Time Fractional Advection Diffusion Equation Research Articles

Cubic B-spline based numerical schemes for delayed time-fractional advection-diffusion equations involving mild singularities

This article presents two efficient layer-adaptive numerical schemes for a class of time-fractional advection-diffusion equations with a large time delay. The fractional derivative of order α with α ∈ (0, 1) is taken in the Caputo sense. The solution to this type of problem generally has a layer due to the mild singularity near the time t = 0. Consequently, the polynomial interpolation discretizing scheme degrades the convergence rate in the case of uniform meshes. In the presence of a singularity, the temporal fractional operator is discretized by employing the L1 technique on a layer-resolving mesh. In contrast, the cubic B-spline collocation method is used in the spatial direction. The convergence analysis and estimation of error are presented for the proposed scheme under reasonable regularity assumptions on the coefficients. The scheme achieves its optimal convergence rate (2 − α) for suitable choice of grading parameter (γ ≥ (2 − α)/α). Furthermore, we modified the proposed scheme by discretizing the fractional operator with the help of the L1-2 technique. The modified scheme gets a quadratic order convergence for γ ≥ 2/α. In addition, we extend the proposed schemes to solve the corresponding semilinear problem. Numerical examples demonstrate the efficiency and applicability of the proposed techniques.

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.

Novel numerical methods based on graded, adaptive and uniform meshes for a time-fractional advection-diffusion equation subjected to weakly singular solution

Novel numerical methods based on graded, adaptive and uniform meshes for a time-fractional advection-diffusion equation subjected to weakly singular solution

A fast difference scheme for the multi-term time fractional advection–diffusion equation with a non-linear source term

In field experiments involving the movement of solutes through heterogeneous porous and fractured media, it has been observed that the contaminant plumes can transition between various diffusive states. This current research is dedicated to the development of a fast difference scheme designed for a multi-term time-fractional order advection–diffusion model that incorporates a nonlinear source term. This model is an instrumental tool in explaining the underlying dynamics of transport. To effectively address the substantial computational and storage challenges inherent in this problem, we present an efficient algorithm specifically designed for the fast computation of the Caputo derivative. This algorithm is based on the summation of exponentials technique. We thoroughly explore the theoretical aspects of our scheme including its unconditional stability and convergence with rigorous proofs. The proposed numerical scheme minimizes the CPU time and storage requirements when compared to the conventional direct difference scheme.

A High Accuracy Numerical Method Based on Interpolation Technique for Time-Fractional Advection-Diffusion Equations

In this paper, the time-fractional advection-diffusion equation (TFADE) is solved by the barycentric Lagrange interpolation collocation method (BLICM). In order to approximate the fractional derivative under the definition of Caputo, BLICM is used to approximate the unknown function. We obtain the discrete scheme of the equation by combining BLICM with the Gauss-Legendre quadrature rule. The convergence rate for the TFADE equation of the BLICM is derived, and the accuracy of the discrete scheme can be improved by modifying the number of Gaussian nodes. To illustrate the efficiency and accuracy of the present method, a few numerical examples are presented and compared with the other existing methods.

Numerical solutions of linear time-fractional advection-diffusion equations with modified Mittag-Leffler operator in a bounded domain

Fractional advection-diffusion equations have demonstrated to be a powerful tool in modeling complex anomalous diffusion in applied science. In this paper, we studied novel linear time-fractional advection-diffusion equations associated with an extension of Mittag-Leffler fractional derivative operator. A useful feature of the used extension is to address the limitations of the Mittag-Leffler fractional derivative model. We, mainly, proposed a numerical approach to provide approximate solutions to linear time-fractional advection-diffusion equations with the studied extended fractional derivative operator. The suggested approach is based on discretizing the studied models with respect to spatio-temporal domain using uniform meshes. A new type of solutions for the studied models was generated numerically using the proposed approach. Besides, a comparative study was conducted to verify the accuracy and feasibility of the proposed approach.

A fractional derivative model for nuclides transport in heterogeneous fractured media

Nuclide transport in fractured media involves the advection, dispersion, adsorption, etc. The dispersion and adsorption properties of the rock matrix have spatial variability, which results in an anomalous transport of nuclides. In this study, a time-fractional advection-diffusion equation (t-FADE) model is utilized to capture the sub-diffusion transport behavior with heavy-tail property, including the breakthrough curves (BTCs) of uranium and thorium transport in granite plates. Moreover, hydrodynamic dispersion of tritiated water, strontium and cesium in granite fractures are also studied. The results indicate that BTCs of nuclides transport in the granite fractures are unimodal and asymmetric. The decrease of the fractional order α reflects the stronger sub-diffusion. Furthermore, small initial velocity enhances sub-diffusion effect of nuclides and lengthens the breakout time of BTCs, which results in obvious heavy-tail phenomena. The analysis results demonstrate that the t-FADE model can accurately describe sub-diffusion behavior of nuclides transport. At last, the advantages of the t-FADE model in prediction and remediation of nuclides contamination are put forward.

Numerical solutions of Atangana-Baleanu time-fractional advection diffusion equation via an extended cubic B-spline technique

The B-spline function is made up of a set of smooth piecewise polynomials that are controlled by a set of control points. A linear combination of B-spline basis of a particular degree can be used to express any spline function of that degree. The spline functions and their derivatives are continuous and depending on the multiplication of knots. In this study, an extended cubic B-spline (ECBS) approximation is used for the numerical solution of time-fractional advection diffusion equation (TFADE) involving Atangana-Baleanu fractional derivative (ABFD). Initially, the non-singular kernel ABFD is discretized using finite difference method (FDM). The spatial derivatives are discretized using ECBS functions. Convergence and stability of the proposed scheme are studied. The results tabulated in tables that show how they accurate and desirable the outcomes are by comparing the obtained approximate results with the available exact solutions. To the authors’ knowledge, this work is the first time to use the ABFD for numerical solution of time-fractional advection diffusion equation using ECBS.

A stable RBF-FD method for solving two-dimensional variable-order time fractional advection-diffusion equation

The meshless method is used in this paper to provide a numerical solution for a two-dimensional variable-order time fractional advection-diffusion equation. The radial basis function and the finite difference scheme on two-dimensional (2-D) arbitrary domains are the foundations of this accurate and reliable meshless approach. The name of this approach is radial basis function-generated finite differences (RBF-FD). The finite difference technique and the RBF-FD approximation are both employed for discrete time and space domains, respectively. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. The advantage of the RBF-FD method is that it has high-order convergence rates and its ability to manage sparse node layouts in arbitrary domains. The use of polyharmonic splines (PHS) as a RBF and the absence of the shape parameter are also a positive points for the mentioned method. The effectiveness and precision of the proposed method are examined with three distinct examples, incorporating two model examples and a real-world application of contamination transportation phenomena.

A novel local Hermite radial basis function‐based differential quadrature method for solving two‐dimensional variable‐order time fractional advection–diffusion equation with Neumann boundary conditions

AbstractA novel Hermite radial basis function‐based differential quadrature (H‐RBF‐DQ) method is presented in this paper based on 2D variable order time fractional advection–diffusion equations with Neumann boundary conditions. The proposed method is designed to treat accurately for derivative boundary conditions, which considerably improve the approximation results and extend the range of applicability for the method of RBF‐DQ. The advantage of the present method is that the Hermite interpolation coefficients are only dependent of the point distribution yielding a substantially better imposition of boundary conditions, even for time evolution. The proposed algorithm is thoroughly validated and is demonstrated to handle the fractional calculus problems with both Dirichlet and Neumann boundaries very well.

Investigation and analysis of the numerical approach to solve the multi-term time-fractional advection-diffusion model

<abstract><p>In this paper, a methodical approach is presented to approximate the multi-term fractional advection-diffusion model (MT-FAD). The Lagrange squared interpolation is used to discretize temporal fractional derivatives, and Legendre polynomials are shifted as an operator to discretize the spatial fractional derivatives. The advantage of these numerical techniques lies in the orthogonality of Legendre polynomials and its matrix operations. A quadratic implicit design as well as its stability and convergence analysis are evaluated. It should be noted that the theoretical proof obtained from this study represents the first results for these numerical schemes. Finally, we provide three numerical examples to verify the validity of the proposed methods and demonstrate their accuracy and effectiveness in comparison with previous studies shown in [W. P. Bu, X. T. Liu, Y. F. Tang, J. Y. Yang, Finite element multigrid method for multi-term time fractional advection diffusion equations, <italic>Int. J. Model. Simul. Sci. Comput.</italic>, <bold>6</bold> (2015), 1540001].</p></abstract>

APPLICATION OF HOSOYA POLYNOMIAL TO SOLVE A CLASS OF TIME-FRACTIONAL DIFFUSION EQUATIONS

In this paper, we study time-fractional diffusion equations such as the time-fractional Kolmogorov equations (TF–KEs) and the time-fractional advection–diffusion equations (TF–ADEs) in the Caputo sense. Here, we have developed the operational matrices (OMs) using the Hosoya polynomial (HP) as basis function for OMs to obtain solution of the TF–KEs and the TF–ADEs. The great benefit of this technique is converting the TF–KEs and the TF–ADEs to algebraic equations, which can be simply solved the problem under study. We provide error bound for the approximation of a bivariate function using the HP. Furthermore, comparison of the numerical results obtained using the proposed technique with the exact solution is done. The results prove that the proposed numerical method is most relevant for solving the TF–KEs and the TF–ADEs and accurate.

Numerical study for a class of time fractional diffusion equations using operational matrices based on Hosoya polynomial

<abstract><p>In this paper, we develop a numerical method by using operational matrices based on Hosoya polynomials of simple paths to find the approximate solution of diffusion equations of fractional order with respect to time. This method is applied to certain diffusion equations like time fractional advection-diffusion equations and time fractional Kolmogorov equations. Here we use the Atangana-Baleanu fractional derivative. With the help of this approach we convert these equations to a set of algebraic equations, which is easier to be solved. Also, the error bound is provided. The obtained numerical solutions using the presented method are compared with the exact solutions. The numerical results show that the suggested method is convenient and accurate.</p></abstract>

Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations

<p style='text-indent:20px;'>We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value <inline-formula><tex-math id="M1">\begin{document}$ \partial_t^{\alpha} u(x, t) = -Au(x, t) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ -A = \sum_{i, j = 1}^d \partial_i(a_{ij}(x) \partial_j) + \sum_{j = 1}^d b_j(x) \partial_j + c(x) $\end{document}</tex-math></inline-formula>. We establish the uniqueness for an inverse problem of determining an order <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> of fractional derivatives by data <inline-formula><tex-math id="M4">\begin{document}$ u(x_0, t) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ 0<t<T $\end{document}</tex-math></inline-formula> at one point <inline-formula><tex-math id="M6">\begin{document}$ x_0 $\end{document}</tex-math></inline-formula> in a spatial domain <inline-formula><tex-math id="M7">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. The uniqueness holds even under assumption that <inline-formula><tex-math id="M8">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ A $\end{document}</tex-math></inline-formula> are unknown, provided that the initial value does not change signs and is not identically zero. The proof is based on the eigenfunction expansions of finitely dimensional approximating solutions, a decay estimate and the asymptotic expansions of the Mittag-Leffler functions for large time.</p>

Solving an inverse problem for a time-fractional advection-diffusion equation with variable coefficients by rationalized Haar wavelet method

In this research work, we use of rationalized Haar wavelet (RHW) method and Crank–Nicolson finite-difference scheme to propose a new idea for solving the inverse time-fractional advection-diffusion equation with variable coefficients. Based on this method, the expansion of the solution is a series of RHW functions with variable coefficients. One of the advantages of the proposed method is the use of operational matrices for single and double integration of basic functions without the need for direct integration. In addition, the numerical estimation of the order of convergence and extrapolation of the results are provided. To reveal the efficiency of our method, the numerical solutions of two examples have been compared with Cubic B-spline function (CBS) and Radial basis function (RBF) methods. Finally, the diagrams of error for the Haar wavelet and RBF methods are shown in the figures.

Numerical solutions of advection diffusion equations involving Atangana–Baleanu time fractional derivative via cubic B-spline approximations

The B-spline or the basis spline function, is a piecewise polynomial function made up of polynomials, its domain is subdivided by knots, and basis functions are non-zero on the entire domain. In this study, a new cubic B-spline (NCBS) approximation together with the θ-weighted scheme is formed to approximate the numerical solution of the time fractional advection diffusion equation (TFADE) involving the Atangana–Baleanu time fractional derivative (ABTFD). The finite difference scheme (FDS) is utilized to discretize the ABTFD with a non-singular kernel. The spatial derivative is discretized by using NCBS functions. The stability analysis of the proposed technique is studied. Convergence analysis of the current technique is also analyzed. The proposed technique is examined on a variety of problems, and the numerical outcomes are contrasted with the previously published technique’s results to ensure the correctness and accuracy of the current technique.

A robust numerical approximation of advection diffusion equations with nonsingular kernel derivative

In this article we aim to approximate linear time fractional advection diffusion equations (TFADE) with Atangana-Baleanu- Caputo(ABC) derivative using local meshless method and Laplace transformation(LT). The method comprises of three steps. In the first step the the time variable is eliminated using LT. In the second step the reduced problem is solved using local meshless method. In the third step the solution of TFADE with ABC derivative is retrieved from local meshless methods solution by representing it as Bromwich integral. We then approximate the integral using some suitable quadrature rule. The stability and convergence of the method are discussed. The local meshless method is utilized to overcome the ill-conditioning issue of the interpolation matrices in global meshless methods and to over come the shape parameters sensitivity. Also in comparison with time stepping methods the LT is employed and contour integration technique is utilized to deal with the ABC derivative, which circumvent the calculation of costly convolution integrals in the approximation of ABC derivative, and also avoids the effect of time step on the stability and accuracy. Some test problems are considered in one and two dimensions to validate the proposed numerical method. The two dimensional problem is solved in regular and irregular domains. The computational experiments confirms that this method is computationally efficient and highly accurate for such type of problems.

Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions

In this paper, we are interested in the study of a Caputo time fractional advection–diffusion equation with nonhomogeneous boundary conditions of integral types ∫01vx,tdx and ∫01xnvx,tdx. The existence and uniqueness of the given problem’s solution is proved using the method of the energy inequalities known as the “a priori estimate” method relying on the range density of the operator generated by the considered problem. The approximate solution for this problem with these new kinds of boundary conditions is established by using a combination of the finite difference method and the numerical integration. Finally, we give some numerical tests to illustrate the usefulness of the obtained results.

Generalized finite difference method with irregular mesh for a class of three-dimensional variable-order time-fractional advection-diffusion equations

Fractional advection-diffusion equation, as a generalization of classical advection-diffusion equation, has been always mentioned to simulate anomalous diffusion in porous media. This work introduces a meshless generalized finite difference method (GFDM) to solve a class of three-dimensional variable-order time fractional advection-diffusion equation (TFADE) in finite domains. Three examples with known analytic solutions in different domains are given to demonstrate that the method is accurate and stable. To reduce computational and storage cost, we discretize the time derivative terms of TFADE by a fast finite difference method (FFDM) based on sum-of-exponentials (SOE) approximation. Meanwhile, discretizing space derivative terms, GFDM generates a linear equation set including function values of neighboring nodes with various weight coefficients. Then the partial derivatives of TFADE are indicated as the linear system above. Also, this paper investigates the irregular mesh in the finite spatial domain, which is more closely meets the description of practice problems. Numerical results indicate that models with irregular mesh can also be simulated by GFDM which maintains high accuracy. Furthermore, the method is stable and accurate in solving three-dimensional irregular domain problems, where the relative errors can be less than 0.01%. This paper shows that FFDM based on SOE approximation can improve computational efficiency, and GFDM can flexibly and efficiently solve three-dimensional variable-order and variable-coefficient TFADE.

An efficient technique based on cubic B-spline functions for solving time-fractional advection diffusion equation involving Atangana-Baleanu derivative.

The present paper deals with cubic B-spline approximation together with theta -weighted scheme to obtain numerical solution of the time fractional advection diffusion equation using Atangana–Baleanu derivative. To discretize the Atangana–Baleanu time derivative containing a non-singular kernel, finite difference scheme is utilized. The cubic basis functions are associated with spatial discretization. The current discretization scheme used in the present study is unconditionally stable and the convergence is of order O(h^2+Delta t^{2}). The proposed scheme is validated through some numerical examples which reveal the current scheme is feasible and quite accurate.