Solution Of Time-fractional Diffusion Equation Research Articles

An $ {\varepsilon} $-approximation solution of time-fractional diffusion equations based on Legendre polynomials

An $ {\varepsilon} $-approximation solution of time-fractional diffusion equations based on Legendre polynomials

A collocation method for time‐fractional diffusion equation on a metric star graph with η$$ \eta $$ edges

A wavelet collocation method based on Haar wavelet is proposed to investigate the numerical solution of time fractional diffusion equation (TFDE) on a metric star graph. We have utilized the Riemann–Liouville definition of fractional integral operator together with Haar wavelet to obtain the Riemann–Liouville fractional integral operator for Haar wavelet (RLFIO‐H). The application of RLFIO‐H and Haar wavelet to TFDE on metric star graph returns a system of linear algebraic equations. Solving this system yields wavelet coefficients and subsequently the solution. The convergence analysis of the proposed method is discussed to establish the theoretical authenticity of the method. The proposed method is tested on four benchmark problems to validate the theoretical findings. Moreover, the comparison of the developed method with the existing method shows the superiority and accuracy of the proposed method. To the best of the author's knowledge, the proposed work is one of the first collocation method in the literature that deals with the solution of TFDE on metric star graph using wavelet numerically.

A NONLOCAL MODELING FOR SOLVING TIME FRACTIONAL DIFFUSION EQUATION ARISING IN FLUID MECHANICS

This study mainly investigates new techniques for obtaining numerical solutions of time-fractional diffusion equations. The fractional derivative term is represented in the Lagrange operational sense. First, we describe the temporal direction of the considered model using the Legendre orthogonal polynomials. Moreover, to archive a full discretization approach a type of nonlocal method has been applied that is known as the nonlocal peridynamic differential operator (PDDO). The PDDO is based on the concept of peridynamic (PD) interactions by proposing the PD functions orthogonal to each term in the Taylor Series Expansion (TSE) of a field variable. The PDDO for numerical integration uses the vicinity of each point (referred to as the horizon, which does not need background mesh). The PDDO is exclusively described in terms of integration (summation) throughout the interaction domain. As a result, it is unsusceptible to singularities caused by discontinuities. It does, however, need the creation of PD functions at each node. We numerically investigate the stability, the convergence of the scheme, which verifies the validity of the proposed method. Numerical results show the simplicity and accuracy of the presented method.

Similarity Solutions to Nonlinear Diffusion/Harry Dym Fractional Equations

By using scalar similarity transformation, nonlinear model of time-fractional diffusion/Harry Dym equation is transformed to corresponding ordinary fractional differential equations, from which a travelling-wave similarity solution of time-fractional Harry Dym equation is presented. Furthermore, numerical solutions of time-fractional diffusion equation are discussed. Again, through another similarity transformation, nonlinear model of space-fractional diffusion/Harry Dym equation is turned into corresponding ordinary differential equations, whose two similarity solutions are also worked out.

A Low-Dimensional Compact Finite Difference Method on Graded Meshes for Time-Fractional Diffusion Equations

Abstract In this paper, we propose an improvement of the classical compact finite difference (CFD) method by using a proper orthogonal decomposition (POD) technique for time-fractional diffusion equations in one- and two-dimensional space. A reduced CFD method is constructed with lower dimensions such that it maintains the accuracy and decreases the computational time in comparison with classical CFD method. Since the solution of time-fractional diffusion equation typically has a weak singularity near the initial time t = 0 {t=0} , the classical L1 scheme on uniform meshes may obtain a scheme with low accuracy. So, we use the L1 scheme on graded meshes for time discretization. Moreover, we provide the error estimation between the reduced CFD method based on POD and classical CFD solutions. Some numerical examples show the effectiveness and accuracy of the proposed method.

Optimal Error Analysis of a FEM for Fractional Diffusion Problems by Energy Arguments

In this article, the piecewise-linear finite element method (FEM) is applied to approximate the solution of time-fractional diffusion equations on bounded convex domains. Standard energy arguments do not provide satisfactory results for such a problem due to the low regularity of its exact solution. Using a delicate energy analysis, a priori optimal error bounds in $$L^2(\varOmega )$$ -, $$H^1(\varOmega )$$ -norms, and a quasi-optimal bound in $$L^{\infty }(\varOmega )$$ -norm are derived for the semidiscrete FEM for cases with smooth and nonsmooth initial data. The main tool of our analysis is based on a repeated use of an integral operator and use of a $$t^m$$ type of weights to take care of the singular behavior of the continuous solution at $$t=0$$ . The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. Numerical experiments are presented to illustrate some of the theoretical results.

On M-Wright transforms and time-fractional diffusion equations

ABSTRACTIn this paper, we consider the Mittag-Leffler operator as an analytical solution of time-fractional diffusion equation in the Caputo sense. This solution is presented by an integral representation in terms of the M-Wright functions and the exponential operators. Further, we study the Mittag-Leffler operators associated with the Legendre and Bessel diffusion equations. Finally, we extend the obtained integral representation for the time-fractional diffusion equation of distributed order.

Generalized Sommerfeld problem for time fractional diffusion equation: analytical and numerical approach

We consider analytical and numerical methods for solving the generalized Sommerfeld problem for time fractional diffusion equations (TFDE). This problem is to find the generalized diffusion coefficient λ of TFDE and the order α of time derivative according to an additional information about a solution of TFDE. In the case of the steady-state anomalous diffusion process, the periodic boundary value problem without initial conditions for TFDE is solved and exact analytical solutions for inverse problems are obtained. When the initial phase of the process is considered, we use the modified Monte Carlo method to obtain the inverse problem data. In this case, inverse problems are formulated as residual function minimization problems, the Levenberg–Marquardt algorithm for residual function minimization is used and numerical results are presented.