Solution Of Nonlinear Diffusion Equation Research Articles

General framework to construct local-energy solutions of nonlinear diffusion equations for growing initial data

This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate solutions. A delicate issue for constructing local-energy solutions resides in the identification of weak limits of nonlinear terms for approximate solutions in a limiting procedure. Indeed, such an identification process often needs the maximal monotonicity of nonlinear elliptic operators (involved in the doubly-nonlinear equations) as well as uniform estimates for approximate solutions; however, even the monotonicity is violated due to a localization of the equations, which is also necessary to derive local-energy estimates for approximate solutions. In the present paper, such an inconsistency is systematically overcome by reducing the original equation to a localized one, where a (no longer monotone) localized elliptic operator is decomposed into the sum of a maximal monotone operator and a perturbation, and by integrating all the other relevant processes. Furthermore, the general framework developed in the present paper is also applied to the Finsler porous medium and fast diffusion equations, which are variants of the classical PME and FDE and also classified as a doubly-nonlinear equation.

Efficient Solutions for Nonlinear Diffusion Equations Appeared as Models of Physical Problems

The differential transform technique (DTM) looks promise for dealing with functional problems. Recent articles have demonstrated the DTM's efficiency in tackling a wide range of issues in many disciplines. In this paper, (DTM) is used to develop approximate, and exact solutions for some nonlinear diffusion equations. Nonlinear diffusion equations are used to describe processes and behaviours in fields of biology, heat transfer, chemical reactions, and mathematical physics. The differential transform method linked with Laplace transform and Pad’e approximation is used to improve some known results. The obtained solutions are compared with the exact known solutions, showing excellent agreement. The differential transformation method was used in conjunction with the use of the Laplace transform and the Pad’e approximation method, for the purpose of improving some calculations in the hope of obtaining a more accurate solution. The results were presented in the form of tables or graphics for the purpose of comparing the calculated solution and comparing it with some of the precise solutions presented previously. The results showed the accuracy of the agreement between the two solutions. This gives us the opportunity to use the method under consideration to find solutions to unknown problems and thus ensure the credibility of the calculated solution.

The process of magnetic flux penetration into superconductors

Abstract In the present paper the magnetic flux penetration dynamics of type-II superconductors in the flux creep regime is studied by analytically solving the nonlinear diffusion equation for the magnetic flux induction, assuming that an applied field parallel to the surface of the sample and using a power-law dependence of the differential resistivity on the magnetic field induction. An exact solution of nonlinear diffusion equation for the magnetic induction B(r, t) is obtained by using a well-known self-similar technique. We study the problem in the framework of a macroscopic approach, in which all length scales are larger than the flux-line spacing; thus, the superconductor is considered as a uniform medium.

New analytical method for solving nonlinear equation in rotating disk electrodes for second-order ECE reactions

The analytical and numerical solution of nonlinear diffusion equations are performed to study the chronoamperometric limiting current generated from the electrochemical reaction in a rotating disk electrode for second-order ECE reactions when the chemical step is irreversible. Simple and closed-form of expressions for the concentration of reactant and the current response are obtained as a function of the rotation rate and diffusion coefficients. The effect of various physical parameters on concentration and current have been computed and shown graphically. Also, the concentration/current expressions here derived using hyperbolic function and padé approximation technique are in satisfactory agreement with numerical results and limiting case results.

Nonlinear diffusion equations with nonlinear gradient noise

We prove the existence and uniqueness of entropy solutions for nonlinear diffusion equations with nonlinear conservative gradient noise. As particular applications our results include stochastic porous media equations, as well as the one-dimensional stochastic mean curvature flow in graph form.

Extensions and solutions for nonlinear diffusion equations and random walks.

We investigate a connection between random walks and nonlinear diffusion equations within the framework proposed by Einstein to explain the Brownian motion. We show here how to properly modify that framework in order to handle different physical scenarios. We obtain solutions for nonlinear diffusion equations that emerge from the random walk approach and analyse possible connections with a generalized thermostatistics formalism. Finally, we conclude that fractal and fractional derivatives may emerge in the context of nonlinear diffusion equations, depending on the choice of distribution functions related to the spreading of systems.

Numerical Solution of Non-Linear Diffusion Equation in Image Blurring Using Two-Point EGSOR Iterative Method

The non-linear diffusion equation is known to be a significant application in solving image processing issues. The equations provided the image filtering techniques that blurring the image without degrade the edge information which is also one of crucial study in computer vision. Nonetheless, an intense amount of computations is needed in filtering the image as the sizes that keep getting bigger. Along these lines, this paper constructs an analysis to speed up the required computation in solving the developed linear system with the faster iterative method, i.e. two-point Explicit Group Successive Over-Relaxation or known as 2-EGSOR. For the performances comparison, the standard Gauss-Seidel (GS), Successive Over-Relaxation (SOR) and 2-EGSOR iterative method will be set to produce almost the similar quality image of Jacobi iterative method measured by using percentage error of the overall pixels difference. Subsequently, it is discovered the 2-EGSOR offers faster approach to blur the image compared the others iterative methods with the least iterations and computational time.

A class of new special solution of nonlinear diffusion equation

One of the most interesting problems of nonlinear differential equations is the construction of partial solutions. A new method is presented in this paper to seek special solutions of nonlinear diffusion equations. This method is based on seeking suitable function to satisfy Bernolli equation. Many new special solutions are obtained.

Lie symmetry analysis, optimal system, exact solutions and conservation laws of a class of high-order nonlinear wave equations

The symmetries, conservation laws and exact solutions to the nonlinear partial differential equations play a significant role in nonlinear science and mathematical physics. Symmetry is derived from physics, and it is a mathematical description for invariance. Symmetry group theory plays an important role in constructing explicit solutions, whether the equations are integrable or not. By using the symmetry method, an original nonlinear system can be reduced to a system with fewer independent variables through any given subgroup. But, since there are almost always an infinite number of such subgroups, it is usually not feasible to list all possible group invariant solutions to the system. It is anticipated to find all those equivalent group invariant solutions, that is to say, to construct the one-dimensional optimal system for the Lie algebra. Construction of explicit forms of conservation laws is meaningful, as they are used for developing the appropriate numerical methods and for making mathematical analyses, in particular, of existence, uniqueness and stability. In addition, the existence of a large number of conservation laws of a partial differential equation (system) is a strong indication of its integrability. The similarity solutions are of importance for investigating the long-time behavior, blow-up profile and asymptotic phenomena of a non-linear system. For instance, in some circumstance, the asymptotic behaviors of finite-mass solutions of non-linear diffusion equation with non-linear source term are described by an explicit self-similar solution, etc. However, how to tackle these matters is a complicated problem that challenges researchers to be solved. In this paper, by using the symmetry method, we obtain the symmetry reduction, optimal systems, and many new exact group invariant solution of a fifth-order nonlinear wave equation. By Lie symmetry analysis method, the point symmetries and an optimal system of the equation are obtained. The exact power series solutions to the equation are provided by the power series method, such solutions can be used for numerical computations in both theory and physical applications conveniently. Finally, a lot of conservation laws of the fifth-order nonlinear wave equation are presented by using the adjoint equation and symmetries of the equation.

Superposition principle and exact solutions of a nonlinear diffusion equation

We present a method for constructing exact solutions of nonlinear diffusion equations in a one-dimensional coordinate space using a special superposition principle. As equations of nonlinear diffusion, we take equations of the form nt − (log n)xx + μn + γn 2 − g = 0, which play an important role in the problem of the emergence of regular structures in nonlinear media under the action of external radiation sources. The method is based on using differential properties of polynomials in functional parameters. We present concrete solutions and analyze some of their common properties.

Multivariate Padé Approximations For Solving Nonlinear Diffusion Equations

Abstract In this paper, multivariate Padé approximation is applied to power series solutions of nonlinear diffusion equations. As it is seen from tables, multivariate Padé approximation (MPA) gives reliable solutions and numerical results.

Sharp Operator Based Edge Detection

Ahmad et al. in their paper [1] for the first time proposed to apply sharp function for classification of images. In continuation of their work, in this paper we investigate the use of sharp function as an edge detector through well known diffusion models. Further, we discuss the formulation of weak solution of nonlinear diffusion equation and prove uniqueness of weak solution of nonlinear problem. The anisotropic generalization of sharp operator based diffusion has also been implemented and tested on various types of images.

Singular phenomena of solutions for nonlinear diffusion equations involving p(x)-Laplace operator and nonlinear sources

The aim of this paper was to study vanishing and blowing-up properties of the solutions to a homogeneous initial Dirichlet problem of a nonlinear diffusion equation involving the p(x)-Laplace operator and a nonlinear source. The authors point out that the results obtained are not trivial generalizations of similar problems in the case of constant exponent because the variable exponent p(x) brings some essential difficulties such as the failure of upper and lower solution method and scaling technique, the existence of a gap between the modular and the norm. To overcome these difficulties, the authors have to improve the regularity of solutions, to construct a new control functional and apply suitable embedding theorems to prove the blowing-up property of the solutions. In addition, the authors utilize an energy estimate method and a comparison principle for ODE to prove that the solution vanishes in finite time. At the same time, the critical extinction exponents and an extinction rate estimate to the solutions are also obtained.

Analytic and numerical solutions of nonlinear diffusion equations via symmetry reductions

Abstract In this article, the authors study analytic and numerical solutions of nonlinear diffusion equations of Fisher’s type with the help of classical Lie symmetry method. Lie symmetries are used to reduce the equations into ordinary differential equations (ODEs). Lie group classification with respect to time dependent coefficient and optimal system of one-dimensional sub-algebras is obtained. Then sub-algebras are used to construct symmetry reduction and analytic solutions. Finally, numerical solutions of nonlinear diffusion equations are obtained by using one of the differential quadrature methods.

Efficient time integration methods based on operator splitting and application to the Westervelt equation

Efficient time integration methods based on operator splitting and application to the Westervelt equation

Numerical Solution of Nonlinear Diffusion Equation with Convection Term by Homotopy Perturbation Method

In this paper, an application of homotopy perturbation method (HPM) is applied to finding the approximate solution of nonlinear diffusion equation with convection term, We obtained the numerically solution and compared with the exact solution.The results reveal that the homotopy perturbation method is very effective, simple and very close to the exact solution.

Splitting-based schemes for numerical solution of nonlinear diffusion equations on a sphere

We provide an advanced study of our recently developed method for the numerical solution of nonlinear diffusion equations on a sphere. In particular, we analyse the method in detail when applied to solving diverse diffusion phenomena, with specific conditions on the smoothness of the solution, the degree of nonlinearity, and the initial data and sources. The main idea of the method consists in splitting the original differential operator by coordinates and subsequent constructing finite difference schemes for the split one-dimensional problems using different coordinate maps for the sphere at the two split time intervals. The essential advantage of this technique is that each split 1D equation can be equipped with a periodic boundary condition, despite the sphere being not a doubly periodic domain. Therefore, unlike the existing methods, this one does not require applying special numerical procedures for careful computing the solution near the poles, which is always a challenge. Each split 1D equation is approximated by a second- or a fourth-order finite difference scheme that keeps all the substantial properties of the differential problem: it is balanced and dissipative, since the spatial finite difference operator is negative definite. The developed algorithm is cheap-to-implement from the computational point of view. The theoretical results are confirmed numerically by simulating various nonlinear diffusion processes. A numerical example, in particular, shows that the competition of the three basic mechanisms—the nonlinear interaction, forcing and dissipation—can generate wave solutions, whose spatial structures on the sphere are subjected to the alternating influence of the processes of self-organisation and self-destruction. The accuracy of the method is evaluated by comparing the numerical solutions versus the analytical ones obtained with specially chosen forcings. The convergence of the numerical solution to the analytics is verified by refining spatial grids.

A stable and accurate scheme for nonlinear diffusion equations: Application to atmospheric boundary layer

Stability concerns are always a factor in the numerical solution of nonlinear diffusion equations, which are a class of equations widely applicable in different fields of science and engineering. In this study, a modified extended backward differentiation formulae (ME BDF) scheme is adapted for the solution of nonlinear diffusion equations, with a special focus on the atmospheric boundary layer diffusion process. The scheme is first implemented and examined for a widely used nonlinear ordinary differential equation, and then extended to a system of two nonlinear diffusion equations. A new temporal filter which leads to significant improvement of numerical results is proposed, and the impact of the filter on the stability and accuracy of the results is investigated. Noteworthy improvements are obtained as compared to other commonly used numerical schemes. Linear stability analysis of the proposed scheme is performed for both systems, and analytical stability limits are presented.

An approximate analytical solution of nonlinear fractional diffusion equation by homotopy analysis method

In the present paper, the approximate analytical solution of a nonlinear diffusion equation with fractional time derivative () and with the diffusion term as () are obtained with the help of analytical method of nonlinear problem called the Homotopy Analysis Method (HAM). By using initial value, the explicit solution of the equation for different particular cases have been derived which demonstrate the effectiveness, validity, potentiality and reliability of the method in reality. Numerical results of the fast and slow diffusion for different particular cases are presented graphically. The numerical solutions show that only a few iterations are needed to obtain accurate approximate solutions. Key words: Partial differential equation, nonlinear fractional diffusion equation, Brownian motion, homotopy analysis method.

SYMMETRY REDUCTIONS AND EXACT SOLUTIONS FOR NONLINEAR DIFFUSION EQUATIONS

The symmetry properties of nonlinear diffusion equations are studied using a Lie group analysis. Reductions and families of exact solutions are found for some of these equations.