Nonlinear Diffusion Equation Research Articles

A modified fixed point method for biochemical transport

This work is devoted to a modified fixed point method applied to the bio-chemical transport equation. To have a good accuracy for the solution we treat, we apply an implicit scheme to this equation and use a modified fixed point technique to linearize the problem of transport equation with a generalized nonlinear reaction and diffusion equation. Next, we apply this methods in particular to the the dynamical system of a bio-chemical process. Eventually, we accelerate these algorithms by the optimized domain decomposition methods.Several test-cases of analytical problems illustrate this approach and show the efficiency of the proposednew method.

An efficient technique for solving fractional-order diffusion equations arising in oil pollution

In this article, non-linear time-fractional diffusion equations are considered to describe oil pollution in the water. The latest technique, fractional reduced differential transform method (FRDTM), is used to acquire approximate solutions of the time fractional-order diffusion equation and two cases of Allen–Cahn equations. The acquired results are collated with the exact solutions and other results from literature for integer-order α, which reveal that the proposed method is effective. Hence, FRDTM can be employed to obtain solutions for different types of nonlinear fractional-order IVPs arising in engineering and science.

An anisotropic diffusion system with nonlinear time-delay structure tensor for image enhancement and segmentation

Anisotropic diffusion model has been established as one of the most significant image processing techniques, because it not only allows for space-variant filtering, but also realizes real direction-specific anisotropic smoothing of image structures. In this paper, an anisotropic nonlinear diffusion system is proposed for image enhancement and segmentation by taking into account the structure tensor constructed from the time-delay regularization and a nonlinear isotropic diffusion equation. Meanwhile, the diffusion system makes full use of the source term which increases the contrast and makes the image be less susceptible to noise, even in high level noise images. The source term also causes the restored image to tend to the binary image, which has proven to be an effective strategy for image segmentation. We analyze the proposed model in the view of the existence and uniqueness of the weak solution by using Galerkin's method. Some other theoretical analysis such as the maximum property is also discussed in the paper. With the appropriate and adaptive parameters, the anisotropic diffusion system achieves a better trade-off between computation time and image processing task. Furthermore, various image experiments illustrate that the proposed algorithm achieves much better efficiency and universality.

On the solvability of an initial-boundary value problem for a non-linear fractional diffusion equation

<abstract><p>In this paper, we consider an initial-boundary value problem for a non-linear fractional diffusion equation on a bounded domain. The fractional derivative is defined in Caputo's sense with respect to the time variable and represents the case of sub-diffusion. Also, the equation involves a second order symmetric uniformly elliptic operator with time-independent coefficients. These initial-boundary value problems arise in applied sciences such as mathematical physics, fluid mechanics, mathematical biology and engineering. By using eigenfunction expansions and Banach fixed point theorem, we establish the existence, uniqueness and regularity properties of the solution of the problem.</p></abstract>

Globally unsolvability of integro-differential diffusion equation and system with exponential nonlinearities

In this paper, the Cauchy problem for an integro-differential diffusion equation and a system with nonlocal nonlinear sources are considered. The results on the existence of local integral solutions and the nonexistence of global weak solutions to the nonlinear integro-differential diffusion equation and system are presented.

Existence, uniqueness and travelling waves to model an invasive specie interaction with heterogeneous reaction and non-linear diffusion

<abstract><p>It is the objective to provide a mathematical treatment of a model to predict the behaviour of an invasive specie proliferating in a domain, but with a certain hostile zone. The behaviour of the invasive is modelled in the frame of a non-linear diffusion (of Porous Medium type) equation with non-Lipschitz and heterogeneous reaction. First of all, the paper examines the existence and uniqueness of solutions together with a comparison principle. Once the regularity principles are shown, the solutions are studied within the Travelling Waves (TW) domain together with stability analysis in the frame of the Geometric Perturbation Theory (GPT). As a remarkable finding, the obtained TW profile follows a potential law in the stable connection that converges to the stationary solution. Such potential law suggests that the pressure induced by the invasive over the hostile area increases over time. Nonetheless, the finite speed, induced by the non-linear diffusion, slows down a possible violent invasion.</p></abstract>

Vortex formation for a non-local interaction model with Newtonian repulsion and superlinear mobility

Abstract We consider density solutions for gradient flow equations of the form u t = ∇ · (γ(u)∇ N(u)), where N is the Newtonian repulsive potential in the whole space ℝ d with the nonlinear convex mobility γ(u) = u α , and α > 1. We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case γ(u) = u. For linear mobility it was shown that there is a special solution in the form of a disk vortex of constant intensity in space u = c 1 t −1 supported in a ball that spreads in time like c 2 t 1/d , thus showing a discontinuous leading front or shock. Our present results are in sharp contrast with the case of concave mobilities of the form γ(u) = u α , with 0 < α < 1 studied in [10]. There, we developed a well-posedness theory of viscosity solutions that are positive everywhere and moreover display a fat tail at infinity. Here, we also develop a well-posedness theory of viscosity solutions that in the radial case leads to a very detailed analysis allowing us to show a waiting time phenomena. This is a typical behaviour for nonlinear degenerate diffusion equations such as the porous medium equation. We will also construct explicit self-similar solutions exhibiting similar vortex-like behaviour characterizing the long time asymptotics of general radial solutions under certain assumptions. Convergent numerical schemes based on the viscosity solution theory are proposed analysing their rate of convergence. We complement our analytical results with numerical simulations illustrating the proven results and showcasing some open problems.

Variational Regularization For The Inverse Problem of a Nonlinear Time-Fractional Diffusion Equations

The nonlinear time-fractional diffusion equations were considered in the 2D domain, and the physical information in initial state <inlineformula><texmath id="M2">\begin{document}$ u(0,y,t) $\end{document}</texmath><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="amm-42-0168_M2.jpg"></graphic><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="amm-42-0168_M2.png"></graphic></alternatives></inlineformula> of the material was recovered from the measured data in the final state. This problem is seriously ill-posed, that is, the solution to this problem does not continuously depend on the measured data. Therefore, a variational regularization method was proposed to construct the approximate solution to the problem, and the convergence error estimates of the exact and approximate solutions were obtained under the assumption of the priori bounds on the exact solutions. Finally, a numerical example was given to verify the effectiveness of the proposed method.

Implementation of the ADMM approach to constrained optimal control problem with a nonlinear time-fractional diffusion equation

In this paper, we study the inverse problem of identifying the parameters in a nonlinear subdiffusion model from an observation defined in the given $ \Omega_1 $ subset of $ \Omega $. The nonlinear subdiffusion model involves a Caputo fractional derivative of order $ \alpha\in (0,1) $ in time. To address our model, we first examine the regularity of the solution for the direct problem using the Mittag-Leffler function. To investigate our inverse parameter problem, we reformulate first it in to Least-Squares optimization problem. Then, we establish the existence of the optimal solution and prove the convexity of the considered cost function by using its first derivative. To solve this problem numerically, we adapt a recent method in the literature known as the alternating direction method of multiplier (ADMM) which we establish its convergence. In order to show the effectiveness of the proposed method we present some numerical experiments.

Method of approximations for the convection-dominated anomalous diffusion equation in a rectangular plate using high-resolution compact discretization

The present method describes the high-resolution compact discretization method for the numerical solution of the nonlinear fractal convection-diffusion model on a rectangular plate by employing the Hausdorff distance metric. Estimation of anomalous diffusion is formulated by averaging forward and backward mesh stencils. The higher-order fractional derivatives are appropriately approximated on a minimum mesh stencil and subsequently considered for designing a numerical method that falls in the scope of expanded accuracy. Compact discretization is an efficient technique for partial differential equations; however, studies that apply high-resolution scheme for fractional-order systems are still uninvestigated. A second and fourth-order numerical method for the fractional-order convection-dominated anomalous diffusion equation in two dimensions is constructed for practical applications. Convergence of high-order method is obtained for the nonlinear partial differential equations employing Hausdorff fractal distance metric. The numerical simulations with fractal Graetz-Nusselt equation, fractal Poisson equation, fractal Schrödinger equation, and anomalous diffusion equations with variable and constant coefficients are considered to illustrate the utility of the numerical method in the context of local fractional partial differential equations.•The paper demonstrates a computational method for the fractal convection-diffusion model on a rectangular plate.•Two numerical methods of order two and four for the mildly nonlinear fractional-order convection-dominated anomalous diffusion equations are proposed.•The high-resolution scheme is computationally efficient and makes use of minimal data storage.Method name: High-order method for 2D convection-dominated anomalous diffusion equation, Graetz-Nusselt equation, Poisson equation, and Schrödinger equation in fractal media

On reducing and finding solutions of nonlinear evolutionary equations via generalized symmetry of ordinary differential equations.

We study symmetry reductions of nonlinear partial differential equations that can be used for describing diffusion processes in heterogeneous medium. We find ansatzes reducing these equations to systems of ordinary differential equations. The ansatzes are constructed using generalized symmetries of second-order ordinary differential equations. The method applied gives the possibility to find exact solutions which cannot be obtained by virtue of the classical Lie method. Such solutions are constructed for nonlinear diffusion equations that are invariant with respect to one-parameter and two-parameter Lie groups of point transformations. We prove a theorem relating the property of invariance of a found solution to the dimension of the Lie algebra admitted by the corresponding equation. We also show that the method is applicable to non-evolutionary partial differential equations and ordinary differential equations.

A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior

&lt;p style='text-indent:20px;'&gt;We study a free boundary problem of a reaction-diffusion equation &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ u_t = \Delta u+f(u) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ t&amp;gt;0,\ |x|&amp;lt;h(t) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; under a radially symmetric environment in &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \mathbb{R}^N $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. The reaction term &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ f $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; has positive bistable nonlinearity, which satisfies &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ f(0) = 0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and allows two positive stable equilibrium states and a positive unstable equilibrium state. The problem models the spread of a biological species, where the free boundary represents the spreading front and is governed by a one-phase Stefan condition. We show multiple spreading phenomena in high space dimensions. More precisely the asymptotic behaviors of solutions are classified into four cases: big spreading, small spreading, transition and vanishing, and sufficient conditions for each dynamical behavior are also given. We determine the spreading speed of the spherical surface &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ \{x\in \mathbb{R}^N:\ |x| = h(t)\} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, which expands to infinity as &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ t\to\infty $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, even when the corresponding semi-wave problem does not admit solutions.&lt;/p&gt;

A new minimizing-movements scheme for curves of maximal slope

Curves of maximal slope are a reference gradient-evolution notion in metric spaces and arise as variational formulation of a vast class of nonlinear diffusion equations. Existence theories for curves of maximal slope are often based on minimizing-movements schemes, most notably on the Euler scheme. We present here an alternative minimizing-movements approach, yielding more regular discretizations, serving as a-posteriori convergence estimator, and allowing for a simple convergence proof.

NUMERICAL QUENCHING FOR A NONLINEAR DIFFUSION EQUATION WITH SINGULAR BOUNDARY FLUX

In this paper, we study the semidiscrete approximation of the solution of a nonlinear diffusion equation with nonlinear source and singular boundary flux. We find some conditions under which the solution of the semidiscrete form quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time to the theoretical one when the mesh size tends to zero. Finally, we give some numerical experiments for a best illustration of our analysis.

Numerical scheme and analytical solutions to the stochastic nonlinear advection diffusion dynamical model

Abstract In this study, we give the numerical scheme to the stochastic nonlinear advection diffusion equation. This models is considered with white noise (or random process) having same intensity by changing frequencies. Furthermore, the stability and consistency of proposed scheme are also discussed. Moreover, it is concerned about the analytical solutions, the Riccati equation mapping method is adopted. The different families of single (shock and singular) and mixed (complex solitary-shock, shock-singular, and double-singular) form solutions are obtained with the different choices of free parameters. The graphical behavior of solutions is also depicted in 3D and corresponding contours.

High-Accuracy Time Discretization of Stochastic Fractional Diffusion Equation

A high-accuracy time discretization is discussed to numerically solve the nonlinear fractional diffusion equation forced by a space-time white noise. The main purpose of this paper is to improve the temporal convergence rate by modifying the semi-implicit Euler scheme. The solution of the equation is only Hölder continuous in time, which is disadvantageous to improve the temporal convergence rate. Firstly, the system is transformed into an equivalent form having better regularity than the original one in time. But the regularity of nonlinear term remains unchanged. Then, combining Lagrange mean value theorem and independent increments of Brownian motion leads to a higher accuracy discretization of nonlinear term which ensures the implementation of the proposed time discretization scheme without loss of convergence rate. Our scheme can improve the convergence rate from $${\min \{\frac{\gamma }{2\alpha },\frac{1}{2}\}}$$ to $${\min \{\frac{\gamma }{\alpha },1\}}$$ in the sense of mean-squared $$L^2$$ -norm. The theoretical error estimates are confirmed by extensive numerical experiments.

A Fast Preconditioned Semi-Implicit Difference Scheme for Strongly Nonlinear Space-Fractional Diffusion Equations

In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respectively. Stability and convergence properties of the proposed scheme are proved with the help of a discrete Grönwall inequality. Moreover, we extend the method to the solution of two-dimensional nonlinear models. A fast matrix-free implementation based on preconditioned Krylov subspace methods is presented for solving the discretized linear systems. The resulting fast preconditioned semi-implicit difference scheme reduces the memory requirement of conventional semi-implicit difference schemes from O(Ns2) to O(Ns) and the computational complexity from O(Ns3) to O(NslogNs) in each iterative step, where Ns is the number of space grid points. Experiments with two numerical examples are shown to support the theoretical findings and to illustrate the efficiency of our proposed method.

Finite volume schemes of evolutionary diffusion equations based on harmonic averaging interpolation

This paper focuses on developing cell-centered finite volume schemes on arbitrary distorted and unstructured meshes for evolutionary diffusion equations. On each mesh cell, by using the cell center and three harmonic averaging points on cell boundaries as the interpolation points, we approximate the normal flux in four different ways. To make these finite volume schemes cell-centered ones, we eliminate the intermediate variables in the flux expression by the harmonic averaging interpolation procedure. We discretize the time derivative term by the backward Euler scheme. Under some proper assumptions on mesh, we prove the stability and error estimate of the scheme theoretically. Considering the continuous, discontinuous, and heterogeneous diffusion tensors, even for the nonlinear diffusion equations, we construct four numerical experiments to investigate the convergence and robustness. Several different unstructured triangular, quadrilateral, and polygonal meshes are adopted. The numerical results imply that the former two schemes are robust for any kind of mesh and any kind of diffusion tensor, and they can obtain second-order convergence in $L^2$ norm and first-order convergence in $H^1$ norm; but the latter two schemes are more dependent on mesh.

Existence and Stability of the Solution to a System of Two Nonlinear Diffusion Equations in a Medium with Discontinuous Characteristics

Existence and Stability of the Solution to a System of Two Nonlinear Diffusion Equations in a Medium with Discontinuous Characteristics

An Efficient Discrete Model to Approximate the Solutions of a Nonlinear Double-Fractional Two-Component Gross–Pitaevskii-Type System

In this work, we introduce and theoretically analyze a relatively simple numerical algorithm to solve a double-fractional condensate model. The mathematical system is a generalization of the famous Gross–Pitaevskii equation, which is a model consisting of two nonlinear complex-valued diffusive differential equations. The continuous model studied in this manuscript is a multidimensional system that includes Riesz-type spatial fractional derivatives. We prove here the relevant features of the numerical algorithm, and illustrative simulations will be shown to verify the quadratic order of convergence in both the space and time variables.