Time-fractional Diffusion Model Research Articles

Simultaneous numerical inversion of space-dependent initial condition and source term in multi-order time-fractional diffusion models

This article deals with a simultaneous reconstruction of unknown initial conditions and space-dependent source function in multi-order time-fractional diffusion problems. We discuss the existence and uniqueness of the direct problem. The problem is presented as a regularized optimization problem and converted into a variational problem. The existence of the minimizer for the optimization problem is demonstrated. For the numerical part, a modified Levenberg-Marquardt regularization approach is constructed to identify the initial condition and source function. Several numerical examples in one and two dimensions are employed to test the performance of the identification technique.

An artificial neural network approach to identify the parameter in a nonlinear subdiffusion model

In this paper, we propose an artificial neural network approach to identify the parameter in a non-linear subdiffusion model from additional data. Instead of determining the parameter in the time fractional diffusion model by its form itself, we approximate it in the form of an artificial neural network. The key point of this approach relies on the approximation capability of neural networks. We formulate this inverse problem as an optimal control one, and we demonstrate the existence of the solution for the control problem and provide a mathematical analysis and the derivation of optimal conditions. Moreover, various numerical tests of the regular and singular examples have shown that the artificial neural network method (ANN) is effective. This is reinforced by its numerical comparison with the gradient descent, the alternating direction multiplier method (ADMM), physics-informed neural network (PINN) and DeepONet method.

Influence of pseudoboehmite on the performance of shaped mordenite catalyst for dimethyl ether carbonylation

As a key step of indirect production of ethanol from syngas, the carbonylation of dimethyl ether (DME) has attracted much attention. It is of great importance to develop shaped catalyst with high efficiency and strong mechanical properties to promote the process industrialization. Extruded mordenite (MOR) has been used as the catalyst for industrial DME carbonylation. In this work, the effect of properties and content of pseudoboehmite as a binder on mechanical strength and catalytic performance of the extruded MOR were systematically studied. Combining TG, XRD, N2 adsorption, and rheological property analysis etc., we found that the interlayer water content of pseudoboehmite affect peptization process, which leads to the difference in mechanical strength and pore structure. Then, the DME carbonylation performance of the extruded MOR was discussed based on textural parameters and acidic properties. Further, DME diffusion experiments were conducted and analyzed by using standard and the time-fractional diffusion models respectively. As a result, the influence of pseudoboehmite on mass transfer performance was quantitatively analyzed. The results will provide important guidance for the development of shaped zeolite catalysts in industry.

A study of distributed‐order time fractional diffusion models with continuous distribution weight functions

AbstractThe distributed‐order time fractional diffusion model with Dirichlet nonhomogeneous boundary conditions on a finite domain is considered. Four choices of continuous distribution weight functions with mean μ and standard deviation σ are investigated to study their impact on both the short‐time and long‐time solution behavior. An implicit numerical method implemented on a graded mesh is proposed to solve the model and the stability and convergence analysis are presented. Semi‐analytic solutions are also derived for these distributions to assess the accuracy of the scheme. Numerical results highlight that the four continuous distribution weight functions produce a short‐time solution behavior that is consistent with those solutions from the classical time fractional partial differential equation with fractional order γ* = μ. There are however long‐time differences in the solution behavior that become more distinguishable as σ increases. In particular, we find a smaller value of σ produces more diffuse profiles and the diffusion rate slows as σ increases. Furthermore, the asymptotic behavior of the solution may be influenced by the time‐fractional orders ranging between the smallest nonzero weight order and mean μ for the continuous uniform and raised cosine distribution weight functions, respectively. Similar findings are also observed for the truncated normal and beta distributions.

WELL-POSEDNESS AND REGULARITY OF CAPUTO–HADAMARD TIME-FRACTIONAL DIFFUSION EQUATIONS

Ultraslow diffusion describes the long-time diffusion of particles whose mean square displacement (MSD) grows logarithmically in time. We prove the well-posedness of a Caputo–Hadamard time-fractional diffusion model in multiple space dimensions, in which the MSD in time grows logarithmically and thus provides adequate descriptions for the ultraslow diffusion processes, as well as the smoothing properties of the solutions.

Variable-Order Fractional Diffusion Model-Based Medical Image Denoising

Fractional differential models are playing a vital role in many applications such as diffusion, probability potential theory, and scattering theory. In this study, the variable-order space and time fractional diffusion model is employed for denoising the medical images. The finite difference approach is implemented to find the numerical solution of the proposed model. Convergence and stability of the numerical method are presented. The experimental outcomes of the variable-order model are analyzed with those of the fractional and integer-order diffusion models. It was noticed that the peak signal-to-noise ratio (PSNR) value is increased considerably for the proposed model.

Recovering multiple fractional orders in time-fractional diffusion in an unknown medium

In this work, we investigate an inverse problem of recovering multiple orders in a time-fractional diffusion model from the data observed at one single point on the boundary. We prove the unique recovery of the orders together with their weights, which does not require a full knowledge of the domain or medium properties, e.g. diffusion and potential coefficients, initial condition and source in the model. The proof is based on Laplace transform and asymptotic expansion. Furthermore, inspired by the analysis, we propose a numerical procedure for recovering these parameters based on a nonlinear least-squares fitting with either fractional polynomials or rational approximations as the model function, and provide numerical experiments to illustrate the approach for small time t .

Local discontinuous Galerkin approximations to variable-order time-fractional diffusion model based on the Caputo–Fabrizio fractional derivative

In this paper, we construct and investigate an accurate numerical scheme for solving a class of variable-order (VO) fractional diffusion equation based on the Caputo–Fabrizio fractional derivative. The scheme is presented by using a finite difference method in temporal variable and a local discontinuous Galerkin method (LDG) in space. For all variable-order α(t)∈(0,1), we derive the stability and L2 convergence of proposed scheme and prove that the method is of accuracy-order O(τ+hk+1), where τ, h and k are temporal step sizes, spatial step sizes and the degree of piecewise Pk polynomials, respectively. Several numerical tests are given to validate the theoretical analysis and efficiency of the proposed algorithm.

A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass transfer

IntroductionDuring the last years the modeling of dynamical phenomena has been advanced by including concepts borrowed from fractional order differential equations. The diffusion process plays an important role not only in heat transfer and fluid flow problems, but also in the modelling of pattern formation that arises in porous media. The modified time-fractional diffusion equation provides a deeper understanding of several dynamic phenomena. ObjectivesThe purpose of the paper is to develop an efficient meshless technique for approximating the modified time-fractional diffusion problem formulated in the Riemann–Liouville sense. MethodsThe temporal discretization is performed by integrating both sides of the modified time-fractional diffusion model. The unconditional stability of the time discretization scheme and the optimal convergence rate are obtained. Then, the spatial derivatives are discretized through a local hybridization of the cubic and Gaussian radial basis function. This hybrid kernel improves the condition of the system matrix. Therefore, the solution of the linear system can be obtained using direct solvers that reduce significantly computational cost. The main idea of the method is to consider the distribution of data points over the local support domain where the number of points is almost constant. ResultsThree examples show that the numerical procedure has good accuracy and applicable over complex domains with various node distributions. Numerical results on regular and irregular domains illustrate the accuracy, efficiency and validity of the technique. ConclusionThis paper adopts a local hybrid kernel meshless approach to solve the modified time-fractional diffusion problem. The main results of the research is the numerical technique with non-uniform distribution in irregular grids.

A computational weighted finite difference method for American and barrier options in subdiffusive Black–Scholes model

Subdiffusion is a well established phenomenon in physics. In this paper we apply the subdiffusive dynamics to analyze financial markets. We focus on the financial aspect of time fractional diffusion model with moving boundary i.e. American and barrier option pricing in the subdiffusive Black–Scholes (B–S) model. Two computational methods for valuing American options in the considered model are proposed - the weighted finite difference (FD) and the Longstaff–Schwartz method. In the article it is also shown how to valuate numerically wide range of barrier options using the FD approach.

Fractional Jacobi Galerkin spectral schemes for multi-dimensional time fractional advection–diffusion–reaction equations

One of the ongoing issues with time fractional diffusion models is the design of efficient high-order numerical schemes for the solutions of limited regularity. We construct in this paper two efficient Galerkin spectral algorithms for solving multi-dimensional time fractional advection–diffusion–reaction equations with constant and variable coefficients. The model solution is discretized in time with a spectral expansion of fractional-order Jacobi orthogonal functions. For the space discretization, the proposed schemes accommodate high-order Jacobi Galerkin spectral discretization. The numerical schemes do not require imposition of artificial smoothness assumptions in time direction as is required for most methods based on polynomial interpolation. We illustrate the flexibility of the algorithms by comparing the standard Jacobi and the fractional Jacobi spectral methods for three numerical examples. The numerical results indicate that the global character of the fractional Jacobi functions makes them well-suited to time fractional diffusion equations because they naturally take the irregular behavior of the solution into account and thus preserve the singularity of the solution.

Fast calculation based on a spatial two‐grid finite element algorithm for a nonlinear space–time fractional diffusion model

AbstractIn this article, a spatial two‐grid finite element (TGFE) algorithm is used to solve a two‐dimensional nonlinear space–time fractional diffusion model and improve the computational efficiency. First, the second‐order backward difference scheme is used to formulate the time approximation, where the time‐fractional derivative is approximated by the weighted and shifted Grünwald difference operator. In order to reduce the computation time of the standard FE method, a TGFE algorithm is developed. The specific algorithm is to iteratively solve a nonlinear system on the coarse grid and then to solve a linear system on the fine grid. We prove the scheme stability of the TGFE algorithm and derive a priori error estimate with the convergence result O(Δt2 + hr + 1 − η + H2r + 2 − 2η). Finally, through a two‐dimensional numerical calculation, we improve the computational efficiency and reduce the computation time by the TGFE algorithm.

A New Kind of Parallel Natural Difference Method for Multi-Term Time Fractional Diffusion Model

Multi-term time fractional diffusion model is not only an important physical subject, but also a practical problem commonly involved in engineering. In this paper, we apply the alternating segment technique to combine the classical explicit and implicit schemes, and propose a parallel nature difference method alternating segment pure explicit–implicit (PASE-I) and alternating segment pure implicit–explicit (PASI-E) difference schemes for multi-term time fractional order diffusion equations. The existence and uniqueness of the solutions are proved, and stability and convergence analysis of the two schemes are also given. Theoretical analyses and numerical experiments show that the PASE-I and PASI-E schemes are unconditionally stable and satisfy second-order accuracy in spatial precision and 2 − α order in time precision. When the computational accuracy is equivalent, the CPU time of the two schemes are reduced by up to 2 / 3 compared with the classical implicit difference method. It indicates that the PASE-I and PASI-E parallel difference methods are efficient and feasible for solving multi-term time fractional diffusion equations.

Implicit sampling for hierarchical Bayesian inversion and applications in fractional multiscale diffusion models

This paper presents an implicit sampling method for hierarchical Bayesian inverse problems. A widely used approach for sampling posterior distribution is Markov chain Monte Carlo (MCMC). However, the samples generated by MCMC are usually strongly correlated, which may lead to a small size of effective samples. The implicit sampling method proposed in Chorin and Tu (2009) can generate independent samples and capture inherent non-Gaussian features of the posterior. In implicit sampling method, the posterior samples are generated by a map and distribution around the maximum a posterior (MAP) point. For practical applications, some parameters in prior density are often unknown and the hierarchical Bayesian formulation is used to estimate the MAP point effectively. It is applied to the Bayesian inverse problems of multi-term time fractional diffusion models in heterogeneous media. To effectively capture the heterogeneity effect, we present a mixed generalized multiscale finite element method (mixed GMsFEM) to substantially speed up the Bayesian inversion. Finally, we carry out a few numerical examples to effectively identify various unknown inputs.

Macroscale modeling the methanol anomalous transport in the porous pellet using the time-fractional diffusion and fractional Brownian motion: A model comparison

In this paper, we present physically-based macroscale models for the simulation of the mass transport process through the solid porous media. The models are based on the standard diffusion, time-fractional diffusion, and the fractional Brownian motion diffusion equations respectively. For the experimental verification of the developed models, the transport process in zeolite ZSM-5 pellet was studied using the methanol as a probing agent. Treating the experimental data by the second Fick's law of diffusion demonstrated no correspondence between the methanol transport and the Fickian concept. Therefore, the models, based on the time-fractional diffusion and the fractional Brownian motion, were adopted for a description of the experimental mass transfer kinetic. Both models provided an excellent correspondence between the experimental data and the relevant asymptotic solutions. For the time-fractional diffusion model, the asymptotic analysis of the experimental data revealed the equivalence of the anomalous diffusion exponents measured at the short and the long times. In contrast, the anomalous diffusion exponents estimated in the frame of the fractional Brownian motion model are different for the short and the long times. These findings allowed us to speculate on the applicability of the certain anomalous diffusion model in the experimental scenario studied. The obtained conclusions are additionally supported by an analysis of the mass transfer data presented in the literature.

Investigation of the anomalous diffusion in the porous media: a spatiotemporal scaling

The methanol and the methane transport in the mesoporous zeolite-alumina pellet was studied. Varying the pellet’s size and the diffusant amount, the values of the diffusion coefficients were estimated. Previously, it had been demonstrated that the methane transport through the zeolite-containing pellet is described by the Fickian diffusion equation, whereas the methanol transport is described by the non-Fickian diffusion equation with either space-time-fractional or time-fractional derivative. In this respect, for calculating the diffusion coefficients, the standard diffusion and the time-fractional diffusion models were used for the methane and the methanol respectively. The relations between the obtained values of the methanol non-Fickian diffusion coefficients (0.0068–0.0276 cm2/sα) measured for unequal pellet sizes and various diffusant amounts were found to follow the temporal scaling with a fractional exponent equal to 1.17 ± 0.03, which corresponds to the time-fractional diffusion equation, in a concise manner. It supported a conclusion that the anomalous diffusion of the methanol is time-fractional. An essential applicability of the approach based on the analysis of the temporal diffusion coefficient scaling was additionally verified using the standard Fickian diffusion coefficients of the methane (0.00094–0.00376 cm2/s). In addition, we demonstrate that the investigation of the anomalous diffusion regime using the spatial scaling of the diffusion coefficient is restricted by the measurement of the diffusion length of a certain diffusant in a porous material.

Application of Nonlinear Time-Fractional Partial Differential Equations to Image Processing via Hybrid Laplace Transform Method

This work considers a hybrid solution method for the time-fractional diffusion model with a cubic nonlinear source term in one and two dimensions. Both Dirichlet and Neumann boundary conditions are considered for each dimensional case. The hybrid method involves a Laplace transformation in the temporal domain which is numerically inverted, and Chebyshev collocation is employed in the spatial domain due to its increased accuracy over a standard finite-difference discretization. Due to the fractional-order derivative we are only able to compare the accuracy of this method with Mathematica’s NDSolve in the case of integer derivatives; however, a detailed discussion of the merits and shortcomings of the proposed hybridization is presented. An application to image processing via a finite-difference discretization is included in order to substantiate the application of this method.

Quenching Phenomenon of a Time-Fractional Kawarada Equation

In this paper, we introduce a class of time-fractional diffusion model with singular source term. The derivative employed in this model is defined in the Caputo sense to fit the conventional initial condition. With assistance of corresponding linear fractional differential equation, we verify that the solution of such model may not be globally well-defined, and the dynamics of this model depends on the order of fractional derivative and the volume of spatial domain. In simulation, a finite difference scheme is implemented and interesting numerical solutions of model are illustrated graphically. Meanwhile, the positivity, monotonicity, and stability of the proposed scheme are proved. Numerical analysis and simulation coincide the theoretical studies of this new model.

Time-fractional diffusion model on dynamical effect of dendritic cells on HIV pathogenesis

Chronic immune activation and viral latency are among the main factors behind the persistence of Human Immunodeficiency Virus (HIV) and other viral diseases. Immune activation is chiefly lead by dendritic cells which detect the presence of Virus (HIV) and notify other immune cells particularly the helper T cells via the process called antigen presentation. However, antigen presentation process increases the chances of T cells to be infected by the viral particles trapped by dendritic cells. In this paper, a time-fractional diffusion model is proposed to investigate the impact of dendritic cells on the spread of HIV among susceptible T cells with respect to time and anomalous diffusion posed by cells crowding. The equilibrium points of the model are obtained and analysed. The disease free steady state proved to be stable both in spatially homogeneous case and in the presence diffusion and chemotaxis. The endemic steady state is stable in the absence of diffusion and chemotaxis, however the presence of diffusion induces instability if a thresshold value is exceeded. A priori estimates were also obtained in the appropriate Sobolev spaces. Furthermore, the numerical experiments were conducted to examine the dynamic behaviour of cells densities with respect to time and the sub-diffusion parameter.

Bayesian Inference Using Intermediate Distribution Based on Coarse Multiscale Model for Time Fractional Diffusion Equations

In the paper, we present a strategy for accelerating posterior inference for unknown inputs in time fractional diffusion models. In many inference problems, the posterior may be concentrated in a s...