Time-fractional Diffusion Equation Research Articles

An α-robust and new two-grid nonuniform L2-1 FEM for nonlinear time-fractional diffusion equation

An α-robust and new two-grid nonuniform L2-1 FEM for nonlinear time-fractional diffusion equation

Lattice Boltzmann method for tempered time-fractional diffusion equation

Tempered fractional calculus, as an extension of fractional calculus, has been successfully applied in numerous scientific and engineering fields. Although several traditional numerical methods have been improved for solving a variety of tempered fractional partial differential equations, solving these equations by the lattice Boltzmann (LB) method is an unresolved issue. This paper is dedicated to presenting a novel LB method for the tempered time-fractional diffusion equation. The tempered time-fractional diffusion equation is first transformed into an integer-order partial differential equation by approximating the tempered fractional derivative term. Then the LB method is proposed to solve the transformed objective equation. The Chapman-Enskog procedure is conducted to confirm that the present LB method can accurately recover the objective equation. Some numerical examples with an analytical solution are employed to validate the present LB method, and a strong consistency is observed between the numerical and analytical solutions. The numerical simulations indicate that the LB method is a second-order accurate scheme. The proposed LB method presents a new approach to solving the tempered time-fractional diffusion equation, which is beneficial for the widespread application of the tempered time-fractional diffusion equation in addressing complex transport problems.

Inverse coefficient problems for one-dimensional time-fractional diffusion equations

We prove the uniqueness in determining a spatially varying zeroth-order coefficient of a one-dimensional time-fractional diffusion equation by initial value and Cauchy data at one end point of the spatial interval.

The Second Critical Exponent for a Time-Fractional Reaction-Diffusion Equation

In this paper, we consider the Cauchy problem of a time-fractional nonlinear diffusion equation. According to Kaplan’s first eigenvalue method, we first prove the blow-up of the solutions in finite time under some sufficient conditions. We next provide sufficient conditions for the existence of global solutions by using the results of Zhang and Sun. In conclusion, we find the second critical exponent for the existence of global and non-global solutions via the decay rates of the initial data at spatial infinity.

Conditional stability and regularization method for inverse source for distributed order time-fractional diffusion equation

Conditional stability and regularization method for inverse source for distributed order time-fractional diffusion equation

Combining approach of collocation and finite difference methods for fractional parabolic PDEs

This research aims to estimate the solutions of fractional-order partial differential equations of spacial fractional and both time-space fractional order. For this, we use finite differences for time derivatives and the well-known collocation method for space derivatives with lower-order Bernstein polynomials as basis functions. We explain the mathematical formulations in detail. Convergence and stability analysis of the space–time fractional diffusion equation with the source term is reported subsequently. Three numerical examples are considered for demonstrating the accuracy and reliability of the proposed method.

Inverse initial‐value problems for time fractional diffusion equations in fractional Sobolev spaces

AbstractWe study the time fractional diffusion equation , , in a bounded domain with an elliptic operator and a locally Lipschitz nonlinearity on fractional Sobolev spaces, subjected to the homogeneous Dirichlet boundary condition. Data have not been measured at the initial time , but at a final time , that is, is given instead of . The problem is, therefore, called an inverse initial‐value problem. We first establish the well‐posedness of this problem on fractional Sobolev spaces and the regularity of the solution by assuming only the local Lipschitz continuity of . Second, an susceptible‐infected (shortly, SI) model with heterogeneity and a Navier–Stokes equation have been exemplified. Finally, a spatial ‐estimate for the solution and its gradient has been provided. The essential tools are asymptotic behaviours of Mittag–Leffler functions, fractional power spaces, fractional Sobolev spaces and embedding, weighted functional spaces, and estimates for heat semigroup.

Sinc-Galerkin method and a higher-order method for a 1D and 2D time-fractional diffusion equations

In this article, a new numerical algorithm for solving a 1-dimensional (1D) and 2-dimensional (2D) time-fractional diffusion equation is proposed. The Sinc-Galerkin scheme is considered for spatial discretization, and a higher-order finite difference formula is considered for temporal discretization. The convergence behavior of the methods is analyzed, and the error bounds are provided. The main objective of this paper is to propose the error bounds for 2D problems by using the Sinc-Galerkin method. The proposed method in terms of convergence is studied by using the characteristics of the Sinc function in detail with optimal rates of exponential convergence for 2D problems. Some numerical experiments validate the theoretical results and present the efficiency of the proposed schemes.

A high-accuracy compact finite difference scheme for time-fractional diffusion equations

A high-accuracy compact finite difference scheme for time-fractional diffusion equations

A strong positivity property and a related inverse source problem for multi-term time-fractional diffusion equations

A strong positivity property and a related inverse source problem for multi-term time-fractional diffusion equations

Numerical Solution of Time-fractional Reaction Diffusion Equation via Elzaki Transform with Residual Power Series Method

Numerical Solution of Time-fractional Reaction Diffusion Equation via Elzaki Transform with Residual Power Series Method

Fractional diffusion equations interpolate between damping and waves

Abstract The behaviour of the solutions of the time-fractional diffusion equation,
based on the Caputo derivative, is studied and its dependence on the fractional
exponent is analysed. The time-fractional convection-diffusion equation is also solved
and an application to Pennes bioheat model is presented. Generically, a wave-like
transport at short times passes over to a diffusion-like behaviour at later times.

Symmetry analysis, optimal system, conservation laws and exact solutions of time-fractional diffusion-type equation

In Liu et al. [On the generalized time fractional diffusion equation: Symmetry analysis, conservation laws, optimal system and exact solutions, Int. J. Geom. Methods Mod. Phys. 17(01) (2020) 2050013], the generalized time-fractional diffusion equation [Formula: see text] is studied by the symmetry analysis method. Liu et al. obtained two group generators [Formula: see text] [Formula: see text] and only one trivial solution [Formula: see text] for the equation. In this paper, we classify the Lie symmetry group admitted by the equation, and for the case [Formula: see text], we obtain a richer set of group generators and some nontrivial solutions, including unprecedented exact solutions and power series solutions. In addition, we construct the one-dimensional optimal system of the Lie symmetry group admitted by the time-fractional diffusion-type equation by Olver’s method, and obtain the conservation laws for all the obtained Lie symmetries using the general method developed by Ibragimov. For the novel exact solutions and power series solutions, we analyze their dynamic behavior graphically.

High-Order Approximation to Caputo Derivative on Graded Mesh and Time-Fractional Diffusion Equation for Nonsmooth Solutions

Abstract In this paper, a high-order approximation to Caputo-type time-fractional diffusion equations (TFDEs) involving an initial-time singularity of the solution is proposed. At first, we employ a numerical algorithm based on the Lagrange polynomial interpolation to approximate the Caputo derivative on the nonuniform mesh. The truncation error rate and the optimal grading constant of the approximation on a graded mesh are obtained as min{4−α,rα} and (4−α)/α, respectively, where α∈(0,1) is the order of fractional derivative and r≥1 is the mesh grading parameter. Using this new approximation, a difference scheme for the Caputo-type time-fractional diffusion equation on the graded temporal mesh is formulated. The scheme proves to be uniquely solvable for general r. Then, we derive the unconditional stability of the scheme on uniform mesh. The convergence of the scheme, in particular for r = 1, is analyzed for nonsmooth solutions and concluded for smooth solutions. Finally, the accuracy of the scheme is verified by analyzing the error through a few numerical examples.

Time-fractional discrete diffusion equation for Schrödinger operator

Time-fractional discrete diffusion equation for Schrödinger operator

Adopted spectral tau approach for the time-fractional diffusion equation via seventh-kind Chebyshev polynomials

This study utilizes a spectral tau method to acquire an accurate numerical solution of the time-fractional diffusion equation. The central point of this approach is to use double basis functions in terms of certain Chebyshev polynomials, namely Chebyshev polynomials of the seventh-kind and their shifted ones. Some new formulas concerned with these polynomials are derived in this study. A rigorous error analysis of the proposed double expansion further corroborates our research. This analysis is based on establishing some inequalities regarding the selected basis functions. Several numerical examples validate the precision and effectiveness of the suggested method.

A spatial sixth-order numerical scheme for solving fractional partial differential equation

In this paper, a spatial sixth-order numerical scheme for solving the time-fractional diffusion equation (TFDE) is proposed. The convergence order of the constructed numerical scheme is O(τ2+h6), where τ and h are the temporal and spatial step sizes, respectively. The stability and error estimation of proposed scheme are given by using Fourier method. Some numerical examples are studied to demonstrate the correctness and effectiveness of the scheme and validate the theoretical analysis.

Numerical solution by finite element method for time Caputo-Fabrizio fractional partial diffusion equation

The foundation of this study lies in the fractional order derivative definition pioneered by Caputo and Fabrizio, with distinct representations for temporal and spatial variables. Our focus is on presenting a finite element scheme to address the time-fractional diffusion equation. This approach contrasts with traditional finite difference methods and variational solutions. We rigorously establish the stability and convergence order of our proposed method. To validate our theoretical assertions, we provide a numerical example demonstrating its efficacy.

Numerical solution to loaded difference scheme for time-fractional diffusion equation with temporal loads

Numerical solution to loaded difference scheme for time-fractional diffusion equation with temporal loads

Regularization methods for the inverse initial value problem for the time-fractional diffusion equation with robin boundary condition ★ ★ The project is supported by Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX23_2549) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20190578).

In the article, we focus on the inverse initial value problem for the time-fractional diffusion equation with robin boundary condition on a cylindrical symmetric field. The ill-posedness of this problem is proved. We introduce the modified Landweber iteration method(MLIM), the Truncated singular value decomposition(TSVD) method for solving it and propose a new regularization method, named as the TSVD-modified Landweber iteration method(TMLIM). The error estimates between the exact solution and the regularized approximate solution are presented by using two regularization parameter selection rules. Finally, numerical examples are provided to demonstrate the effectiveness and feasibility of the regularization methods.