Time-fractional Diffusion Equation Research Articles

Quasilinear Fractional Order Equations and Fractional Powers of Sectorial Operators

The fractional powers of generators for analytic operator semigroups are used for the proof of the existence and uniqueness of a solution of the Cauchy problem to a first order semilinear equation in a Banach space. Here, we use an analogous construction of fractional powers Aγ for an operator A such that −A generates analytic resolving families of operators for a fractional order equation. Under the condition of local Lipschitz continuity with respect to the graph norm of Aγ for some γ∈(0,1) of a nonlinear operator, we prove the local unique solvability of the Cauchy problem to a fractional order quasilinear equation in a Banach space with several Gerasimov–Caputo fractional derivatives in the nonlinear part. An analogous nonlocal Lipschitz condition is used to obtain a theorem of the nonlocal unique solvability of the Cauchy problem. Abstract results are applied to study an initial-boundary value problem for a time-fractional order nonlinear diffusion equation.

Analysis of Numerical Method for Diffusion Equation with Time-Fractional Caputo–Fabrizio Derivative

In this paper, we propose a high-precision discrete scheme for the time-fractional diffusion equation (TFDE) with Caputo-Fabrizio type. First, a special discrete scheme of C-F derivative is used in time direction and a compact difference operator is used in space direction. Second, we discuss the convergence of the proposed method in discrete L 1 -norm and L 2 -norm. The convergence order of our discrete scheme is O τ 2 + h 4 , where τ and h are the temporal and spatial step sizes, respectively. The aim of this paper is to show that fractional operator without singular term is very useful for improving the accuracy of discrete scheme.

A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives

This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order 0<α<1, and 1<β<2. The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered.

Determination of a Nonlinear Coefficient in a Time-Fractional Diffusion Equation

In this paper, we study direct and inverse problems for a nonlinear time fractional diffusion equation. We prove that the direct problem has a unique weak solution and the solution depends continuously on the coefficient. Then we show that the inverse problem has a quasi-solution. The direct problem is solved by the method of lines using an operator approach. A quasi-Newton optimization method is used for the numerical solution to the inverse problem. The Tikhonov regularization is used to overcome the ill-posedness of the inverse problem. Numerical examples with noise-free and noisy data illustrate the applicability and accuracy of the proposed method to some extent.

Numerical approximation of the stochastic equation driven by the fractional noise

We consider the time-fractional nonlinear stochastic fourth-order reaction diffusion equation driven by the fractional noise (FSRDE). The FSRDE is discretized by using the mixed finite element method in space and the FBT-θ method in time. Some good techniques are proposed to prove the important lemmas in the proof of the strong convergence and the weak convergence. By proving the error estimates on the strong and weak convergence, we find the strong convergence and the weak convergence have the same convergence rate, and the convergence order is related to the fractional noise derivative β but not to the time fractional derivative α and the discrete format coefficient θ. Finally, the theorems on the strong and weak convergence are verified by the numerical tests.

Analysis of Time Fractional Diffusion Equation Arising in Ocean Pollution with Different Kernels

Analysis of Time Fractional Diffusion Equation Arising in Ocean Pollution with Different Kernels

Improved Meshless Finite Integration Method for Solving Time Fractional Diffusion Equations

In this paper, a new method named Improved Finite Integration Method (IFIM) is proposed for solving Time Fractional Diffusion Equations (TFDEs). In the IFIM, the Extended Simpson’s Rule (ESR) is employed for numerical quadrature in spatial discretization. Besides, the Piecewise Quadratic Interpolation (PQI) in sense of the Hadamard finite-part integral is utilized for time discretization. Compared with the primary Finite Integration Method (FIM) with Trapezoidal rule which uses the finite difference scheme to address the time discretization, the combination of ESR and PQI in IFIM will lead to a better performance in solving TFDEs. Numerical examples are performed and compared to show the superiority of IFIM. It can also be found that the IFIM is able to obtain a higher accuracy without losing the stability and efficiency.

A Preconditioned Iterative Method for a Multi-State Time-Fractional Linear Complementary Problem in Option Pricing

Fractional derivatives and regime-switching models are widely used in various fields of finance because they can describe the nonlocal properties of the solutions and the changes in the market status, respectively. The regime-switching time-fractional diffusion equations that combine both advantages are also used in European option pricing; however, to our knowledge, American option pricing based on such models and their numerical methods is yet to be studied. Hence, a fast algorithm for solving the multi-state time-fractional linear complementary problem arising from the regime-switching time-fractional American option pricing models is developed in this paper. To construct the solution strategy, the original problem has been converted into a Hamilton–Jacobi–Bellman equation, and a nonlinear finite difference scheme has been proposed to discretize the problem with stability analysis. A policy-Krylov subspace method is developed to solve the nonlinear scheme. Further, to accelerate the convergence rate of the Krylov method, a tri-diagonal preconditioner is proposed with condition number analysis. Numerical experiments are presented to demonstrate the validity of the proposed nonlinear scheme and the efficiency of the proposed preconditioned policy-Krylov subspace method.

L1/LDG Method for Caputo-Hadamard Time Fractional Diffusion Equation.

In this paper, a class of discrete Gronwall inequalities is proposed. It is efficiently applied to analyzing the constructed L1/local discontinuous Galerkin (LDG) finite element methods which are used for numerically solving the Caputo-Hadamard time fractional diffusion equation. The derived numerical methods are shown to be -robust using the newly established Gronwall inequalities, that is, it remains valid when . Numerical experiments are given to demonstrate the theoretical statements.

Petrov-Galerkin Lucas Polynomials Procedure for the Time-Fractional Diffusion Equation

Herein, we build and implement a combination of Lucas polynomials basis that fulfills the spatial homogenous boundary conditions. This basis is then used to solve the time-fractional diffusion equation spectrally. The elements of all spectral matrices are explicitly obtained in terms of the Gauss hypergeometric function. The convergence and error analysis of the proposed Lucas expansion is studied. Numerical results indicate the high accuracy and applicability of the suggested algorithm.

On the Fractional Diffusion Equation Associated With Exponential Source and Operator With Exponential Kernel

AbstractIn this paper, we investigate the well-posedness of mild solutions of the time-fractional diffusion equation with an exponential source function and the Caputo-Fabrizio derivative of a fractional order α∈(0,1). Some linear estimates of the solution kernels on Hilbert scale spaces are constructed using a spectrum of the Dirichlet Laplacian. Based on the Banach fixed point theorem, the global existence and uniqueness of the small-data mild solution are approved. This work is considered the first study on the time-fractional diffusion equation with a nonlinear function for all common dimensions of 1, 2, and 3.

Green’s Functions on Various Time Scales for the Time-Fractional Reaction-Diffusion Equation

The time-fractional diffusion equation coupled with a first-order irreversible reaction is investigated by employing integral transforms. We derive Green’s functions for short and long times via approximations of the Mittag-Leffler function. The time value for which the crossover between short- and long-time asymptotic holds is presented in explicit form. Based on the developed Green’s functions, the exact analytic asymptotic solutions of the time-fractional reaction-diffusion equation are obtained. The applicability of the obtained solutions is demonstrated via quantification of the reaction-diffusion kinetics during heterogeneous catalytic chitin conversion to chitosan.

Continuity of the Solution to a Stochastic Time-fractional Diffusion Equations in the Spatial Domain with Locally Lipschitz Sources

Continuity of the Solution to a Stochastic Time-fractional Diffusion Equations in the Spatial Domain with Locally Lipschitz Sources

Glioblastoma multiforme growth prediction using a Proliferation-Invasion model based on nonlinear time-fractional 2D diffusion equation

The glioblastoma multiforme (GBM) is the fast-growing and aggressive type of tumor of the brain or spinal cord that is associated with high morbidity and mortality. Therefore, accurate knowledge of the tumor growth process in different time intervals seems necessary to adopt optimal treatment methods. In this study, with a combination of available early stages imaging data and using image processing methods, a time fractional reaction–diffusion equation based on the moving least squares method was implemented for predicting the growth of the GBM tumor model implemented in the mouse brain. The obtained results demonstrate that the Proliferation-Invasion model implemented by a time fractional form shows more agreement with the experimental results. As a result, using a fractional derivative of order α=0.9 results in the lowest prediction error of the tumor surface (less than 1%). In addition, to reducing the cost and side effects of theranostic methods, the presented model will have the possibility to consider the effects of therapeutic factors, such as hyperthermia, radiography, and surgery, on tumor growth. The ability to use the patient’s characteristics in diagnostic and treatment considerations and the development of personalized medicine can also be considered one of the capabilities of the presented model.

Reconstruction of the Source Term in a Time-Fractional Diffusion Equation from Partial Domain Measurement

Reconstruction of the Source Term in a Time-Fractional Diffusion Equation from Partial Domain Measurement

Regularization of the Final Value Problem for the Time-Fractional Diffusion Equation

We consider the backward problem of reconstructing the initial condition of a nonhomogeneous time-fractional diffusion equation from final measurements. The proposed method relies on the eigenfunction expansion of the forward solution and the Tikhonov regularization to control the instability of the underlying inverse problem. We establish stability results and we provide convergence rates under a priori and a posteriori parameter choice rules. The resulting algorithm is robust and computationally inexpensive. Two examples are included to illustrate the effectiveness and accuracy of the proposed method.

A novel numerical technique for solving time fractional nonlinear diffusion equations involving weak singularities

In this work, an efficient numerical approximation for the solution of the time fractional nonlinear diffuse interface model is studied. The solution to this problem has a weak singularity near the initial time . The fractional order nonlinear diffusion model is transformed into a system of nonlinear functional equations. The Daftardar–Gejji and Jafari method is employed to solve the corresponding nonlinear system. The L1 scheme is used to discretize the Caputo fractional derivative on a graded mesh in the time direction. In contrast, the spatial derivative is approximated by applying a classical central finite difference scheme to a uniform mesh. The convergence analysis and the error bounds are carried out. The analysis and the computational findings exhibit the effectiveness of the proposed method.

Initial-boundary value problems for coupled systems of time-fractional diffusion equations

This article deals with the initial-boundary value problem for a moderately coupled system of time-fractional diffusion equations. Defining the mild solution, we establish fundamental unique existence, limited smoothing property and long-time asymptotic behavior of the solution, which mostly inherit those of a single equation. Owing to the coupling effect, we also obtain the uniqueness for an inverse problem on determining all the fractional orders by the single point observation of a single component of the solution.

Cauchy problems for the time-fractional degenerate diffusion equations

This paper is devoted to the Cauchy problems for the one-dimensional linear time-fractional diffusion equations with $\partial^{\alpha}_{t}$ the Caputo fractional derivative of order $\alpha\in(0,1)$ in the variable t and time-degenerate diffusive coefficients $t^{\beta}$ with $\beta >1-\alpha$. The solutions of  Cauchy problems for the one-dimensional time-fractional degenerate diffusion equations with the time-fractional derivative $\partial^{\alpha}_{t}$ of order $\alpha\in(0,1)$  in the variable $t$, are shown. In the "Problem statement and main results" section of the paper, the solution of the time-fractional degenerate diffusion equation in a variable coefficient with two different initial conditions are considered. In this work, a solution is found by using the Kilbas-Saigo function $E_{\alpha,m,l}(z)$ and applying the Fourier transform $F$ and inverse Fourier transform $\mathcal{F}^{-1}$. Convergence of solution of problem 1 and problem 2 are proven using Plancherel theorem. The existence and uniqueness of the solution of the problem are confirmed.&nbsp

Uniqueness of the potential in a time-fractional diffusion equation

Abstract This article concerns the uniqueness of an inverse coefficient problem of identifying a spatially varying potential in a one-dimensional time-fractional diffusion equation. The input sources are given by a complete system in L 2 ⁢ ( 0 , 1 ) {L^{2}(0,1)} , and measurements are observed at the end point of the spatial interval. Firstly, we provide the positive lower bound of the Green function for the differential operator with different boundary conditions. Then, based on the positive lower bound estimation of the Green function, the relationship between the Green function, the solution of the forward problem, and the potential, such measurements uniquely determine the potential on the entire interval under different boundary conditions.