Time-fractional Diffusion Equation Research Articles

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

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

An α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Robust and Second-Order Accurate Scheme for a Subdiffusion Equation

We investigate a second-order accurate time-stepping scheme for solving a time-fractional diffusion equation with a Caputo 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}. The basic idea of our scheme is based on local integration followed by linear interpolation. It reduces to the standard Crank–Nicolson scheme in the classical diffusion case, that is, as α→1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\rightarrow 1$$\\end{document}. Using a novel approach, we show that the proposed scheme is α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-robust and second-order accurate in the L2(L2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(L^2)$$\\end{document}-norm, assuming a suitable time-graded mesh. For completeness, we use the Galerkin finite element method for the spatial discretization and discuss the error analysis under reasonable regularity assumptions on the given data. Some numerical results are presented at the end.

A direct sampling method for time-fractional diffusion equation

This paper introduces a direct sampling method tailored for identifying the location of the source term within a time-fractional diffusion equation (TFDE). The key aspect of our approach involves the utilization of a versatile family of index functions, which can be chosen according to the specific characteristics of the source term. Recognizing the key role of the TFDE’s fundamental solution within the index function, we further enhance our method by deriving its asymptotic expansions. This advancement not only enhances the accuracy, but also significantly improves the computational efficiency of our method. To validate the effectiveness and robustness of the proposed sampling method, we conduct a series of comprehensive numerical experiments.

Numerical algorithm for a general fractional diffusion equation

In this manuscript, the analytical solution to the general time fractional diffusion equation followed by the regularity analysis of its solution is presented. Further, a hybrid efficient numerical algorithm for a general fractional derivative of order α∈(0,1) is designed. The physical application of the developed approximation formula is employed to design a hybrid numerical algorithm to determine the numerical solution of the corresponding general time fractional diffusion equation. The proposed numerical techniques are subjected to solvability, numerical stability, and convergence analysis. Numerical example is utilized to verify that (3−α)-th order convergence is attained in time which is higher than the existing scheme (Xu et al., 2013). In the spatial direction, the fourth order convergence is obtained utilizing the compact finite difference method. Besides, such a hybrid numerical algorithm is also used in the stochastic model.

Error analysis of a fully discrete method for time-fractional diffusion equations with a tempered fractional Gaussian noise

We develop and analyze a fully discrete method, in order to solve numerically time-fractional diffusion equations driven by additive tempered fractional Gaussian noise. The stochastic model involves a Caputo fractional derivative of order α∈(0,1). We discuss the regularity results of the solution by using the inversion of Laplace transform, the Wright function and covariance function of stochastic process. A spectral Galerkin method is applied to approximate the stochastic model in space. Time discretization is achieved by using firstly a Riemann–Liouville derivative to reformulate the spatial semi-discrete form, and then applying the Grünwald–Letnikov formula. We derive the error estimates of the discrete methods, based on regularity results of the solution. Finally, extensive numerical experiments are presented to confirm our theoretical analysis.

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

This paper explores an inverse source problem related to the heat equation, incorporating nonlocal boundary conditions and featuring two-term time-fractional derivatives. The task is to identify a source term that is independent of the spatial variable, as well as to define the temperature distribution based on energy measurements. Since the stated problem cannot be solved by direct use of the generalized Fourier method, we divide the problem into two sub-problems. The well-posedness of each problem is established through the application of the generalized Fourier method.

Unique recovery of the initial state of distributed order time fractional diffusion equation

We use Phragm\'en-Lindel\"of-Liouville argument to prove the uniqueness for the determining the initial state of solution for the time fractional diffusion equation with distributed order derivative. Several numerical experiments are presented to show the accuracy and efficiency of the algorithm.

Recovering Source Function and Kernel for a Time-fractional Diffusion Equation in the Bounded Domain

Recovering Source Function and Kernel for a Time-fractional Diffusion Equation in the Bounded Domain

IDENTIFICATION OF A TIME-DEPENDENT SOURCE TERM IN A NONLOCAL PROBLEM FOR TIME FRACTIONAL DIFFUSION EQUATION

This paper is concerned with the inverse problem of recovering the time dependent source term in a time fractional diffusion equation, in the case of nonlocal boundary condition and integral overdetermination condition. The boundary conditions of this problem are regular but not strongly regular. The existence and uniqueness of the solution are established by applying generalized Fourier method based on the expansion in terms of root functions of a spectral problem, weakly singular Volterra integral equation and fractional type Gronwall’s inequality. Moreover, we show its continuous dependence on the data.

SIMULTANEOUS INVERSION OF THE SOURCE TERM AND INITIAL VALUE OF THE TIME FRACTIONAL DIFFUSION EQUATION

In this paper, the problem we investigate is to simultaneously identify the source term and initial value of the time fractional diffusion equation. This problem is ill-posed, i.e., the solution (if exists) does not depend on the measurable data. We give the conditional stability result under the a-priori bound assumption for the exact solution. The modified Tikhonov regularization method is used to solve this problem, and under the a-priori and the a-posteriori selection rule for the regularization parameter, the convergence error estimations for this method are obtained. Finally, numerical example is given to prove the effectiveness of this regularization method.

Numerical analysis for optimal quadratic spline collocation method in two space dimensions with application to nonlinear time-fractional diffusion equation

Numerical analysis for optimal quadratic spline collocation method in two space dimensions with application to nonlinear time-fractional diffusion equation

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.

On approximation for time-fractional stochastic diffusion equations on the unit sphere

This paper develops a two-stage stochastic model to investigate the evolution of random fields on the unit sphere S2 in R3. The model is defined by a time-fractional stochastic diffusion equation on S2 governed by a diffusion operator with a time-fractional derivative defined in the Riemann–Liouville sense. In the first stage, the model is characterized by a homogeneous problem with an isotropic Gaussian random field on S2 as an initial condition. In the second stage, the model becomes an inhomogeneous problem driven by a time-delayed Brownian motion on S2. The solution to the model is given in the form of an expansion in terms of complex spherical harmonics. An approximation to the solution is given by truncating the expansion of the solution at degree L≥1. The rate of convergence of the truncation errors as a function of L and the mean square errors as a function of time are also derived. It is shown that the convergence rates depend not only on the decay of the angular power spectrum of the driving noise and the initial condition, but also on the order of the fractional derivative. We study sample properties of the stochastic solution and show that the solution is an isotropic Hölder continuous random field. Numerical examples and simulations inspired by the cosmic microwave background (CMB) are given to illustrate the theoretical findings.

Existence of solutions to a nonlinear fractional diffusion equation with exponential growth

In this paper, we study a Cauchy problem for a space–time fractional diffusion equation with exponential nonlinearity. Based on the standard Lp-Lq estimates of strongly continuous semigroup generated by fractional Laplace operator, we investigate the existence of global solutions for initial data with small norm in Orlicz space exp Lp(Rd) and a time weighted Lr(Rd) space. In the framework of the Hölder interpolation inequality, we also discuss the existence of local solutions without the Orlicz space.

Stochastic modeling of injection induced seismicity based on the continuous time random walk model

The spatiotemporal evolution of earthquakes induced by fluid injections into the subsurface can be erratic owing to the complexity of the physical process. To effectively mitigate the associated hazard and to draft appropriate regulatory strategies, a detailed understanding of how induced seismicity may evolve is needed. In this work, we build on the well-established continuous-time random walk (CTRW) theory to develop a purely stochastic framework that can delineate the essential characteristics of this process. We use data from the 2003 and 2012 hydraulic stimulations in the Cooper Basin geothermal field that induced thousands of microearthquakes to test and demonstrate the applicability of the model. Induced seismicity in the Cooper Basin shows all the characteristics of subdiffusion, as indicated by the fractional order power-law growth of the mean square displacement with time and broad waiting-time distributions with algebraic tails. We further use an appropriate master equation and the time-fractional diffusion equation to map the spatiotemporal evolution of seismicity. The results show good agreement between the model and the data regarding the peak earthquake concentration close to the two injection wells and the stretched exponential relaxation of seismicity with distance, suggesting that the CTRW model can be efficiently incorporated into induced seismicity forecasting.

Operational matrix based numerical scheme for the solution of time fractional diffusion equations

Operational matrix based numerical scheme for the solution of time fractional diffusion equations

Non-Overlapping Domain Decomposition for 1D Optimal Control Problems Governed by Time-Fractional Diffusion Equations on Coupled Domains: Optimality System and Virtual Controls

We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method.

A fast linearized virtual element method on graded meshes for nonlinear time-fractional diffusion equations

A fast linearized virtual element method on graded meshes for nonlinear time-fractional diffusion equations

Unconditionally optimal time two-mesh mixed finite element algorithm for a nonlinear fourth-order distributed-order time fractional diffusion equation

In this article, a fast second-order time two-mesh mixed finite element (TT-MMFE) algorithm is considered to numerically solve the nonlinear fourth-order distributed-order time fractional diffusion equation, where the generated formula for the fractional BDF2 and fractional trapezoidal rule through a free parameter θ (FBT-θ formula) combined with the composite trapezoid formula for the distributed-order derivative is used to discretize the time direction, and the MFE method is utilized to approximate the space direction, respectively. Further, to improve the computing efficiency of our numerical method, the TT-M algorithm is applied to deal with the nonlinear system. For the case of nonsmooth solution, the corrected TT-MMFE scheme by adding starting parts is designed. The existence and uniqueness of the numerical solution are proved in detail. Different from the traditional finite element analysis, the error between the exact solution and the finite element solution is split into temporal error and spatial error by introducing a so-called the error splitting technique, which helps us obtain the spatial error independent of the time step. Finally, numerical examples are given to validate that the TT-MMFE method can reduce computing time, and the corrected TT-MMFE scheme by adding starting parts can recover the convergence rate.

Finite Representations of the Wright Function

The two-parameter Wright special function is an interesting mathematical object that arises in the theory of the space and time-fractional diffusion equations. Moreover, many other special functions are particular instantiations of the Wright function. The article demonstrates finite representations of the Wright function in terms of sums of generalized hypergeometric functions, which in turn provide connections with the theory of the Gaussian, Airy, Bessel, and Error functions, etc. The main application of the presented results is envisioned in computer algebra for testing numerical algorithms for the evaluation of the Wright function.