Nonlinear Diffusion Equation Research Articles

Complicated Asymptotic Behavior of Solutions for the Cauchy Problem of Doubly Nonlinear Diffusion Equation

Complicated Asymptotic Behavior of Solutions for the Cauchy Problem of Doubly Nonlinear Diffusion Equation

Mathematical Simulation of Honeycomb Weathering via Moisture Transport and Salt Deposition

Honeycomb weathering is a common phenomenon found on various rock surfaces all around the world. However, honeycomb formation mechanisms are still poorly understood. In this study, we propose a model describing moisture transport within the sandstone and erosion resulting from salt deposition during evaporation of moisture off the rock surface. The moisture transport model is based on the non-linear diffusion equation, where the volumetric moisture content is a combined parameter accounting for the moisture and gas (vapor) content. The moisture transport model accounts for the several-orders-of-magnitude decrease in moisture diffusivity, observed during drying. It was assumed that erosion occurs when the evaporation front is located close to the rock surface. The depth of erosion is proportional to the moisture flow rate through the drying surface. The ABAQUS finite-element software suite was used for numerical solution of the non-linear diffusion equation. The iterative scheme of erosion simulation for different drying cycles was implemented using the Python programming language. Computations were conducted in the 2D setting for the square model with dimensions of 50 mm × 50 mm. Simulation results demonstrate the possibility of obtaining various landform shapes (honeycombs, tafoni) by varying only the value of the distribution of moisture content at the bottom side, simulating the rate of internal wetting of rock.

Diffusion - Convection Equation of Galactic Cosmic Rays (GCR) in the Atmosphere and Its Analytical, Numerical Solutions by Using Finite Elements Method Using Parker Transport Equation

This study depicts that to find the Analytical and numerical solutions of Diffusion – Convections Equations of Galactic Cosmic Rays (GCRs) by Finite Element method (FEM) and also to find the Energy Equation of GCRs by using a part of Parker’s transport equation. This considers moreover centres on the exactness and acknowledgment of the FEM strategy by utilizing dissemination blunder, scattering mistake and add up to blunder investigation. The comes about are depicted both graphically and in a unthinkable frame, which essentially guarantees the method’s legitimacy and the algorithm’s proficiency to maintain the exactness, effortlessness, and nonlinear Convection-Diffusion Equation conditions. The proposed method may be connected for tackling any nonlinear convection diffusion equation. We concentrate on assaying the confluence and stability of the nonlinear parabolic partial differential equation. This study focuses on the delicacy and acceptance of FEM method by exercising dispersion error dissipation error, and total error analysis.

Correction: Analytical Solution of the Nonlinear Hydraulic Diffusivity Equation with Depletion‑Dependent Permeability in Oil Wells Near Sealing Faults

Correction: Analytical Solution of the Nonlinear Hydraulic Diffusivity Equation with Depletion‑Dependent Permeability in Oil Wells Near Sealing Faults

An image-based meso-scale model for the hygro-mechanical time-dependent analysis of concrete

A new computational framework is developed in this paper for investigating the time-dependent behaviour of concrete including creep, shrinkage and cracking. The developed model aims to explain certain aspects of the time-dependent cracking and creep of concrete that cannot be captured using homogeneous models. The model is based on the scaled boundary finite element method, and it is coupled with a quadtree decomposition algorithm which converts digital images of concrete meso-structures into meshes. Concrete is treated as a two-phase composite which consists of elastic aggregates and mortar that is subjected to time-dependent deformation. The basic creep behaviour is treated as viscoelastic, which is modelled based on a rate-type rheological model corresponding to a Kelvin chain. Drying creep is modelled using a viscous unit which depends on the stress level, and drying shrinkage is stress independent. Both drying creep and drying shrinkage are related to the internal humidity. The humidity distribution within concrete is determined using a diffusion analysis. The moisture movement within mortar is governed by a nonlinear diffusion equation, whereas the aggregates are assumed impermeable. The cracking of concrete is explicitly modelled on the meso-scale through coupling of the continuum damage model for cracking within the mortar phase, and the cohesive zone model for debonding between aggregates and mortar. The proposed model is verified by simulating well-documented experimental studies in the literature. The capability of the proposed model in simulating the time-dependent behaviour of concrete and capturing the crack patterns has also been demonstrated.

Quasilinear Fractional Order Equations and Fractional Powers of Sectorial Operators

The fractional powers of generators for analytic operator semigroups are used for the proof of the existence and uniqueness of a solution of the Cauchy problem to a first order semilinear equation in a Banach space. Here, we use an analogous construction of fractional powers Aγ for an operator A such that −A generates analytic resolving families of operators for a fractional order equation. Under the condition of local Lipschitz continuity with respect to the graph norm of Aγ for some γ∈(0,1) of a nonlinear operator, we prove the local unique solvability of the Cauchy problem to a fractional order quasilinear equation in a Banach space with several Gerasimov–Caputo fractional derivatives in the nonlinear part. An analogous nonlocal Lipschitz condition is used to obtain a theorem of the nonlocal unique solvability of the Cauchy problem. Abstract results are applied to study an initial-boundary value problem for a time-fractional order nonlinear diffusion equation.

New results for convergence problem of fractional diffusion equations when order approach to $1^-$

This work studies the convergence problem for a class of fractional diffusion equations in which the time-derivative order approaches $1^-$. Up to now, few works have investigated this topic. The purpose of the article consists of three main contents. The first result is related to the convergence of the Caputo derivative and the Mittag-Leffler operators when $\alpha \to 1^-$. The second is to investigate the convergence problem for a linear fractional diffusion equation on $L^p$ spaces. And last result is concerned with the convergence problem for nonlinear fractional diffusion equations. The main analysis and techniques of the paper involve the evaluation related to Riemann-Liouville integration, Caputo derivative and Sobolev embeddings. Our analysis provides a complete and detailed answer to the convergence problem as fractional order tends to $1^-$.

Determination of a Nonlinear Coefficient in a Time-Fractional Diffusion Equation

In this paper, we study direct and inverse problems for a nonlinear time fractional diffusion equation. We prove that the direct problem has a unique weak solution and the solution depends continuously on the coefficient. Then we show that the inverse problem has a quasi-solution. The direct problem is solved by the method of lines using an operator approach. A quasi-Newton optimization method is used for the numerical solution to the inverse problem. The Tikhonov regularization is used to overcome the ill-posedness of the inverse problem. Numerical examples with noise-free and noisy data illustrate the applicability and accuracy of the proposed method to some extent.

Numerical approximation of the stochastic equation driven by the fractional noise

We consider the time-fractional nonlinear stochastic fourth-order reaction diffusion equation driven by the fractional noise (FSRDE). The FSRDE is discretized by using the mixed finite element method in space and the FBT-θ method in time. Some good techniques are proposed to prove the important lemmas in the proof of the strong convergence and the weak convergence. By proving the error estimates on the strong and weak convergence, we find the strong convergence and the weak convergence have the same convergence rate, and the convergence order is related to the fractional noise derivative β but not to the time fractional derivative α and the discrete format coefficient θ. Finally, the theorems on the strong and weak convergence are verified by the numerical tests.

Analytical Solution of the Nonlinear Hydraulic Diffusivity Equation with Depletion-Dependent Permeability in Oil Wells Near Sealing Faults

Analytical Solution of the Nonlinear Hydraulic Diffusivity Equation with Depletion-Dependent Permeability in Oil Wells Near Sealing Faults

The Quasi-Boundary Regularization Method for Recovering the Initial Value in a Nonlinear Time–Space Fractional Diffusion Equation

In this paper, we consider the inverse problem for identifying the initial value problem of the time–space fractional nonlinear diffusion equation. The uniqueness of the solution is proved by taking the fixed point theorem of Banach compression, and the ill-posedness of the problem is analyzed through the exact solution. The quasi-boundary regularization method is chosen to solve the ill-posed problem, and the error estimate between the regularization solution and the exact solution is given. Moreover, several numerical examples are chosen to prove the effectiveness of the quasi-boundary regularization method. Finally, our method can be used to solve high dimensional time–space fractional nonlinear diffusion equation, especially in cylindrical and spherical symmetric regions.

Glioblastoma multiforme growth prediction using a Proliferation-Invasion model based on nonlinear time-fractional 2D diffusion equation

The glioblastoma multiforme (GBM) is the fast-growing and aggressive type of tumor of the brain or spinal cord that is associated with high morbidity and mortality. Therefore, accurate knowledge of the tumor growth process in different time intervals seems necessary to adopt optimal treatment methods. In this study, with a combination of available early stages imaging data and using image processing methods, a time fractional reaction–diffusion equation based on the moving least squares method was implemented for predicting the growth of the GBM tumor model implemented in the mouse brain. The obtained results demonstrate that the Proliferation-Invasion model implemented by a time fractional form shows more agreement with the experimental results. As a result, using a fractional derivative of order α=0.9 results in the lowest prediction error of the tumor surface (less than 1%). In addition, to reducing the cost and side effects of theranostic methods, the presented model will have the possibility to consider the effects of therapeutic factors, such as hyperthermia, radiography, and surgery, on tumor growth. The ability to use the patient’s characteristics in diagnostic and treatment considerations and the development of personalized medicine can also be considered one of the capabilities of the presented model.

A novel numerical technique for solving time fractional nonlinear diffusion equations involving weak singularities

In this work, an efficient numerical approximation for the solution of the time fractional nonlinear diffuse interface model is studied. The solution to this problem has a weak singularity near the initial time . The fractional order nonlinear diffusion model is transformed into a system of nonlinear functional equations. The Daftardar–Gejji and Jafari method is employed to solve the corresponding nonlinear system. The L1 scheme is used to discretize the Caputo fractional derivative on a graded mesh in the time direction. In contrast, the spatial derivative is approximated by applying a classical central finite difference scheme to a uniform mesh. The convergence analysis and the error bounds are carried out. The analysis and the computational findings exhibit the effectiveness of the proposed method.

Numerical solution of two‐dimensional nonlinear Riesz space‐fractional reaction–advection–diffusion equation using fast compact implicit integration factor method

AbstractIn the present article, a finite domain is considered to find the numerical solution of a two‐dimensional nonlinear fractional‐order partial differential equation (FPDE) with Riesz space fractional derivative (RSFD). Here two types of FPDE–RSFD are considered, the first one is a two‐dimensional nonlinear Riesz space‐fractional reaction–diffusion equation (RSFRDE) and the second one is a two‐dimensional nonlinear Riesz space‐fractional reaction‐advection‐diffusion equation (RSFRADE). SFRDE is obtained by simply replacing second‐order derivative term of the standard nonlinear diffusion equation by the Riesz fractional derivative of order whereas the SFRADE is obtained by replacing the first‐order and second‐order space derivatives from the standard order advection–dispersion equation with the Riesz fractional derivatives of order . A numerical method is provided to deal with the RSFD with the weighted and shifted Grünwald–Letnikov (WSGD) approximations, for the spatial discretization. The SFRDE and SFRADE are transformed into a system of ordinary differential equations (ODEs), which have been solved using a fast compact implicit integration factor (FcIIF) with nonuniform time meshes. Finally, the demonstration of the validation and effectiveness of the numerical method is given by considering some existing models.

Topology optimization method with nonlinear diffusion

This paper is concerned with topology optimization based on a level set method using (doubly) nonlinear diffusion equations. Topology optimization using the level set method is called level set-based topology optimization, which is possible to determine optimal configurations that minimize objective functionals by updating level set functions. In this paper, as an update equation for level set functions, (doubly) nonlinear diffusion equations with reaction terms are derived, and then the singularity and degeneracy of the diffusion coefficient are applied to obtain fast convergence of configurations and damping oscillation on boundary structures. In particular, the reaction terms in the proposed method do not depend on the topological derivatives, and therefore, sensitivity analysis to determine a descent direction for objective functionals is relaxed. Furthermore, a numerical algorithm for the proposed method is constructed and applied to typical minimization problems to show numerical validity. This paper is a justification and generalization of the method using reaction–diffusion equations developed by one of the authors in Yamada et al. (2010).

Any order spectral volume methods for diffusion equations using the local discontinuous Galerkin formulation

In this paper, we present and study two spectral volume (SV) schemes of arbitrary order for diffusion equations by using the local discontinuous Galerkin formulation to discretize the viscous flux. The basic idea of the scheme is to rewrite the diffusion equation into an equivalent first-order system first, and then use the SV method to solve the system. The SV scheme is designed with control volumes constructed by using the Gauss points or Radau points in subintervals of the underlying meshes, which leads to two SV schemes referred to as LSV and RSV schemes, respectively. The stability analysis for the linear diffusion equations based on alternating fluxes are provided, and optimal error estimates are established for both the exact solution and the auxiliary variable. Furthermore, a rigorous mathematical proof are given to demonstrate that the proposed RSV method is identical to the standard LDG method when applied to constant diffusion problems. Numerical experiments are presented to demonstrate the stability, accuracy and performance of the two SV schemes for both linear and nonlinear diffusion equations.

Separable solutions to nonlinear anisotropic diffusion equation in elliptic coordinates.

Two- and three-dimensional exact solutions of the nonlinear diffusion equation are proved to exist in elliptic coordinates subject to an arbitrary piecewise constant azimuthal anisotropy. Degrees of freedom traditionally used to satisfy boundary conditions are instead employed to ensure continuity and conservation of mass across contiguity surfaces between subdomains of distinct diffusivities. Not all degrees of freedom are exhausted thereby, and conditions are given for the inclusion of higher harmonics. Degrees of freedom associated with one isotropic subdomain are always available to satisfy boundary conditions. The second harmonic is pivotal in the solution construction as well as the identification of partial symmetries in the domain partition. The anisotropy gives rise to an unconventional mixed type critical point that combines saddle and node-like characteristics. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Mechanical formation damage management in oil wells with finite extent hydraulic fractures and source effects using an asymptotic-perturbation method

The petroleum industry has used hydraulic fracturing technology to solve various problems and provide an understanding of many complex issues. This work presents a new analytical model to evaluate the effective permeability response during the oil’s production through a vertical well fully penetrating a pressure-sensitive reservoir with a finite extent hydraulic fracture. The nonlinear hydraulic diffusivity equation (NHDE) is solved analytically through an integro-differential model coupled with Green’s function (GF) related to an oil source plan representing a finite extent hydraulic fracture. A new effective hydraulic diffusivity deviator factor is also derived to represent the permeability loss as a function of the pore pressure throughout the well-reservoir life cycle. For this oil flow modeling, the general solution is expressed by the sum of the linear solution (constant permeability) plus a corrective term given by the combination of the exponential function Ei(xD,yD,tD) and the complementary error function erfc(xD,yD,tD). The proposed model is implemented in Matlab software to evaluate the effect of the effective deviator factor in the permeability response over the oil’s production period. The model calibration is performed through a numerical oil flow simulator, widely used in the petroleum industry, and the results were accurate.

Space-time homogenization for nonlinear diffusion

The present paper is concerned with a space-time homogenization problem for nonlinear diffusion equations with periodically oscillating (in space and time) coefficients. Main results consist of a homogenization theorem (i.e., convergence of solutions as the period of oscillation goes to zero) as well as a characterization of homogenized equations. In particular, homogenized matrices are described in terms of solutions to cell-problems, which have different forms depending on the log-ratio of the spatial and temporal periods of the coefficients. At a critical ratio, the cell problem turns out to be a parabolic equation in microscopic variables (as in linear diffusion) and also involves the limit of solutions, which is a function of macroscopic variables. The latter feature stems from the nonlinearity of the equation, and moreover, some strong interplay between microscopic and macroscopic structures can explicitly be seen for the nonlinear diffusion. As for the other ratios, the cell problems are always elliptic (in micro-variable only) and do not involve any macroscopic variables, and hence, micro- and macrostructures are weakly interacting each other. Proofs of the main results are based on the two-scale convergence theory (for space-time homogenization). Furthermore, finer asymptotics of gradients, diffusion fluxes and time-derivatives with certain corrector terms are provided, and a qualitative analysis on homogenized matrices is also performed.

Exponential Tracking and Disturbance Rejection for Nonlinear Reaction Convection Diffusion Equations via Boundary Control

This paper studies the problem of exponential tracking and disturbance rejection for nonlinear reaction convection diffusion equations via Dirichlet or Neumann boundary control. Disturbances are present both in the equation and on the boundary. The tracking problem is simply a nonlinear dynamic regulator problem. If an initial condition is not compatible with a reference, the nonlinear dynamic regulator problem does not have a solution. Thus, we split the tracking problem into two problems to relax the compatibility condition. We choose one problem to be exponentially stable and leave everything else in a dynamic regulator problem. Then a controller is just a solution component of the dynamic regulator problem, while the tracking error is given by the solution of the exponentially stable problem. The controller can be obtained by integrating the dynamic regulator equation and using the reference condition. The dynamic regulator problem with the obtained controller becomes an initial boundary value problem. We then transfer the initial boundary value problem into an equivalent integral equation and prove that the integral equation has a unique global solution if the nonlinearity is globally Lipschitz. In addition, a numerical example is given to illustrate the theoretical results.