Li-Yau estimates and Harnack inequalities for nonlinear slow diffusion equations on a smooth metric measure space

Regularity Properties of Flows Through Porous Media: A Counterexample

Background. The concept of fractal is one of the main paradigms of modern theoretical and experimental physics, radiophysics and radar, and fractional calculus is the mathematical basis of fractal physics, geothermal energy and space electrodynamics. We investigate the solvability of the Cauchy problem for linear and nonlinear inhomogeneous pseudodifferential diffusion equations. The equation contains a fractional derivative of a Riemann–Liouville time variable defined by Caputo and a pseudodifferential operator that acts on spatial variables and is constructed in a homogeneous, non-negative homogeneous order, a non-smooth character at the origin, smooth enough outside. The heterogeneity of the equation depends on the temporal and spatial variables and permits the Laplace transform of the temporal variable. The initial condition contains a restricted function. Objective. To show that the homotopy perturbation transform method (HPTM) is easily applied to linear and nonlinear inhomogeneous pseudodifferential diffusion equations. To prove the solvability and obtain the solution formula for the Cauchy problem series for the given linear and nonlinear diffusion equations. Methods. The problem is solved by the NPTM method, which combines a Laplace transform with a time variable and a homotopy perturbation method (HPM). After the Laplace transform, we obtain an integral equation which is solved as a series by degrees of the entered parameter with unknown coefficients. Substituting the input formula for the solution into the integral equation, we equate the expressions to equal parameter degrees and obtain formulas for unknown coefficients. When solving the nonlinear equation, we use a special polynomial which is included in the decomposition coefficients of the nonlinear function and allows the homotopy perturbation method to be applied as well for nonlinear non-uniform pseudodifferential diffusion equation. Results. The result is a solution of the Cauchy problem for the investigated diffusion equation, which is represented as a series of terms whose functions are found from the parametric series. Conclusions. In this paper we first prove the solvability and obtain the formula for solving the Cauchy problem as a series for linear and nonlinear inhomogeneous pseudodifferential equations.

Modelling and analysis of microseismic signatures based on the nonlinear pore pressure diffusion equation in 2D is studied. The nonlinear diffusion equation is solved by using a finite element method on an irregular grid. Following the seismicity-based reservoir characterization approach microseismic earthquakes are triggered and analyzed regarding their spatio-temporal distribution. We find that there are significant differences in the microseismic signatures of linear and nonlinear pore pressure diffusion. It is shown that the microseismic event distribution is governed by two event triggering processes. One process interpreted as a fracturing front changing significantly the fluid transport properties and induces a large number of microearthquakes. Another triggering front that induces only a small number of microearthquakes ahead of the fracturing front. For the Fenton Hill hydrofrac experiment we demonstrate the existence of these nonlinear pore pressure diffusion signatures.

Two-velocity kinetic models are used to derive, in the appropriate limit, the equations which govern the macroscopic density of fluid systems. Such equations are obtained from an asymptotic expansion in powers of a small parameter related to the microscopic mean free path. It is shown that the density of a fluid interacting with a non-equilibrium background satisfies a linear diffusion equation, and that the hierarchy of equations arising from the asymptotic expansion can be completely solved by a recursive scheme. For a system of interacting particles, a nonlinear diffusion equation is obtained and some of its solutions are analysed. Finally, the density of a system of particles undergoing chemical reactions is shown to satisfy a nonlinear equation which formally coincides with the reaction-diffusion equation proposed ad hoc at the macroscopic level.

<i>Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems</i>

Purpose The purpose of this paper is to introduce a new homotopy perturbation method (NHPM) to solve Cauchy problem of unidimensional non-linear diffusion equation. Design/methodology/approach In this paper a modified version of HPM, which the authors call NHPM, has been presented; this technique performs much better than the HPM. HPM and NHPM start by considering a homotopy, and the solution of the problem under study is assumed to be as the summation of a power series in p, the difference between two methods starts from the form of initial approximation of the solution. Findings In this article, the authors have applied the NHPM for solving nonlinear Cauchy diffusion equation. In comparison with the homotopy perturbation method (HPM), in the present method, the authors achieve exact solutions while HPM does not lead to exact solutions. The authors believe that the new method is a promising technique in finding the exact solutions for a wide variety of mathematical problems. Originality/value The basic idea described in this paper is expected to be further employed to solve other functional equations.

This paper is concerned with the critical sharp travelling wave for doubly nonlinear diffusion equation with time delay, where the doubly nonlinear degenerate diffusion is defined by ((um)xp−2(um)x)x with m > 0 and p > 1. The doubly nonlinear diffusion equation is proved to admit a unique sharp type travelling wave for the degenerate case m(p − 1) > 1, the so-called slow-diffusion case. This sharp travelling wave associated with the minimal wave speed c*(m, p, r) is monotonically increasing, where the minimal wave speed satisfies c*(m, p, r) < c*(m, p, 0) for any time delay r > 0. The sharp front is C 1-smooth for 1p−1<m<pp−1 , and piecewise smooth for m⩾pp−1 . Our results indicate that time delay slows down the minimal travelling wave speed for the doubly nonlinear degenerate diffusion equations. The approach adopted for proof is the phase transform method combining the variational method. The main technical issue for the proof is to overcome the obstacle caused by the doubly nonlinear degenerate diffusion.

This article presents new gradient estimates for positive solutions to the nonlinear fast diffusion equation on smooth metric measure spaces, involving the f-Laplacian. The gradient estimates of interest are of Hamilton-Souplet-Zhang or elliptic type and are established using different methods and techniques. Various implications, notably to parabolic Liouville type results and characterisation of ancient solutions are given. The problem is considered in the general setting where the metric and potential evolve under a super flow involving the Bakry-Émery m-Ricci curvature tensor. The curious interplay between geometry, nonlinearity, and evolution – and their intricate roles in the estimates and the maximum exponent range of fast diffusion – is at the core of the investigation.

Correspondence between frame shrinkage and high-order nonlinear diffusion

Fractional and nonlinear diffusion equation: additional results

Rice is the main food for nearly half of the world's population. Cooking of rice consumes more energy in households as well as in industries which use rice as an input material for preparing certain foods. In order to reduce the energy spent for cooking and to obtain perfectly and uniformly cooked rice, soaking before cooking is traditionally practiced in some countries. This paper depicts a study of soaking rice prior to cooking. The proposed method uses two models which describe the moisture level when soaking at room temperature less than the gelatinization temperature and the moisture transfer model at temperatures above gelatinization and therefore cooking the rice grain. The moisture movements through the rice grain when soaking and when cooking are developed using the non-linear moisture diffusion equation. Both models are solved numerically using a finite element package in an elliptical geometry which is adjustable depends on the type of the rice. The validity of the proposed model is verified by the good fit between the experimental and theoretical curves and it is observed that the non-linear diffusion equation is a better way to describe the moisture transfer model in a rice grain.

Image segmentation is the problem of partitioning a picture into several homogeneous regions. We describe a class of very effective image segmentation algorithms that have recently been shown to have potential for the analysis of microstructure imagery, where different homogeneous regions may correspond to different physical characteristics of the material. The methods treat image pixel intensities as an initial temperature distribution over a rectangular domain, and perform image filtering by solving a nonlinear heat diffusion equation forward in time. The nonlinear diffusion equation is designed to attract to a piecewise constant solution which provides an accurate segmentation of the initial input image. We illustrate our methods on several alloy micrographs.

A nonlinear complex diffusion equation was obtained by changing the diffusion coefficient of the commonly used nonlinear diffusion equation into a complex one. Then the complex diffusion equation was resolved and utilized in image edge detection. In each round of iteration, the image parts of the solution derived from the complex diffusion equation were used to eliminate fake edges and isolated points to find real edges. A lot of image segmentation tests have been performed, and the results also were compared with those done by Canny detector, showing the effectiveness of the proposed algorithm.

Modeling anomalous heat diffusion: Comparing fractional derivative and non-linear diffusivity treatments

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.