Multi-term Time-fractional Diffusion Equation Research Articles

Efficient computational hybrid method for the solution of 2D multi-term fractional order advection-diffusion equation

Abstract Multi-term time-fractional advection diffusion equations are vital for simulating a wide range of physical phenomena, including fluid dynamics and environmental transport processes. However, due to their natural complexity, these equations pose challenges for conventional numerical approaches. In this article, we develop a high order accurate method to solve the multi-term time-fractional advection diffusion equations. We combine the Laplace transform (LT) to integrate the considered equations in time, with Chebyshev spectral method (CSM) for spatial terms The proposed method produces highly accurate solutions with remarkably low computational cost as compared to finite difference method. The propose numerical scheme first employs the LT which reduces the considered problem into a finite set of elliptic equations which may be solved in parallel. Then, the CSM is employed for the disctrezation of spatial operators, which makes it possibly to accurately represent the solution chebyshev grid. Finally, numerical inversion of LT is used to convert the obtain solution from the Laplace domain into the real domain. This work utilizes the modified Talbot’s method and Stehfest’s method for numerical inversion of the LT. To measure the performance, efficiency, and accuracy of the suggested approach, numerical approximations of three models are acquired and verified against the exact solution. The outcomes presented in tables and figures demonstrate that the modified Talbot’s method performed better as compared to Stehfest’s method.

Analysis of a fast element-free Galerkin method for the multi-term time-fractional diffusion equation

In this paper, a fast element-free Galerkin (EFG) method is proposed for solving the multi-term time-fractional diffusion equation (TFDE). Through the use of the multi-term L2−1σ formula to discrete the multi-term Caputo time-fractional derivative, a fast second-order scheme is presented for the temporal discretization, and then the EFG method is used to perform the spatial discretization. The stability of the temporal discretization scheme is discussed. By verifying the fractional Grönwall inequality, the global error of the proposed EFG method for the multi-term TFDE is analyzed in theory. Numerical results validate the theoretical results and the effectiveness of the method.

Recovering the temperature distribution for multi-term time-fractional sideways diffusion equations

Recovering the temperature distribution for multi-term time-fractional sideways diffusion equations

A Novel Accurate Method for Multi-Term Time-Fractional Nonlinear Diffusion Equations in Arbitrary Domains

A Novel Accurate Method for Multi-Term Time-Fractional Nonlinear Diffusion Equations in Arbitrary Domains

Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives

<abstract><p>In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders $ \alpha_i\in(0, 1) $, $ i = 1, 2, \cdots, n $). The proposed method employs a fast finite difference scheme to approximate multi-term fractional derivatives in time, requiring only $ O(1) $ storage and $ O(N_T) $ computational complexity, where $ N_T $ denotes the total number of time steps. Then we use a Legendre spectral collocation method for spatial discretization. The stability and convergence of the scheme have been thoroughly discussed and rigorously established. We demonstrate that the proposed scheme is unconditionally stable and convergent with an order of $ O\left(\left(\Delta t\right)^{2}+N^{-m}\right) $, where $ \Delta t $, $ N $, and $ m $ represent the timestep size, polynomial degree, and regularity in the spatial variable of the exact solution, respectively. Numerical results are presented to validate the theoretical predictions.</p></abstract>

Inverse source problem for multi-term time-fractional diffusion equation with nonlocal boundary conditions

<abstract><p>In this paper, we consider an inverse source problem with nonlocal boundary conditions for the heat equation involving multi-term time-fractional derivatives. We determine a source term independent of the space variable, and the temperature distribution from the energy measurement. We reduce the solution of the inverse problem to finding solutions to two problems. The well-posedness of each problem is shown using the generalized Fourier method.</p></abstract>

Solutions of a three-dimensional multi-term fractional anomalous solute transport model for contamination in groundwater

The study presents a meshless computational approach for simulating the three-dimensional multi-term time-fractional mobile-immobile diffusion equation in the Caputo sense. The methodology combines a stable Crank-Nicolson time-integration scheme with the definition of the Caputo derivative to discretize the problem in the temporal direction. The spatial function derivative is approximated using the inverse multiquadric radial basis function. The solution is approximated on a set of scattered or uniform nodes, resulting in a sparse and well-conditioned coefficient matrix. The study highlights the advantages of meshless method, particularly their simplicity of implementation in higher dimensions. To validate the accuracy and efficacy of the proposed method, we performed numerical simulations and compared them with analytical solutions for various test problems. These simulations were carried out on computational domains of both rectangular and non-rectangular shapes. The research highlights the potential of meshless techniques in solving complex diffusion problems and its successful applications in groundwater contamination and other relevant fields.

Error estimate of a transformed L1 scheme for a multi-term time-fractional diffusion equation by using discrete comparison principle

This work is concerned with the multi-term time-fractional diffusion equation ∑j=0JbjDtαju−pΔu+c(x,t)u=f, where Dtαj is the Caputo derivative with 1>α0>α1>⋯>αJ>0 and bj, p are positive constants. The solution of this problem usually has a weak singularity near the initial time. To handle such difficulty, a smoothing transformation t=s1/α0 is applied so that an equivalent re-scaled fractional differential equation is obtained. Then the equivalent equation is solved by the transformed L1 scheme of the Caputo derivative and the standard 3-point discretization of the spatial derivative on uniform meshes both in time and space direction. The α-robust error estimate with the temporal convergence order O(Nα0−2) is given by using the discrete comparison principle, which does not blow up as α→1−. Finally, numerical results are given to confirm our error analysis.

Solving time-fractional diffusion equations with a singular source term

This article deals with linear time-fractional diffusion equations with time-dependent singular source term. Whether the order of the time-fractional derivative is multi-term, distributed or space-dependent, we prove that the system admits a unique weak solution enjoying a Duhamel representation, provided that the time-dependence of the source term is a distribution. As an application, the square integrable space-dependent part and the distributional time-dependent part of the source term of a multi-term time-fractional diffusion equation are simultaneously recovered by partial internal observation of the solution.

Preconditioning techniques of all-at-once systems for multi-term time-fractional diffusion equations

Preconditioning techniques of all-at-once systems for multi-term time-fractional diffusion equations

Simultaneous Recovery of Two Time-Dependent Coefficients in a Multi-Term Time-Fractional Diffusion Equation

Abstract This paper deals with an inverse problem on simultaneously determining a time-dependent potential term and a time source function from two-point measured data in a multi-term time-fractional diffusion equation. First we study the existence, uniqueness and some regularities of the solution for the direct problem by using the fixed point theorem. Then a nice conditional stability estimate of inversion coefficients problem is obtained based on the regularity of the solution to the direct problem and a fine property of the Caputo fractional derivative. In addition, the ill-posedness of the inverse problem is illustrated and we transfer the inverse problem into a variational problem. Moreover, the existence and convergence of the minimizer for the variational problem are given. Finally, we use a modified Levenberg–Marquardt method to reconstruct numerically the approximate functions of two unknown time-dependent coefficients effectively. Numerical experiments for three examples in one- and two-dimensional cases are provided to show the validity and robustness of the proposed method.

Two-grid fully discrete finite element algorithms on temporal graded meshes for nonlinear multi-term time-fractional diffusion equations with variable coefficient

In this paper, we are concerned with two-grid fully discrete finite element algorithms for solving the nonlinear multi-term time-fractional diffusion equation with a variable coefficient. Taking into account the weak singularity of the solution at the initial time t=0, we employ graded meshes in time, which is an effective method to compensate for the lack of smoothness. A fully discrete implicit finite element scheme is established with the L1 method on temporally graded meshes; the stability and optimal error estimates are theoretically proved. To improve the computational efficiency, we propose a two-grid algorithm based on the fully discrete scheme, and then the second algorithm is constructed to further improve the optimal convergence order. Similar properties as standard FEM are presented for the two algorithms. A significant fact is that this technique reduces the computation time without loss of numerical precision. Several numerical examples are provided to support our theoretical findings and verify the performance of the proposed algorithms.

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>

Unique Continuation for the Multi-term Time Fractional Diffusion Equation and its Numerical Simulation

This paper deals with the unique continuation principle of solutions for a one-dimensional anomalous diffusion equation with multi-term time fractional derivatives. The proof is mainly based on the Laplace transform and Theta function method. Numerically, we reformulate the unique continuation as an optimization problem, and propose an iterative thresholding algorithm to simulate it numerically. Finally, several numerical experiments are presented to show the accuracy and efficiency of the algorithm.

Uniqueness of orders and parameters in multi-term time-fractional diffusion equations by short-time behavior

As the most significant difference from parabolic equations, long-time or short-time behavior of solutions to time-fractional evolution equations is dominated by the fractional orders, whose unique determination has been frequently investigated in literature. Unlike all the existing results, in this article we prove the uniqueness of orders and parameters (up to a multiplier for the latter) only by principal terms of asymptotic expansions of solutions near t = 0 at a single spatial point. Moreover, we discover special conditions on unknown initial values or source terms for the coincidence of observation data. As a byproduct, we can even conclude the uniqueness for initial values or source terms by the same data. The proof relies on the asymptotic expansion after taking the Laplace transform and the completeness of generalized eigenfunctions.

Discrete comparison principle of a finite difference method for the multi-term time fractional diffusion equation

Discrete comparison principle of a finite difference method for the multi-term time fractional diffusion equation

A weak Galerkin finite element method on temporal graded meshes for the multi-term time fractional diffusion equations

We consider the multi-term time fractional diffusion equation on a bounded convex domain. We analyze the L1 method on a graded mesh in time to compensate for the weak singularity of the solution and a weak Galerkin finite element method in space discretization. The stability analyses are presented for both semi-discrete and fully-discrete schemes and we prove that two schemes are unconditionally stable. Error estimates in L2-norm and a discrete H1 equivalent norm for both schemes are rigorously derived. Further we discuss the optimal spatial order error estimates in L2-norm. Finally, we give some numerical experiments to show the efficiency of the proposed method.

Existence and uniqueness results for a multi-parameters nonlocal diffusion equation

This paper is devoted to identifying a time-dependent source term for a multi-term time-fractional diffusion equation with a nonlocal dynamic boundary and integral type over-determination condition. The time-fractional derivatives are considered in Caputo's sense. By applying Fourier's method we obtained multi-term ordinary fractional order differential equation which has been reduced to an algebraic equation by using Laplace transform. Inverse Laplace transform is used to obtain the solution of multi-term ordinary fractional order differential equation which involves multinomial Mittag-Leffler functions. Under some regularity and consistency conditions on the data, unique existence and stability of the regular solution of the inverse problem is proved.

A high-order unconditionally stable numerical method for a class of multi-term time-fractional diffusion equation arising in the solute transport models

In this paper, we study a high-order unconditionally stable numerical method to approximate the class of multi-term time-fractional diffusion equations. This type of problem appears in the modelling of transport of certain quantities such as heat, mass, energy, solutes in ground water and soils. The multi-term time-fractional derivative is approximated by using the Crank–Nicolson method for the Caputo's time derivative. The space derivative is approximated by using the collocation method based on quintic B-spline basis functions. We have established the stability and convergence analysis of the proposed numerical scheme thoroughly, and it is shown that the order of convergence in space variable is almost four and in the time variable is . To prove the accuracy and efficiency of the developed method, we consider four numerical examples and perform the numerical simulation. The developed algorithm works well andvalidate the theoretical results. The developed method is fourth-order convergent in the space variable, which is almost two orders of magnitude higher than the other spline collocation methods.

Galerkin operational approach for multi-dimensions fractional differential equations

The current manuscript introduces a novel numerical treatment for multi-term fractional differential equations with variable coefficients. The spectral Galerkin approach is developed with the operational matrix of fractional derivatives to get a system of algebraic equations that can be solved using a suitable technique. The operational matrix is introduced based on a combination of the shifted Jacobi polynomials. The proposed approach is applied also for 1+1 and 2+1 multi-term time-fractional diffusion and diffusion-wave equations. In addition, the convergence analysis for the presented approaches is investigated. Some specific numerical examples are given to ascertain the wide applicability and the good efficiency of the suggested algorithms, and to confirm that nonlocal numerical methods are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account.