Time-fractional Diffusion Problem Research Articles

Numerical solution of multi-dimensional time-fractional diffusion problems using an integral approach.

This paper presents a significant scheme to drive the numerical solution of multi-dimensional diffusion problems where the fractional derivatives are taken in Caputo sense. The Mohand homotopy integral transform scheme (MHITS) is the composition of Mohand integral transform (MIT) and the homotopy perturbation scheme (HPS) which can be used to investigate the numerical solution in the form of convergence series. This approach does not require any presumptions, limitations on elements, or any other hypothesis. The primary objective of this strategy is to perform its direct implementation to the recurrence relation. This method produces results in the form of a convergent series, which accurately predicts the exact results. Graphical results and plot error distribution show an excellent agreement between MHITS results and the exact solution.

On the inverse problem of time dependent coefficient in a time fractional diffusion problem by sinc wavelet collocation method

The object of this study is to establish the unknown function in a time fractional diffusion problem and the solution as well by utilizing Sinc wavelet collocation method (SWCM) and residual power series method (RPSM) together. SWCM enables us to convert time fractional diffusion problem into a system of fractional ordinary differential and algebraic equations. At this stage, the unknown function and the solution are constructed in the series form by employing RPSM. The novelty of this study is that the combination of SWCM and RPSM is utilized to establish the solution of inverse coefficient problem for the first time. Demonstrative examples are presented to articulate the implementation and importance of the proposed method.

Fast High-Order Compact Finite Difference Methods Based on the Averaged L1 Formula for a Time-Fractional Mobile-Immobile Diffusion Problem

Fast High-Order Compact Finite Difference Methods Based on the Averaged L1 Formula for a Time-Fractional Mobile-Immobile Diffusion Problem

α-Robust Error Analysis of L2-1σ Scheme on Graded Mesh for Time-Fractional Nonlocal Diffusion Equation

Abstract In this work, a time-fractional nonlocal diffusion equation is considered. Based on the L2-1σ scheme on a graded mesh in time and the standard finite element method (FEM) in space, the fully-discrete L2-1σ finite element method is investigated for a time-fractional nonlocal diffusion problem. We prove the existence and uniqueness of fully-discrete solution. The α-robust error bounds are derived, i.e., bounds remain valid as α→1−, where α ∈(0,1) is the order of a temporal fractional derivative. The numerical experiments are presented to justify the theoretical findings.

A Symmetry of Boundary Functions Method for Solving the Backward Time-Fractional Diffusion Problems

In the paper, we develop three new methods for estimating unknown initial temperature in a backward time-fractional diffusion problem, which is transformed to a space-dependent inverse source problem for a new variable in the first method. Then, the initial temperature can be recovered by solving a second-order boundary value problem. The boundary functions and a unique zero element constitute a group symmetry. We derive energetic boundary functions in the symmetry group as the bases to retrieve the source term as an unknown function of space and time. In the second method, the solution bases are energetic boundary functions, and then by collocating the governing equation we obtain the expansion coefficients for retrieving the entire solution and initial temperature. For the first two methods, boundary fluxes are over-specified to retrieve the initial condition. In the third method, we give two boundary conditions and a final time temperature to construct the bases in another symmetry group; the governing equation is collocated to a linear system to obtain the whole solution (initial temperature involved). These three methods are assessed and compared by numerical experiments.

طريقة بيتروف جاليركين الخطية المنفصلة لمعادلات الانتشار الكسرية للزمن

We propose and analyze piecwise linear discontinuous Petrov-Galerkin method in time combined with a standard conforming finite element method in space for the numerical solution of time-fractional diffusion problems of order 0 < μ < 1. We prove the stability of the exact solution. The existence, uniqueness and stability of approximate solutions will be proved. We employ a non-uniform mesh based on concentrating the cells near the singularity. The advantage of employing a non-uniform mesh is improving the accuracy of the approximate solution. Numerical experiments indicate the error in L∞(0, T ; L2(Ω))-norm is of order kmin(γ(1- μ),2) + h2, where k denotes the maximum time steps and h is the maximum diameter of the elements of the (quasi-uniform) spatial mesh and γ > 0.

A new analytical approach for the local radial point interpolation discretisation in space and applications to high-order in time schemes for two-dimensional fractional PDEs

In weak-form formulations of the local radial point interpolation method (LRPIM) for the solution of partial differential equations, space discretisation matrices have most often been obtained entirely through numerical integration. This work introduces a novel approach which derives closed-form expressions for obtaining the entries of the discretisation matrix for the solution of two-dimensional time-fractional diffusion problems. This analytical approach also yields a closed-form formula for the approximation of the Laplacian. These techniques are then applied for developing LRPIM-based numerical algorithms. Since the exact solutions usually have unbounded first-order time derivatives at time zero, a graded mesh is employed for a high-order in time approximation of the Caputo derivative. The analytical shape functions are used to develop a weak-form algorithm and a strong-form algorithm is developed using the analytical approximation of the Laplacian. We demonstrate that computed solutions obtained using the weak-form and strong-form algorithms have the accuracy levels consistent with the theoretical accuracy in space. An appropriate choice of the mesh grading parameter yields a high-order convergence rate in time. The unconditional stability and convergence of a LRPIM strong form algorithm on a uniform temporal mesh is established under the assumption that the exact solution has sufficient regularity.

On the Inverse Problem of Time-Dependent Coefficient in a Time Fractional Diffusion Problem by Newly Defined Monic Laquerre Wavelets

Abstract The primary aim of this research is to establish the time-dependent diffusion coefficient in a one-dimensional time fractional diffusion equation in Caputo sense by means of newly defined Monic Laquerre wavelets (MLW) and collocation points. We first give the definition of MLW by taking Monic Laquerre's polynomials into account. Later, time fractional diffusion problem is reduced into a system of ordinary fractional and algebraic equations by utilizing MLW. The residual power series method (RPSM) and the overdetermined data are applied to this system to determine the solution and the unknown time-dependent coefficient together in series form. In the end, illustrative examples are presented to show the stability and accuracy of the proposed wavelet method for the inverse problem of determining unknown time-dependent coefficient in fractional diffusion problems. The reliability of the proposed algorithm for the inverse problems is supported by high degree of accuracy in the given examples.

Collocation-Based Approximation for a Time-Fractional Sub-Diffusion Model

We consider the numerical solution of a one-dimensional time-fractional diffusion problem, where the order of the Caputo time derivative belongs to (0, 1). Using the technique of the method of lines, we first develop from the original problem a system of fractional ordinary differential equations. Using an integral equation reformulation of this system, we study the regularity properties of the exact solution of the system of fractional differential equations and construct a piecewise polynomial collocation method to solve it numerically. We also investigate the convergence and the convergence order of the proposed method. To conclude, we present the results of some numerical experiments.

On regularization results for a two-dimensional nonlinear time-fractional inverse diffusion problem

Ill-posed inverse problems arise in a variety of practically important applications including image restoration, medicine, ecology as well as many other branches of pure and applied sciences. In this paper, we consider an ill-posed inverse problem for a two-dimensional nonlinear time-fractional inverse diffusion model involving the Caputo time-fractional derivative as follows:{Dtβu(x,y,t)=−a(x)(ux(x,y,t)+uy(x,y,t))+ℓ(x,y,t,u(x,y,t)),x>0,y>0,t>0,u(x,y,0)=0,x>0,y>0,limx→∞⁡u(x,y,t)=0,y>0,t>0,u(1,y,t)=u1(y,t),y>0,t>0, where β∈(0,1) is fixed, a is a space-dependent diffusion coefficient, ℓ is a nonlinear function satisfying a Lipschitz condition, and the data u1 is given approximately. We want to determine u(x,y,t) for 0≤x<1. There exists a vast literature on regularization results related to the considered problem. We are concerned with regularization results for the nonlinear case. The first regularization results in such a case were studied by Tuan, Hoan and Tatar in [17]. While these results apply to a constant diffusivity coefficient in the one-dimensional setting, the results of Vo and co-worker in [18], more generally, apply to the case of the space-dependent diffusion coefficient in the two-dimensional setting. In all these papers relatively strong a priori assumptions on the regularity of the exact solution are made. In this work, we weaken such a priori assumptions via two regularization strategies based on the integral equation method. We obtain several explicit error estimates including an error estimate of the Hölder-Logarithmic type for all x∈[0,1), which can be seen as the improvement and the generalization of error estimates presented by Zheng and Wei in [20,21]. Eventually, several numerical examples are presented for the purpose of illustrating the theoretical results.

Reconstruction of pointwise sources in a time-fractional diffusion equation

This paper is concerned with an inverse pointwise source problem for the time-fractional diffusion equation in the two-dimensional case. The source term to be identified models the action of a finite number of small particles. Each particle is assumed to be no larger than a single point, characterized by its location and intensity. Both theoretical and numerical aspects are discussed. In the theoretical part, we analyse the well-posedness of the Dirac time-fractional diffusion problem. For the inverse problem, we establish that the unknown point sources can be uniquely identified from local measured data and we derive a local Lipschitz stability result. In the numerical part, we develop a fast and accurate reconstruction approach. The unknown pointwise sources are characterized as solution to an optimization problem minimizing a tracking-type functional. A noniterative reconstruction algorithm is devised, allowing us to determine the number, locations and intensities of the pointwise sources. The efficiency of the proposed approach is confirmed by some numerical examples.

On the critical behavior for time-fractional reaction diffusion problems

We first investigate the existence and nonexistence of weak solutions to the time-fractional reaction diffusion problem \begin{document}$ \frac{\partial^\alpha u}{\partial t^\alpha}-\frac{\partial^2 u}{\partial x^2}+u\geq x^{-a}|u|^p, \, \, t&gt;0, \, x\in (0, 1], \quad u(0, x) = u_0(x), \, \, x\in (0, 1] $\end{document} under the inhomogeneous Dirichlet boundary condition \begin{document}$ u(t, 1) = \delta, \quad t&gt;0, $\end{document} where $ u = u(t, x) $, $ 0&lt;\alpha&lt;1 $, $ \frac{\partial^\alpha }{\partial t^\alpha} $ is the time-Caputo fractional derivative of order $ \alpha $, $ a\geq 0 $, $ p&gt;1 $ and $ \delta&gt;0 $. We show that, if $ a\leq 2 $, the existence holds for all $ p&gt;1 $ while if $ a&gt;2 $, then the dividing line with respect to existence or nonexistence is given by the critical exponent $ p^* = a-1 $. The proof of the nonexistence result is based on nonlinear capacity estimates specifically adapted to the nonlocal nature of the problem, the modified Helmholtz operator $ -\frac{\partial^2}{\partial x^2}+I $, and the considered boundary condition. The existence part is proved by the construction of explicit solutions. We next extend our study to the case of systems.

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

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

Numerical analysis of multi-dimensional time-fractional diffusion problems under the Atangana-Baleanu Caputo derivative.

This paper presents the Elzaki homotopy perturbation transform scheme (EHPTS) to analyze the approximate solution of the multi-dimensional fractional diffusion equation. The Atangana-Baleanu derivative is considered in the Caputo sense. First, we apply Elzaki transform (ET) to obtain a recurrence relation without any assumption or restrictive variable. Then, this relation becomes very easy to handle for the implementation of the homotopy perturbation scheme (HPS). We observe that HPS produces the iterations in the form of convergence series that approaches the precise solution. We provide the graphical representation in 2D plot distribution and 3D surface solution. The error analysis shows that the solution derived by EHPTS is very close to the exact solution. The obtained series shows that EHPTS is a very simple, straightforward, and efficient tool for other problems of fractional derivatives.

$L2-1_\sigma$ Finite Element Method for Time-Fractional Diffusion Problems with Discontinuous Coefficients

$L2-1_\sigma$ Finite Element Method for Time-Fractional Diffusion Problems with Discontinuous Coefficients

Immersed finite element method for time fractional diffusion problems with discontinuous coefficients

In this article, we present the immersed finite element (IFE) method to solve time fractional diffusion equation with discontinuous coefficients, in which the Caputo fractional derivative is approximated by nonuniform L1 scheme to deal with singularity of solution. For interface problems caused by discontinuous coefficients in space, we adopt the nonconforming immersed finite element method to discrete. Then the stabilities under L2(Ω) norm and broken H1(Ω) seminorm of fully discrete scheme are analyzed. Based on these results, error estimates in L2(Ω) norm and broken H1(Ω) seminorm of fully immersed finite element scheme are obtained. Finally, several numerical examples are presented to illustrate the theoretical results.

Numerical solution of two-dimensional inverse time-fractional diffusion problem with non-local boundary condition using a-polynomials

Numerical solution of two-dimensional inverse time-fractional diffusion problem with non-local boundary condition using a-polynomials

Asymptotic profile for a two-terms time fractional diffusion problem

We consider the Cauchy-type problem associated to the time fractional partial differential equation: ∂tu+∂tβu-Δu=g(t,x),t>0,x∈Rnu(0,x)=u0(x),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\partial _t u+\\partial _t^{\\beta }u-\\varDelta u=g(t,x), &{} t>0, \\ x\\in {\\mathbb {R}}^n \\\\ u(0,x)=u_0(x), \\end{array}\\right. } \\end{aligned}$$\\end{document}with beta in (0,1), where the fractional derivative partial _t^{beta } is in Caputo sense. We provide a sufficient condition on the right-hand term g(t, x) to obtain a solution in {mathcal {C}}_b([0,infty ),H^s). We exploit a dissipative-smoothing effect which allows to describe the asymptotic profile of the solution in low space dimension.

GL1 Scheme for Solving a Class of Generalized Time-Fractional Diffusion Equations

In this paper, a numerical scheme based on a general temporal mesh is constructed for a generalized time-fractional diffusion problem of order α. The main idea involves the generalized linear interpolation and so we term the numerical scheme the gL1 scheme. The stability and convergence of the numerical scheme are analyzed using the energy method. It is proven that the temporal convergence order is (2−α) for a general temporal mesh. Simulation is carried out to verify the efficiency of the proposed numerical scheme.

A transformed [formula omitted] method for solving the multi-term time-fractional diffusion problem

In this paper, we present a novel scheme for solving a time-fractional initial–boundary value problem, where the equation contains a sum of Caputo derivatives with orders between 0 and 1. In order to overcome the difficulty of initial layer, we introduce a change of variable in the temporal direction and investigate the regularity of the solutions of the resulting system. A modified L1 approximation is used to approximate the Caputo derivatives and a standard Galerkin-Spectral method is applied to approximate the spatial derivatives. Unconditional stability and convergence of the fully-discrete scheme are proved by applying a novel discrete fractional Grönwall inequality. Finally, numerical examples are given to confirm our theoretical results.