Time Fractional Advection Diffusion Equation Research Articles

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.

Advection–diffusion in a porous medium with fractal geometry: fractional transport and crossovers on time scales

In a porous fractal medium, the transport dynamics is sometimes anomalous as well as the crossover between numerous transport regimes occurs. In this paper, we experimentally investigate the mass transfer of the diffusing agents of various classes in the composite porous particle with fractal geometry. It is shown that transport mechanisms differ at short and long times. At the beginning, pure advection is observed, whereas the longtime transport follows a convective mechanism. Moreover, the longtime transport experiences either Fickian or non-Fickian kinetics depending on the diffusing agent. The non-Fickian transport is justified for the diffusing agent with higher adsorption energy. Therefore, we speculate that non-Fickian transport arises due to the strong irreversible adsorption sticking of the diffusing molecules on the surface of the porous particle. For the distinguishing of the transport regimes, an approach admitting the transformations of the experimental data and the relevant analytic solutions in either semi-logarithmic or logarithmic coordinates is developed. The time-fractional advection–diffusion equation is used on a phenomenological basis to describe the experimental data exhibiting non-Fickian kinetics. The obtained anomalous diffusion exponent corresponds to the superdiffusive transport.

The meshless approach for solving 2D variable-order time-fractional advection–diffusion equation arising in anomalous transport

In this paper, with the aim of extending an elegant and straightforward numerical approximation to describe one of the most common physical phenomena has been undertaken. In this regard, the generalization of advection–diffusion equation, namely, the time-fractional advection–diffusion equation with understanding sense variable-order fractional derivative, is taken into consideration. An efficient and accurate approach is relying on the Kansa scheme and finite difference method to provide a mathematical framework to treat the spatial discretization and temporal term, respectively. The meshless collocation approach is utilized for interior scattered points and those on the boundary. Thus, the problem under consideration is reduced to a system of linear algebraic equations. The use of the radial basis function as shape function brings many advantages for proposal numerical method in terms of improved accuracy by setting an appropriate shape parameter and applied for solving high-dimensional models without extra cost. The validity and accuracy of the proposed approach is investigated by four various examples involving three benchmark examples and a practical application of pollution transfer phenomena.

A fast and high-order numerical method for nonlinear fractional-order differential equations with non-singular kernel

Efficient and fast explicit methods are proposed to solve nonlinear Caputo-Fabrizio fractional differential equations, where Caputo-Fabrizio operator is a new proposed fractional derivative with a smooth kernel. The proposed methods produce the second-order for linear interpolation and the third-order accuracy for quadratic interpolation, respectively. The convergence analysis is proved by using discrete Gronwall's inequality. Furthermore, applying the recurrence relation of the memory term, it reduces CPU time executed the proposed methods. The proposed fast algorithm requires approximately O(N) arithmetic operations while O(N2) is required in case of the regular predictor-corrector schemes, where N is the total number of the time step. The following numerical examples demonstrate the accuracy of the proposed methods as well as the efficiency: nonlinear fractional differential equations, time-fraction sub-diffusion, and time-fractional advection-diffusion equation. Numerical experiments also verify the theoretical convergence rates.

A meshless method in reproducing kernel space for solving variable-order time fractional advection–diffusion equations on arbitrary domain

In this paper, a meshless method in reproducing kernel space is proposed for solving VOTFA-DE on arbitrary domain. Advantages of the meshless method proposed could avoid effectively difficulties of constructing shape functions using known Mercer kernel and deal with arbitrary domains. And the accuracy is verified by two examples.

An unconditionally stable algorithm for multiterm time fractional advection–diffusion equation with variable coefficients and convergence analysis

AbstractThis paper focuses on the numerical solution of the variable coefficient multiterm time fractional advection–diffusion equation via exponential B‐splines. We discretize the temporal part by using the Crank–Nicolson method and spatial part by the exponential B‐splines. The unconditional stability is obtained by the Von‐Neumann method. The convergence rates are also studied. Numerical simulations confirm the theoretically expected accuracy in both time and space directions. A comparative analysis with the other methods shows the superiority of the proposed algorithm.

Numerical study for time fractional stochastic semi linear advection diffusion equations

In this work, a stochastic fractional advection diffusion model with multiplicative noise is studied numerically. The Galerkin finite element method in space and finite difference in time are used, where the fractional derivative is in Caputo sense. The error analysis is investigated via Galerkin finite element method. In terms of the Mittag Leffler function, the mild solution is obtained. For the error estimates, the strong convergence for the semi and fully discrete schemes are proved in a semigroup structure. Finally, two numerical examples are given to confirm the theoretical results.

A pseudo-spectral based method for time-fractional advection-diffusion equation

In this paper, a pseudo-spectral method with the Lagrange polynomial basis is proposed to solve the time-fractional advection-diffusion equation. A semi-discrete approximation scheme is used for conversion of this equation to a system of ordinary fractional differential equations. Also, to protect the high accuracy of the spectral approximation, the Mittag-Leffler function is used for the integration along the time variable. Some examples are performed to illustrate the accuracy and efficiency of the proposed method.

Shifted Chebyshev collocation of the fourth kind with convergence analysis for the space–time fractional advection-diffusion equation

The main purpose of this paper is to design a numerical method for solving the space–time fractional advection-diffusion equation (STFADE). First, a finite difference scheme is applied to obtain the semi-discrete in time variable with convergence order $$\mathcal {O}(\tau ^{2-\beta })$$ . In the next, to discrete the spatial fractional derivative, the Chebyshev collocation method of the fourth kind has been applied. This discrete scheme is based on the closed formula for the spatial fractional derivative. Besides, the time-discrete scheme has studied in the $$L_{2}$$ space by the energy method and we have proved the unconditional stability and convergence order. Finally, we solve three examples by the proposed method and the obtained results are compared with other numerical problems. The numerical results show that our method is much more accurate than existing techniques in the literature.

A computational approach for solving time fractional differential equation via spline functions

A computational approach based on finite difference scheme and a redefined extended B-spline functions is presented to study the approximate solution of time fractional advection diffusion equation. The Caputo time-fractional derivative and redefined extended B-spline functions have been used for the time and spatial discretization, respectively. The numerical scheme is shown to be O(h2+Δt2-α) accurate and unconditionally stable. The proposed method is tested through some numerical experiments involving homogeneous/non-homogeneous boundary conditions which concluded that it is more accurate than existing methods. The simulation results show superior agreement with the exact solution as compared to existing methods.

Computationally Efficient Hybrid Method for the Numerical Solution of the 2D Time Fractional Advection-Diffusion Equation

In this paper, a hybrid method based on the Laplace transform and implicit finite difference scheme is applied to obtain the numerical solution of the two-dimensional time fractional advection-diffusion equation (2D-TFADE). Some of the major limitations in computing the numerical solution for fractional differential equations (FDEs) in multi-dimensional space are the huge computational cost and storage requirement, which are O(N^2) cost and O(MN) storage, N and M are the total number of time levels and space grid points, respectively. The proposed method reduced the computational complexity efficiently as it requires only O(N) computational cost and O(M) storage with reasonable accuracy when numerically solving the TFADE. The method’s stability and convergence are also investigated. The Results of numerical experiments of the proposed method are obtained and found to compare well with the results of existing standard finite difference scheme.

On Numerical Solution Of The Time Fractional Advection-Diffusion Equation Involving Atangana-Baleanu-Caputo Derivative

Abstract A powerful algorithm is proposed to get the solutions of the time fractional Advection-Diffusion equation(TFADE): A B C D 0 + , t β u ( x , t ) = ζ u x x ( x , t ) − κ u x ( x , t ) + $^{ABC}\mathcal{D}_{0^+,t}^{\beta}u(x,t) =\zeta u_{xx}(x,t)- \kappa u_x(x,t)+$ F(x, t), 0 < β ≤ 1. The time-fractional derivative A B C D 0 + , t β u ( x , t ) $^{ABC}\mathcal{D}_{0^+,t}^{\beta}u(x,t)$ is described in the Atangana-Baleanu Caputo concept. The basis of our approach is transforming the original equation into a new equation by imposing a transformation involving a fictitious coordinate. Then, a geometric scheme namely the group preserving scheme (GPS) is implemented to solve the new equation by taking an initial guess. Moreover, in order to present the power of the presented approach some examples are solved, successfully.

An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions

In this paper, a linearized L1 Legendre–Galerkin spectral method for the two-dimensional nonlinear time fractional advection–diffusion equation is derived. The numerical method is stable without the CFL conditions based on the error splitting argument technique and the discrete fractional Gronwall type inequality. We also consider the case of non-smooth solutions by adding some correction terms. The numerical experiments are given to verify the theoretical analysis and the effectiveness of the correction method.

A two-grid MMOC finite element method for nonlinear variable-order time-fractional mobile/immobile advection–diffusion equations

A fully discrete two-grid modified method of characteristics (MMOC) scheme is proposed for nonlinear variable-order time-fractional advection–diffusion equations in two space dimensions. The MMOC is used to handle the advection-dominated transport and the two-grid method is designed for efficiently solving the resulting nonlinear system. Optimal L2 error estimates are derived for both the MMOC scheme and the corresponding two-grid MMOC scheme. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed method.

A Numerical approach of fractional advection-diffusion equation with Atangana–Baleanu derivative

Abstract In the current paper, a new approach is applied to solve time fractional advection-diffusion equation. The utilized fractional derivative operator is the Atangana–Baleanu (AB) derivative in Caputo sense. The mentioned fractional derivative involves the Mittag–Leffler function as the kernel that is both non-singular and non-local. A new operational matrix of AB fractional integration is obtained for the Bernstein polynomials (Bps). By applying the aforesaid matrix, the considered problems are reduced to a system of equations. The approximate solution is derived by solving the yielded system. Also, the error bound is studied. The obtained results show that the applied scheme is simple and powerful tool in finding numerical solutions of fractional equations.

On an optimal control problem of time-fractional advection-diffusion equation

<p style='text-indent:20px;'>We consider an optimal control problem of an advection-diffusion equation with Caputo time-fractional derivative. By convex duality method we obtain as optimality condition a forward-backward coupled system. We then prove the existence of a solution to this coupled system using Schauder fixed point theorem. The uniqueness of the solution is also established under certain monotonicity condition on the cost functional.

Space‐time finite element adaptive AMG for multi‐term time fractional advection diffusion equations

In this study, we construct a space‐time finite element (FE) scheme and furnish cost‐efficient approximations for one‐dimensional multi‐term time fractional advection diffusion equations on a bounded domain Ω. Firstly, a fully discrete scheme is obtained by the linear FE method in both temporal and spatial directions, and many characterizations on the resulting matrix are established. Secondly, the associated condition number estimation is derived, where τn and h, respectively, are the current time and space step sizes. Next, we propose a lossless in robustness and adaptive algebraic multigrid (AMG) with computational cost and matrix‐free storage, in contrast to classical AMG with , where M is the number of spatial segments. Meanwhile, uniform convergence analyses on the two‐level V(0,1)‐cycle of classical and adaptive AMGs are provided. Finally, we demonstrate the optimal error order of space‐time FE approximations in the L2(Ω) norm sense, present theoretical confirmations and predictable behaviors of the proposed algorithm in numerical experiments.

A fractional PDE for first passage time of time-changed Brownian motion and its numerical solution

We show that the First-Passage-Time probability distribution of a Lévy time-changed Brownian motion with drift is solution of a time fractional advection-diffusion equation subject to initial and boundary conditions; the Caputo fractional derivative with respect to time is considered. We propose a high order compact implicit discretization scheme for solving this fractional PDE problem and we show that it preserves the structural properties (non-negativity, boundedness, time monotonicity) of the theoretical solution, having to be a probability distribution. Numerical experiments confirming such findings are reported. Simulations of the sample paths of the considered process are also performed and used to both provide suitable boundary conditions and to validate the numerical results.