The linear coefficients are estimated along with nonlinear fractional parameters of the Volterra kernels by nonlinear programming techniques. The fractional Volterra series are first used to identify thermal diffusion in an iron sample with data generated using the finite element method and temperature variations up to 700K. For that purpose, the thermal properties of the iron sample have been characterized. Then, the fractional Volterra series are used to identify the thermal diffusion with experimental data obtained by injecting a heat flux generated by a 200W laser beam in the iron sample with temperature variations of 150K. It is shown that the identified model is always more accurate than the finite element model because it allows, in a single experiment, to take into account system uncertainties.

Doubly Nonlinear Equations as Convex Minimization

We present a variational reformulation of a class of doubly nonlinear parabolic equations as (limits of) constrained convex minimization problems. In particular, an $\varepsilon$-dependent family of weighted energy-dissipation (WED) functionals on entire trajectories is introduced and proved to admit minimizers. These minimizers converge to solutions of the original doubly nonlinear equation as $\varepsilon \to 0$. The argument relies on the suitable dualization of the former analysis of [G. Akagi and U. Stefanelli, J. Funct. Anal., 260 (2011), pp. 2541--2578] and results in a considerable extension of the possible application range of the WED functional approach to nonlinear diffusion phenomena, including the Stefan problem and the porous media equation.

Long-time self-similarity of classical solutions to the bipolar quantum hydrodynamic models

In this paper, a bipolar transient quantum hydrodynamic model (BQHD) for charge density, current density and electric field is considered on the one-dimensional real line. This model takes the form of the classical Euler–Poisson system with additional dispersion caused by the quantum (Bohn) potential. We investigate the long-time behavior of the BQHD model and show the asymptotical self-similarity property of the global smooth solution. Namely, both of the charge densities tend to a nonlinear diffusion wave in large time, which is not a solution to the BQHD equation, but to the combined quasi-neutral, relaxation and semiclassical limiting model. Next, as a by-product, we can compare the large-time behavior of the bipolar quantum hydrodynamic models and of the corresponding classical bipolar hydrodynamic models. As far as we know, the nonlinear diffusion phenomena about the 1D BQHD is new.

An Integral Equation Formulation for Solving Reaction-Diffusion-Convection Boundary-Value Problems

Many interesting problems that include convective transport arise in chemical reactor engineering (for example, tubular reactors). To solve these boundary-value problems, finite-difference schemes with a type of discretization of the convection term have been traditionally used. Some controversy about the discretization form of the convection term has arisen because of the different possibilities, including backward, forward and central discretizations. To overcome this problem, the usage of Green's function formulations for the numerical solution of typical chemical engineering problems with both linear and nonlinear kinetics, diffusion and convection phenomena, is presented. A distinctive feature of the proposed scheme is that boundary conditions are exactly incorporated. The results show that the integral formulation is, in general, superior in accuracy to the different finite-differences schemes. That is, more accurate calculations of the performance factor are obtained in terms of less mesh points and computer time.

Duality Theorem for a Three-Phase Partition Problem

In some nonlinear diffusive phenomena, the systems have three or more stable states. Sternberg and Zeimer established the existence of minimal solutions for the problem of partitioning a certain domain Ω⊂ℝ2 into three subdomains having least interfacial area. Ikota and Yanagida investigated stability and instability for stationary curves with one triple junction and for stationary binary-tree type interfaces. In this paper, we introduce a new concept of separation of three convex sets by a triangle, define a dual problem to the three-phase partition problem, and present a duality theorem.

Exact solutions for two-dimensional steady flows of a power-law liquid on an incline

Under assumptions that are not too restrictive it is possible to reduce the equations that describe steady viscous gravity flows of a power-law liquid on an inclined plane to an equivalent problem consisting of an unsteady one-dimensional nonlinear diffusion process. In a paper dealing with the steady spreading flow of a Herschel–Buckley liquid, Wilson and Burgess [“The steady, spreading flow of a rivulet of mud,” J. Non-Newtonian Fluid Mech. 79, 77 (1998)] noticed a formal analogy between the steady, two-dimensional viscous gravity flows of a power-law liquid on an incline and a one-dimensional time-dependent nonlinear diffusion phenomena; however, they did not pursue the matter further. Here we develop the analogy and show how it can be used to find a large number of exact solutions representing steady two-dimensional flows of power-law liquids, based on the available knowledge concerning nonlinear diffusion. We describe flows whose widths stay constant until a certain distance from the source, which are analogous to the well-known waiting-time solutions of nonlinear diffusion. We then introduce a phase-plane formalism that allows us to find self-similar solutions and we give as examples three different currents limited laterally by a wall that ends abruptly and currents on an inclined stripe. Finally we describe the two-dimensional currents that are analogous to the traveling wave solutions of the nonlinear diffusion equation. The approximations involved in the analogy are essentially equivalent to those of the lubrication theory, so that they do not impose restrictions more severe than those usually present in problems of this type. The present theory does not include surface tension effects, which implies that the appropriate Bond number must be large.

Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors

The asymptotic behavior of classical solutions of the bipolar hydrodynamical model for semiconductors is considered in the present paper. This system takes the form of Euler–Poisson with electric field and frictional damping added to the momentum equation. The global existence of classical solutions is proven, and the nonlinear diffusive phenomena is observed in large time in the sense that both densities of electron and hole tend to the same unique nonlinear diffusive wave.

New Model for Nonlinear Adsorption and Diffusion Based on a Quartic Concentration Profile Approximation

A new model that describes nonlinear adsorption and pore and surface diffusion (including concentration-dependent diffusion) in a single particle was derived using an intraparticle quartic concentration profile approximation and a summation of the gas and adsorbed phases in the material balances. Analytical expressions were obtained for the fractional uptake and adsorbed-phase concentration profile, and a differential expression was obtained for the rate of accumulation in the adsorbed phase that can be used as an accurate simplification in fixed-bed adsorption models. The new model was tested against the exact numerical solution for a variety of adsorbate−adsorbent systems and a broad range of bulk concentrations, with the results always being far superior to those obtained with the LDF approximation and even those obtained with a similarly derived model based on a parabolic profile approximation. This was especially the case at short times, where the other models failed to capture the behavior of the exact solution. The success of this new model was attributed to a set of assumptions that allowed more of the physics to be retained, while still making the solution tractable. This approach led to very useful and also very accurate analytical and differential expressions for nonlinear adsorption and diffusion phenomena.

Nonlinear diffusive phenomena of entropy weak solutions for a system of quasilinear hyperbolic conservation laws with damping

We consider a system of isentropic flow through porous media and show that nonlinear diffusive phenomena occur even if there are shocks in the flow.

Identification of source terms in nonlinear convection diffusion phenomena by sinc collocation-interpolation methods

The solution of an inverse problem related to the identification of a source term for convection diffusion equations in one space dimension is dealt with in this paper. The solution technique is based on collocation-interpolation methods which use sinc functions and on a suitable application of domain decomposition methods.

Analysis of nonlinear diffusion phenomena in the compacted bentonite backfill material

Analysis of nonlinear diffusion phenomena in the compacted bentonite backfill material

Nonlinear Diffusive Waves

This monograph deals with Burgers' equation and its generalisations. Such equations describe a wide variety of nonlinear diffusive phenomena, for instance, in nonlinear acoustics, laser physics, plasmas and atmospheric physics. The Burgers equation also has mathematical interest as a canonical nonlinear parabolic differential equation that can be exactly linearised. It is closely related to equations that display soliton behaviour and its study has helped elucidate other such nonlinear behaviour. The approach adopted here is applied mathematical. The author discusses fully the mathematical properties of standard nonlinear diffusion equations, and contrasts them with those of Burgers' equation. Of particular mathematical interest is the treatment of self-similar solutions as intermediate asymptotics for a large class of initial value problems whose solutions evolve into self-similar forms. This is achieved both analytically and numerically.

Pattern selection and instability in nonlinear wave equation—an aspect of soliton and chaos

Pattern selection problems are found in a variety of phenomena. Fluid dynamical systems and nonlinear diffusion phenomena give typical examples of pattern formation problems in dissipative systems. In some cases the dissipation reduces the effective dimension of the system, and this fact leads to several strikingly universal behaviors which were initially found in simple model systems with a few degrees of freedom. Nonlinear wave equations themselves, however, describes systems without dissipation in which the situation is more complicated. In spite of this complexity, many completely integrable systems are known in nonlinear wave equations, where neither ergodicity nor chaos is expected. With addition of small perturbation to completely integrable systems, one can see the growth of instability and the role of coherent structures in the pattern selection problem. Two aspects are briefly discussed in the following sections.

Stability and instability conditions for nonlinear evolutional equations in Hilbert spaces

Sufficient conditions for stability, global asymptotic stability, and explosive instability are established for a class of nonlinear evolutional equations defined in Hilbert spaces by using certain relations between an abstract function and its Gateaux differential. These results are applied to specific forms of nonlinear evolutional equations arising from physics, in particular, a finite-dimensional system of complex ordinary differential equations, functional differential equations, and systems of complex partial differential equations describing nonlinear diffusion or wave phenomena.