Time-fractional Diffusion Research Articles

Stable Computational Techniques for the Advection–Dispersion Variable Order Model

In this paper, a novel approximation to the Caputo time fractional derivative of variable order (VO) [Formula: see text] ([Formula: see text]) is established. Since time fractional derivatives are integral, with a weakly singular kernel the discretization on the uniform mesh may not lead to satisfactory accuracy. So, numerical approximation is constructed on nonuniform meshes based on Newton interpolation polynomial. As a result, the novel approximation can be viewed as the modification of the existing method [Shen, S., Liu, F., Chen, J., Turner, I. and Anh, V. [2012] “Numerical techniques for the variable order time fractional diffusion equation,” Appl. Math. Comput. 218(22), 10861–10870]. The truncation errors and some significant properties of coefficients are analyzed. Furthermore, we derived Scheme-I and Scheme-II with second-order accuracy in spatial direction to solve the mobile–immobile advection–dispersion model (MIMADM) based on the VO Caputo derivative. The designed scheme converts the considered problems into a linear system of equations. Moreover, the stability properties of Scheme-I are discussed in detail. Ultimately, the proposed scheme is examined on several text examples demonstrating the high precision and effectiveness of the scheme. Also, comparative study between the outcomes achieved by implementing the suggested scheme with other numerical scheme in the existing literature is provided and also the acquired numerical results of these applications indicate that the suggested technique performs extremely well in terms of reliability and capability.

An inverse problem of determining the fractional order in the TFDE using the measurement at one space-time point

In this paper we are concerned with an inverse problem for determining the order of time fractional derivative in a time fractional diffusion equation (TFDE in short), where the available measurement is given at a single space-time point. The inverse problem is transformed to a nonlinear algebraic equation of the fractional order based on the solution to the forward problem. We prove that the algebraic equation possesses a unique solution by the strict monotonicity of the Mittag-Leffler function, and the inverse problem is of uniqueness. Numerical examples are presented to show the unique solvability of the inverse problem.

A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives

This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order 0<α<1, and 1<β<2. The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered.

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.

Analysis of Time Fractional Diffusion Equation Arising in Ocean Pollution with Different Kernels

Analysis of Time Fractional Diffusion Equation Arising in Ocean Pollution with Different Kernels

Improved Meshless Finite Integration Method for Solving Time Fractional Diffusion Equations

In this paper, a new method named Improved Finite Integration Method (IFIM) is proposed for solving Time Fractional Diffusion Equations (TFDEs). In the IFIM, the Extended Simpson’s Rule (ESR) is employed for numerical quadrature in spatial discretization. Besides, the Piecewise Quadratic Interpolation (PQI) in sense of the Hadamard finite-part integral is utilized for time discretization. Compared with the primary Finite Integration Method (FIM) with Trapezoidal rule which uses the finite difference scheme to address the time discretization, the combination of ESR and PQI in IFIM will lead to a better performance in solving TFDEs. Numerical examples are performed and compared to show the superiority of IFIM. It can also be found that the IFIM is able to obtain a higher accuracy without losing the stability and efficiency.

L1/LDG Method for Caputo-Hadamard Time Fractional Diffusion Equation.

In this paper, a class of discrete Gronwall inequalities is proposed. It is efficiently applied to analyzing the constructed L1/local discontinuous Galerkin (LDG) finite element methods which are used for numerically solving the Caputo-Hadamard time fractional diffusion equation. The derived numerical methods are shown to be -robust using the newly established Gronwall inequalities, that is, it remains valid when . Numerical experiments are given to demonstrate the theoretical statements.

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.

Approximate Solution of Sub diffusion Bio heat Transfer Equation

In this paper, author’s study sub diffusion bio heat transfer model and developed explicit finite difference scheme for time fractional sub diffusion bio heat transfer equation by using caputo fabrizio fractional derivative. Also discussed conditional stability and convergence of developed scheme. Furthermore numerical solution of time fractional sub diffusion bio heat transfer equation is obtained and it is represented graphically by Python.

The Two Stage Moisture Diffusion Model for Non-Fickian Behaviors of 3D Woven Composite Exposed Based on Time Fractional Diffusion Equation

The moisture diffusion behaviors of 3D woven composites exhibit non-Fickian properties when they are exposed to a hydrothermal environment. Although some experimental works have been undertaken to investigate this phenomenon, very few mathematical works on non-Fickian moisture diffusion predictions of 3D woven composites are available in the literature. To capture the non-Fickian behavior of moisture diffusion in 3D woven composites, this study first utilized a time fractional diffusion equation to derive the percentage of moisture content of a homogeneous material under hydrothermal conditions. A two-stage moisture diffusion model was subsequently developed based on the moisture diffusion mechanics of both neat resin and 3D woven composites, which describes the initial fast diffusion and the long-term slow diffusion stages. Notably, the model incorporated fractional order parameters to account for the nonlinear property of moisture diffusion in composites. Finally, the weight gain curves of neat resin and the 3D woven composite were calculated to verify the fractional diffusion model, and the predicted moisture uptake curves were all in good agreement with the experimental results. It is important to note that when the fractional order parameter α < 1, the initial moisture uptake will become larger with a later slow down process. This phenomenon can better describe non-Fickian behavior caused by initial voids or complicated structures.

Error analysis of a finite difference scheme on a modified graded mesh for a time-fractional diffusion equation

This paper is concerned with a finite difference scheme on a new modified graded mesh for a time fractional diffusion equation with a Caputo fractional derivative of order α∈(0,1). At first, the construction and some basic properties of this new modified graded mesh are investigated and on its basis the L1 scheme is applied to approximate the Caputo derivative. Meanwhile, the standard center finite difference scheme on a uniform mesh is used to discretize the diffusion term. Then, stability and convergence of the proposed scheme in the maximum norm are proved. The convergence result shows that on this modified graded mesh one attains an optimal 2−α rate for the L1 scheme. Finally, the presented theoretical results are supported by some numerical experiments.

A stability result of a fractional heat equation and time fractional diffusion equations governed by fractional fluxes in the Heisenberg group

A stability result of a fractional heat equation and time fractional diffusion equations governed by fractional fluxes in the Heisenberg group

The long time error estimates for the second order backward difference approximation to sub-diffusion equations with boundary time delay and feedback gain

We consider time fractional diffusion initial boundary value problem with boundary time delay and feedback gain involving two Caputo fractional derivatives in time. When both the time delay and feedback gain are taken as appropriately small such that the characteristic equation has no solution in the location of the right half plane of the imaginary axis, we derive the solution’s existence, uniqueness and representation. The second-order backward difference time discrete schemes are investigated in time discretization. The lt∞(0,∞;L2) error analysis of the numerical schemes is derived when both the time delay and feedback gain are appropriately small such that the characteristic equation of the discrete problem has no solution in a strip of the right half plane of the imaginary axis, and the order of boundary Caputo fractional derivative is equal to the order or half order of interior domain Caputo fractional derivative. The numerical examples illustrate its accuracy, efficiency and robustness, and are consistent with the theoretical results.

A computational approach based on the fractional Euler functions and Chebyshev cardinal functions for distributed-order time fractional 2D diffusion equation

In this paper, the distributed-order time fractional diffusion equation is introduced and studied. The Caputo fractional derivative is utilized to define this distributed-order fractional derivative. A hybrid approach based on the fractional Euler functions and 2D Chebyshev cardinal functions is proposed to derive a numerical solution for the problem under consideration. It should be noted that the Chebyshev cardinal functions process many useful properties, such as orthogonality, cardinality and spectral accuracy. To construct the hybrid method, fractional derivative operational matrix of the fractional Euler functions and partial derivatives operational matrices of the 2D Chebyshev cardinal functions are obtained. Using the obtained operational matrices and the Gauss–Legendre quadrature formula as well as the collocation approach, an algebraic system of equations is derived instead of the main problem that can be solved easily. The accuracy of the approach is tested numerically by solving three examples. The reported results confirm that the established hybrid scheme is highly accurate in providing acceptable results.

A collocation method for time‐fractional diffusion equation on a metric star graph with η$$ \eta $$ edges

A wavelet collocation method based on Haar wavelet is proposed to investigate the numerical solution of time fractional diffusion equation (TFDE) on a metric star graph. We have utilized the Riemann–Liouville definition of fractional integral operator together with Haar wavelet to obtain the Riemann–Liouville fractional integral operator for Haar wavelet (RLFIO‐H). The application of RLFIO‐H and Haar wavelet to TFDE on metric star graph returns a system of linear algebraic equations. Solving this system yields wavelet coefficients and subsequently the solution. The convergence analysis of the proposed method is discussed to establish the theoretical authenticity of the method. The proposed method is tested on four benchmark problems to validate the theoretical findings. Moreover, the comparison of the developed method with the existing method shows the superiority and accuracy of the proposed method. To the best of the author's knowledge, the proposed work is one of the first collocation method in the literature that deals with the solution of TFDE on metric star graph using wavelet numerically.

Threshold Results for the Existence of Global and Blow-Up Solutions to a Time Fractional Diffusion System with a Nonlinear Memory Term in a Bounded Domain

In this paper, we consider a time fractional diffusion system with a nonlinear memory term in a bounded domain. We mainly prove some blow-up and global existence results for this problem. Moreover, we also give the decay estimates of the global solutions. Our proof relies on the eigenfunction method combined with the asymptotic behavior of the solution of a fractional differential inequality system, the estimates of the solution operators and the asymptotic behavior of the Mittag–Leffler function. In particular, we give the critical exponents of this problem in different cases. Our results show that, in some cases, whether one of the initial values is identically equal to zero has a great influence on blow-up and global existence of the solutions for this problem, which is a remarkable property of time fractional diffusion systems because the classical diffusion systems can not admit this property.

Investigation and analysis of the numerical approach to solve the multi-term time-fractional advection-diffusion model

<abstract><p>In this paper, a methodical approach is presented to approximate the multi-term fractional advection-diffusion model (MT-FAD). The Lagrange squared interpolation is used to discretize temporal fractional derivatives, and Legendre polynomials are shifted as an operator to discretize the spatial fractional derivatives. The advantage of these numerical techniques lies in the orthogonality of Legendre polynomials and its matrix operations. A quadratic implicit design as well as its stability and convergence analysis are evaluated. It should be noted that the theoretical proof obtained from this study represents the first results for these numerical schemes. Finally, we provide three numerical examples to verify the validity of the proposed methods and demonstrate their accuracy and effectiveness in comparison with previous studies shown in [W. P. Bu, X. T. Liu, Y. F. Tang, J. Y. Yang, Finite element multigrid method for multi-term time fractional advection diffusion equations, <italic>Int. J. Model. Simul. Sci. Comput.</italic>, <bold>6</bold> (2015), 1540001].</p></abstract>

Numerical study for a class of time fractional diffusion equations using operational matrices based on Hosoya polynomial

<abstract><p>In this paper, we develop a numerical method by using operational matrices based on Hosoya polynomials of simple paths to find the approximate solution of diffusion equations of fractional order with respect to time. This method is applied to certain diffusion equations like time fractional advection-diffusion equations and time fractional Kolmogorov equations. Here we use the Atangana-Baleanu fractional derivative. With the help of this approach we convert these equations to a set of algebraic equations, which is easier to be solved. Also, the error bound is provided. The obtained numerical solutions using the presented method are compared with the exact solutions. The numerical results show that the suggested method is convenient and accurate.</p></abstract>

Error estimate of L1-ADI scheme for two-dimensional multi-term time fractional diffusion equation

<abstract><p>A two-dimensional multi-term time fractional diffusion equation $ {D}_{t}^{\mathit{\boldsymbol{\alpha}}} u(x, y, t)- \Delta u(x, y, t) = f(x, y, t) $ is considered in this paper, where $ {D}_{t}^{\mathit{\boldsymbol{\alpha}}} $ is the multi-term time Caputo fractional derivative. To solve the equation numerically, L1 discretisation to each fractional derivative is used on a graded temporal mesh, together with a standard finite difference method for the spatial derivatives on a uniform spatial mesh. We provide a rigorous stability and convergence analysis of a fully discrete L1-ADI scheme for solving the multi-term time fractional diffusion problem. Numerical results show that the error estimate is sharp.</p></abstract>

Exponential Tikhonov Regularization Method for Solving an Inverse Source Problem of Time Fractional Diffusion Equation

Exponential Tikhonov Regularization Method for Solving an Inverse Source Problem of Time Fractional Diffusion Equation