Time Fractional Sub-diffusion Equation Research Articles

A Comparative Study of Efficient Numerical Schemes for Time-Fractional Subdiffusion Equation Involving Singularity

A Comparative Study of Efficient Numerical Schemes for Time-Fractional Subdiffusion Equation Involving Singularity

Higher Order Stable Numerical Algorithm for the Variable Order Time-Fractional Sub-diffusion Equation

Higher Order Stable Numerical Algorithm for the Variable Order Time-Fractional Sub-diffusion Equation

Numerical Study of Multi-Term Time-Fractional Sub-Diffusion Equation Using Hybrid L1 Scheme with Quintic Hermite Splines

Anomalous diffusion of particles has been described by the time-fractional reaction–diffusion equation. A hybrid formulation of numerical technique is proposed to solve the time-fractional-order reaction–diffusion (FRD) equation numerically. The technique comprises the semi-discretization of the time variable using an L1 finite-difference scheme and space discretization using the quintic Hermite spline collocation method. The hybrid technique reduces the problem to an iterative scheme of an algebraic system of equations. The stability analysis of the proposed numerical scheme and the optimal error bounds for the approximate solution are also studied. A comparative study of the obtained results and an error analysis of approximation show the efficiency, accuracy, and effectiveness of the technique.

On the Solution Existence for Collocation Discretizations of Time-Fractional Subdiffusion Equations

Time-fractional parabolic equations with a Caputo time derivative of order α∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1)$$\\end{document} are discretized in time using continuous collocation methods. For such discretizations, we give sufficient conditions for existence and uniqueness of their solutions. Two approaches are explored: the Lax–Milgram Theorem and the eigenfunction expansion. The resulting sufficient conditions, which involve certain m×m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m\ imes m$$\\end{document} matrices (where m is the order of the collocation scheme), are verified both analytically, for all m≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m\\ge 1$$\\end{document} and all sets of collocation points, and computationally, for all m≤20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ m\\le 20$$\\end{document}. The semilinear case is also addressed.

Numerical treatment of time-fractional sub-diffusion equation using p-fractional linear multistep methods

In this paper, a kind of the differential equation including a time-fractional sub-diffusion equation is considered. Through this memorandum, a well-known technique, in the time direction is adopted by the p-fractional linear multistep method (p-FLMM) according to the q-fractional backward difference formula (q-FBDF) of implicit type for q = 1, 2, 3, and the spatial direction is approximated by the second-order central difference method. The stability properties of the proposed method can be investigated in combination with the Fourier technique and Z -transformation and also its convergence is studied by using the truncated error maximum. It is shown that the method is unconditionally stable and the orders of convergence are O ( τ p + h 2 ) for 1 ≤ p ≤ 4 , in which p is the order of accuracy in the time direction and τ and h determine temporal and spatial stepsizes, respectively. Some numerical experiments are included to demonstrate the validity and applicability of the scheme.

An element-free Galerkin method for the time-fractional subdiffusion equations

In this paper, an element-free Galerkin (EFG) method is developed for the numerical analysis of the time-fractional subdiffusion equation. By using the L2−1σ formula to approximate the Caputo fractional derivative, a second-order accurate scheme is proposed to achieve temporal discretization. Then, time-independent integer-order boundary value problems are formed, and a stabilized EFG method is applied to establish the discretize linear algebraic systems. Error of the proposed meshless method is proved theoretically. Numerical results show the convergence and effectiveness of the method.

A cardinal approach for two-dimensional modified anomalous space–time fractional sub-diffusion equation

In this paper, the modified anomalous space–time fractional sub-diffusion equation in two dimensions is introduced. The Chebyshev cardinal polynomials (as an appropriate family of basis functions) are successfully used to make a computational technique for this equation. To this end, the time and space fractional derivative matrices of these polynomials are obtained at first. Then, by approximating the unknown solution via the expressed polynomials and applying these matrices, as well as the collocation technique, solving the problem turns into solving an algebraic system of equations. The capability of the approach is checked by solving two test problems.

Breaking barriers: Robert Denis Mare and research on social stratification

In this paper, the modified anomalous space–time fractional sub-diffusion equation in two dimensions is introduced. The Chebyshev cardinal polynomials (as an appropriate family of basis functions) are successfully used to make a computational technique for this equation. To this end, the time and space fractional derivative matrices of these polynomials are obtained at first. Then, by approximating the unknown solution via the expressed polynomials and applying these matrices, as well as the collocation technique, solving the problem turns into solving an algebraic system of equations. The capability of the approach is checked by solving two test problems.

A Corrected L1 Method for a Time-Fractional Subdiffusion Equation

A Corrected L1 Method for a Time-Fractional Subdiffusion Equation

A new fourth-order grouping iterative method for the time fractional sub-diffusion equation having a weak singularity at initial time

<abstract><p>A new fourth-order explicit grouping iterative method is constructed for the numerical solution of the fractional sub-diffusion equation. The discretization of the equation is based on fourth-order finite difference method. Captive fractional discretization having functions with a weak singularity at $ t = 0 $ is used for time and similarly, the space derivative is approximated with the help of fourth-order approximation. Furthermore, the convergence and stability of the scheme are analyzed. Finally, the accuracy and validity are investigated by some numerical examples.</p></abstract>

Modelling, analysis, and numerical methods for a geometric inverse source problem in variable-order time-fractional subdiffusion

There exist research works on studying time-dependent integer-order and time-fractional constant-order geometric inverse source problems in the literature. The time-fractional variable-order geometric inverse source problems although also have important physical applications have not been studied mathematically and numerically in literature. The aim of this work is to study an inverse source problem associated with a variable-order time-fractional subdiffusion equation. We first build a mathematical model and show existence of the optimal shape for shape reconstruction of the source support. Then, shape sensitivity analysis is performed to propose a shape gradient optimization algorithm allowing deformations for numerically solving the model problem. In order to reconstruct the source support with topology unknown a priori, moreover, we build a phase-field model and propose a gradient algorithm allowing both shape and topological changes by a phase-field method. A variety of numerical examples are presented to demonstrate effectiveness of the two algorithms.

A high-order stabilizer-free weak Galerkin finite element method on nonuniform time meshes for subdiffusion problems

<abstract><p>We present a stabilizer-free weak Galerkin finite element method (SFWG-FEM) with polynomial reduction on a quasi-uniform mesh in space and Alikhanov's higher order L2-$ 1_\sigma $ scheme for discretization of the Caputo fractional derivative in time on suitable graded meshes for solving time-fractional subdiffusion equations. Typical solutions of such problems have a singularity at the starting point since the integer-order temporal derivatives of the solution blow up at the initial point. Optimal error bounds in $ H^1 $ norm and $ L^2 $ norm are proven for the semi-discrete numerical scheme. Furthermore, we have obtained the values of user-chosen mesh grading constant $ r $, which gives the optimal convergence rate in time for the fully discrete scheme. The optimal rate of convergence of order $ \mathcal{O}(h^{k+1}+M^{-2}) $ in the $ L^\infty(L^2) $-norm has been established. We give several numerical examples to confirm the theory presented in this work.</p></abstract>

Mittag–Leffler stability of numerical solutions to time fractional ODEs

The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix. Very different from the exponential decay of solutions to classical ODEs, solutions of F-ODEs decay only polynomially, leading to the so-called Mittag-Leffler stability, which was already extended to semi-linear F-ODEs with small perturbations. This work is mainly devoted to the qualitative analysis of the long-time behavior of numerical solutions. By applying the singularity analysis of generating functions developed by Flajolet and Odlyzko (SIAM J. Disc. Math. 3(2), 216–240, 1990), we are able to prove that both $$\mathcal {L}$$ 1 scheme and strong A-stable fractional linear multistep methods (F-LMMs) can preserve the numerical Mittag-Leffler stability for linear homogeneous F-ODEs exactly as in the continuous case. Through an improved estimate of the discrete fractional resolvent operator, we show that strong A-stable F-LMMs are also Mittag-Leffler stable for semi-linear F-ODEs under small perturbations. For the numerical schemes based on $$\alpha$$ -difference approximation to Caputo derivative, we establish the Mittag-Leffler stability for semi-linear problems by making use of properties of the Poisson transformation. The new results and analyses provide not only the rigorous justifications and explanations of the Mittag-Leffler stability of numerical solutions with exact decay rate, but also establish some close connection between the continuous and discrete F-ODEs. Numerical experiments are presented for several typical time fractional evolutional equations, including time fractional sub-diffusion equations and semi-linear F-ODEs. All the numerical results exhibit the typical long-time polynomial decay rate, which is fully consistent with our theoretical predictions.

Error estimate of the fast L1 method for time-fractional subdiffusion equations

This paper analyzes the convergence of the fast L1 method for the semi-linear time-factional subdiffusion equations. The tool we use is the generalized discrete Gronwall inequality. We show that the difference between the solution of the L1 method and the solution of the fast L1 method can be arbitrarily small and is independent of the sizes of the time and/or space grids. Our proof is very simple and is helpful to be used to simplify the convergence analysis of the fast methods for time-fractional evolution equations.

Fast Second-Order Evaluation for Variable-Order Caputo Fractional Derivative with Applications to Fractional Sub-Diffusion Equations

In this paper, we propose a fast second-order approximation to the variable-order (VO) Caputo fractional derivative, which is developed based on $L2$-$1_\sigma$ formula and the exponential-sum-approximation technique. The fast evaluation method can achieve the second-order accuracy and further reduce the computational cost and the acting memory for the VO Caputo fractional derivative. This fast algorithm is applied to construct a relevant fast temporal second-order and spatial fourth-order scheme ($FL2$-$1_{\sigma}$ scheme) for the multi-dimensional VO time-fractional sub-diffusion equations. Theoretically, $FL2$-$1_{\sigma}$ scheme is proved to fulfill the similar properties of the coefficients as those of the well-studied $L2$-$1_\sigma$ scheme. Therefore, $FL2$-$1_{\sigma}$ scheme is strictly proved to be unconditionally stable and convergent. A sharp decrease in the computational cost and the acting memory is shown in the numerical examples to demonstrate the efficiency of the proposed method.

High-order finite difference method based on linear barycentric rational interpolation for Caputo type sub-diffusion equation

The main aim of this paper is to develop a class of high-order finite difference method for the numerical solution of Caputo type time-fractional sub-diffusion equation. In the time direction, the Caputo derivative is discretized by employing a numerical technique based on the fractional linear barycentric rational interpolation method (FLBRI). The stability properties and the convergence analysis of the proposed scheme are considered based on the energy method. As a result, the order of convergence is O(Δt)p+(Δx)2 for 1≤p≤7, in which Δt is the temporal step size, Δx is the spatial step size, and p is the order of accuracy in the time direction. Finally, numerical experiment is provided to show that the results confirm the theoretical analysis, perfectly.

INITIAL-BOUNDARY VALUE PROBLEM FOR A SUBDIFFUSION EQUATION WITH CAPUTO DERIVATIVE

We investigate an initial-boundary value problem for a time-fractional subdiffusion equation with the Caputo derivatives on $N$-dimensional torus by the classical Fourier method. Since our solution is established on the eigenfunction expansion of elliptic operator, the method proposed in this article can be used to an arbitrary domain and an elliptic operator with variable coefficients. It should be noted that the conditions for the existence of a solution to the initial-boundary value problem found in the article cannot be weakened, and the article provides a corresponding example.

A SPECTRAL APPROACH FOR TIME-FRACTIONAL DIFFUSION AND SUBDIFFUSION EQUATIONS IN A LARGE INTERVAL

In this paper, we concentrate on a class of time-fractional diffusion and subdiffusion equations. To solve the mentioned problems, we construct twodimensional Genocchi-fractional Laguerre functions (G-FLFs). Then, the pseudooperational matrices are used to convert the proposed equations to systems of algebraic equations. The properties of pseudo-operational matrices have reflected well in the process of the numerical technique and create an approximate solution with high precision. Finally, several examples are presented to illustrate the accuracy and effectiveness of the technique.

Subdiffusion Equations with a Source Term and Their Extensions

We propose an alternative derivation of the subdiffusion equation with a source term. Basing on this approach, we obtain the variable-order and the distributed-order equations that reduce to the well-known Riemann–Liouville and Caputo forms of the time-fractional subdiffusion equation. We find out that the naive inclusion of sources/sinks as a separate term is correct for the equations with the Riemann–Liouville derivative. However, for the equations with the Caputo derivative, the source term must go under a fractional integral.

On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.