An integrable fourth-order nonlinear evolution equation applied to thermal grooving of metal surfaces

The fourth-order nonlinear partial differential equation forsurface diffusion is approximated by a new integrable nonlinearevolution equation. Exact solutions are obtained for thermalgrooving, subject to boundary conditions representing a sectionof a grain boundary. When the slope m of the groove centre islarge, the linear model grossly overestimates the groove depth.In the linear model dimensionless groove depth increases linearlywith m, but in the nonlinear model it approaches an upper limitA nontrivial similarity solution is found for the limiting caseof a thermal groove whose central slope is vertical.     

一种特殊条件下的扩散方程的数值解法及物理分析

利用有限差分方法中的预估校正格式求解裂变产物在燃料芯块中的扩散方程,并进行了灵活的边界条件处理,取得了令人满意的计算精度,为有效开展对反应堆燃料元件破损探测信号的定量分析提供了条件.对数值计算过程进行了一定的物理分析,从物理意义上解释了数值计算过程中一些特殊处理的合理性.     

Amplitude Modulation of Nonlinear Waves in Fluid-Filled Tapered Tubes

We study the modulation of nonlinear waves in fluid-filled prestressed tapered tubes. For this, we obtain the nonlinear dynamical equations of motion of a prestressed tapered tube filled with an incompressible inviscid fluid. Assuming that the tapering angle is small and using the reductive perturbation method, we study the amplitude modulation of nonlinear waves and obtain the nonlinear Schrödinger equation with variable coefficients as the evolution equation. A traveling-wave type of solution of such a nonlinear equation with variable coefficients is obtained, and we observe that in contrast to the case of a constant tube radius, the speed of the wave is variable. Namely, the wave speed increases with distance for narrowing tubes and decreases for expanding tubes.     

Convergence properties of a class of boundary element approximations to linear diffusion problems with localized nonlinear reactions

We consider a boundary element (BE) Algorithm for solving linear diffusion desorption problems with localized nonlinear reactions. The proposed BE algorithm provides an elegant representation of the effect of localized nonlinear reactions, which enables the effects of arbitrarily oriented defect structures to be incorporated into BE models without having to perform severe mesh deformations. We propose a one-step recursion procedure to advance the BE solution of linear diffusion localized nonlinear reaction problems and investigate its convergence properties. The separation of the linear and nonlinear effects by the boundary integral formulation enables us to consider the convergence properties of approximations to the linear terms and nonlinear terms of the boundary integral equation separately. For the linear terms we investigate how the degree of piecewise polynomial collocation in space and the size of the spatial mesh relative to the time step affects the accumulation of errors in the one-step recursion scheme. We develop a novel convergence analysis that combines asymptotic methods with Lax's Equivalence Theorem. We identify a dimensionless meshing parameter θ whose magnitudé governs the performance of the one-step BE schemes. In particular, we show that piecewise constant (PWC) and piecewise linear (PWL) BE schemes are conditionally convergent, have lower asymptotic bounds placed on the size of time steps, and which display excess numerical diffusion when small time steps are used. There is no asymptotic bound on how large the tie steps can be–this allows the solution to be advanced in fewer, larger time steps. The piecewise quadratic (PWQ) BE scheme is shown to be unconditionally convergent; there is no asymptotic restriction on the relative sizes of the time and spatial meshing and no numerical diffusion. We verify the theoretical convergence properties in numerical examples. This analysis provides useful information about the appropriate degree of spatial piecewise polynomial and the meshing strategy for a given problem. For the nonlinear terms we investigate the convergence of an explicit algorithm to advance the solution at an active site forward in time by means of Caratheodory iteration combined with piecewise linear interpolation. We consider a model problem comprising a singular nonlinear Volterra equation that represents the effect of the term in the BE formulation that is due to a single defect. We prove the convergence of the piecewise linear Caratheodory iteration algorithm to a solution of the model problem for as long as such a solution can be shown to exist. This analysis provides a theoretical justification for the use of piecewise linear Caratheodory iterates for advancing the effects of localized reactions.     

A series solution of nonlinear injection–extraction well equation

In this paper, the groundwater flow will be investigated. The tracer concentration is calculated for the saturated–unsaturated aquifer. A nonlinear diffusion equation is derived in a single injection–extraction well. A numerical procedure based on Adomian-decomposition method (ADM) is proposed for solving this nonlinear diffusion equation and finally this method is examined for some test problems.     

Nonlinear diffusion waves for hyperbolic p-system with nonlinear damping

This paper is concerned with the p-system of hyperbolic conservation laws with nonlinear damping. When the constant states are small, the solutions of the Cauchy problem for the damped p-system globally exist and converge to their corresponding nonlinear diffusion waves, which are the solutions of the corresponding nonlinear parabolic equation given by the Darcy's law. The optimal convergence rates are also obtained. In order to overcome the difficulty caused by the nonlinear damping, a couple of correction functions have been technically constructed. The approach adopted is the elementary energy method together with the technique of approximating Green function. On the other hand, when the constant states are large, the solutions of the Cauchy problem for the p-system will blow up at a finite time.     

On an Optimal Filtration Problem for One-Dimensional Diffusion Processes

We find a method that reduces the solution of a problem of nonlinear filtration of one-dimensional diffusion processes to the solution of a linear parabolic equation with constant diffusion coefficients whose remaining coefficients are random and depend on the trajectory of the observable process. The method consists in reducing the initial filtration problem to a simpler problem with identity diffusion matrix and subsequently reducing the solution of the parabolic Itô equation for the filtered density to solving the above-mentioned parabolic equation. In addition, the filtered densities of both problems are connected by a sufficiently simple formula.     

Modeling and computer simulation of the identification problem related to the sludge concentration in a settler

The mathematical model of sludge particles settling in the water treatment plant (settler) is considered. In the case of the residence time of sludge particles in the settler the model leads to a nonlinear age-dependent transport–diffusion with a nonlocal additional condition. This problem is formulated as an identification/optimal control problem, where the sludge concentration is assumed to be a control. For the case of constant (“average”) velocity, as a characterization of the optimal control problem two necessary conditions are obtained. These conditions permit reducing the nonlinear coupled two-dimensional problem to the two-point boundary value problem for the second order nonlinear ordinary differential equation, and then, to a nonlinear equation, with respect to sludge concentration. For the solution of the problem an iteration algorithm is derived. Convergence of the iteration algorithm is analyzed theoretically, as well as on test examples. The numerical procedure for the considered problem is demonstrated on concrete examples.     

Applications of He’s principles to partial differential equations

This paper applies the variational iteration method (VIM) and semi-inverse variational principle to obtain solutions of linear and nonlinear partial differential equations. The nonlinear model is considered from gas dynamics, fluid dynamics and Burgers equation. The linear model is the heat transfer (diffusion) equation. Results show that variational iteration method is a powerful mathematical tool for solving linear and nonlinear partial differential equations, and therefore, can be widely applied to engineering problems.     

New exact solutions for some power law nonlinear diffusion equation

An extended auxiliary equation method for exact traveling wave solutions of constant coefficient nonlinear partial differential equations of evolution is proposed. This, together with a convenient characterization, affords new exact traveling wave solutions of some classes of nonlinear power law diffusion equations to be obtained.     

Numerical simulations of water–gas flow in heterogeneous porous media with discontinuous capillary pressures by the concept of global pressure

We present an approach and numerical results for a new formulation modeling immiscible compressible two-phase flow in heterogeneous porous media with discontinuous capillary pressures. The main feature of this model is the introduction of a new global pressure, and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation equation) with nonlinear transmission conditions at the interfaces that separate different media. The resulting system is discretized using a vertex-centred finite volume method combined with pressure and flux interface conditions for the treatment of heterogeneities. An implicit Euler approach is used for time discretization. A Godunov-type method is used to treat the convection terms, and the diffusion terms are discretized by piecewise linear conforming finite elements. We present numerical simulations for three one-dimensional benchmark tests to demonstrate the ability of the method to approximate solutions of water–gas equations efficiently and accurately in nuclear underground waste disposal situations.     

Bäcklund transformation of fractional Riccati equation and its applications to the space–time FDEs

In this paper, the Bäcklund transformation of fractional Riccati equation is presented to establish traveling wave solutions for two nonlinear space–time fractional differential equations in the sense of modified Riemann–Liouville derivatives, namely, the space–time fractional generalized reaction duffing equation and the space–time fractional diffusion reaction equation with cubic nonlinearity. The proposed method is effective and convenient for solving nonlinear evolution equations with fractional order. Copyright © 2014 John Wiley & Sons, Ltd.     

一类非线性扩散方程解的blow up速率估计

利用F riedm an-M cleod方法和变动尺度方法研究了一类具有非线性边界条件的非线性扩散方程解的b low up问题,证明了解在有限时间b low up,并且得到了b low up速率估计.     

NONLINEAR DIFFUSIVE PHENOMENA OF NONLINEAR HYPERBOLIC SYSTEMS

The authors study the nonlinear hyperbolic system which describes the motion of isentropic gas flow with external friction acting on it, such as a flow through porous media, and show the nonlinear diffusive phenomena for the large time behavior of solutions for this system by proving that the solutions tend to those of a nonlinear diffusion equation time-asymptotically.     

Analysis of the time fractional nonlinear diffusion equation from diffusion process

Under investigation in this paper is a time fractional nonlinear diffusion equation which can be utilized to express various diffusion processes. The symmetry of this considered equation has been obtained via fractional Lie group approach with the sense of Riemann-Liouville (R-L) fractional derivative. Based on the symmetry, this equation can be changed into an ordinary differential equation of fractional order. Moreover, some new invariant solutions of this considered equation are found. Lastly, utilising the Noether theorem and the general form of Noether type theorem, the conservation laws are yielded to the time fractional nonlinear diffusion equation, respectively. Our discovery that there are no conservation laws under the general form of Noether type theorem case. This result tells us the symmetry of this equation is not variational symmetry of the considered functional. These rich results can give us more information to interpret this equation.     

Using group theoretic method to solve multi-dimensional diffusion equation

The nonlinear diffusion equation arises in many important areas of science and technology such as modeling of dopant diffusion in semiconductors. We give analytical solution to N-dimensional radially symmetric nonlinear diffusion equation of the form$\text{∂}\text{∂r}\text{D(C)}\text{∂C}\text{∂r}\text{+}\text{N−1}\text{r}\text{D(C)}\text{∂C}\text{∂r}\text{=}\text{∂C}\text{∂t}\text{,}$where C(r,t) is the concentration and D(C) is diffusion coefficient.The transformation group theoretic approach is applied to present an analysis of the nonlinear diffusion equation. The one-parameter group transformation reduces the number of independent variables by one and the governing partial differential equation with the boundary conditions reduce to an ordinary differential equation with the appropriate boundary conditions. Effect of the time “t” and the number of dimension “N” on the concentration diffusion function C(r,t) has been studied and the results are plotted.     

Analysis of a Free Boundary Problem Modeling the Growth of Spherically Symmetric Tumors with Angiogenesis

This paper is concerned with a free boundary problem modeling the growth of a spherically symmetric tumor with angiogenesis. The unknown nutrient concentration \(\sigma =\sigma (r,t)\) occupies the unknown tumor region \(r< R(t)\) and satisfies a nonlinear reaction diffusion equation, and the unknown tumor radius \(R=R(t)\) satisfies a nonlinear integro-differential equation. Unlike existing literatures on this topic where Dirichlet boundary condition for \(\sigma \) is imposed, in this paper the model uses the Robin boundary condition for \(\sigma \). We prove existence and uniqueness of a global in-time classical solution (\(\sigma (r,t),R(t)\)) for arbitrary \(c>0\) and establish asymptotic stability of the unique stationary solution (\(\sigma _{s}(r),R_{s}\)) for sufficiently small \(c\), where \(c\) is a positive constant reflecting the ratio between nutrient diffusion scale and the tumor cell-doubling scale.

     

A numerical scheme based on a solution of nonlinear advection–diffusion equations

A mathematical formulation of the two-dimensional Cole–Hopf transformation is investigated in detail. By making use of the Cole–Hopf transformation, a nonlinear two-dimensional unsteady advection–diffusion equation is transformed into a linear equation, and the transformed equation is solved by the spectral method previously proposed by one of the authors. Thus a solution to initial value problems of nonlinear two-dimensional unsteady advection–diffusion equations is derived. On the base of the solution, a numerical scheme explicit with respect to time is presented for nonlinear advection–diffusion equations. Numerical experiments show that the present scheme possesses the total variation diminishing properties and gives solutions with good quality.     

Symmetry reduced and new exact non-traveling wave solutions of potential Kadomtsev-Petviashvili equation with p-power

With the aid of Maple symbolic computation and Lie group method, PKPp equation is reduced to some (1 + 1)-dimensional partial differential equations, in which there are linear PDE with constant coefficients, nonlinear PDE with constant coefficients, and nonlinear PDE with variable coefficients. Using the separation of variables, homoclinic test technique and auxiliary equation methods, we obtain new abundant exact non-traveling solution with arbitrary functions for the PKPp.     

The nonlinear diffusion equation in cylindrical coordinates

Nonlinear corrections to some classical solutions of the linear diffusion equation in cylindrical coordinates are studied within quadratic approximation. When cylindrical coordinates are used, we try to find a nonlinear correction using quadratic polynomials of Bessel functions whose coefficients are Laurent polynomials of radius. This usual perturbation technique inevitably leads to a series of overdetermined systems of linear algebraic equations for the unknown coefficients (in contrast with the Cartesian coordinates). Using a computer algebra system, we show that all these overdetermined systems become compatible if we formally add one function on radius W(r). Solutions can be constructed as linear combinations of these quadratic polynomials of the Bessel functions and the functions W(r) and W′(r). This gives a series of solutions to the nonlinear diffusion equation; these are found with the same accuracy as the equation is derived. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 235–245, 2007.