Computational approaches to time-fractional partial differential equations

Wellposedness and regularity of the variable-order time-fractional diffusion equations

We first show that the infinitesimal generator of Lie symmetry of a time-fractional partial differential equation (PDE) takes a unified and simple form, and then separate the Lie symmetry condition into two distinct parts, where one is a linear time-fractional PDE and the other is an integer-order PDE that dominates the leading position, even completely determining the symmetry for a particular type of time-fractional PDE. Moreover, we show that a linear time-fractional PDE always admits an infinite-dimensional Lie algebra of an infinitesimal generator, just as the case for a linear PDE and a nonlinear time-fractional PDE admits, at most, finite-dimensional Lie algebra. Thus, there exists no invertible mapping that converts a nonlinear time-fractional PDE to a linear one. We illustrate the results by considering two examples.

A method is presented to derive the Lie point symmetries of time fractional partial differential equations in the sense of Riemann-Liouville fractional derivative. The applicability of the method has been illustrated through time fractional Burgers-Korteweg-de Vries with time dependent variable coefficients, time fractional dissipative Zabolotskaya-Khokhlov equation, time fractional generalised Benjamin equation and time fractional diffusion equation with variable coefficients. Using the obtained Lie point symmetries, it is shown that each of the above mentioned time fractional partial differential equations can be transformed into a ordinary differential equations of fractional order. Exact solutions of the above mentioned time fractional equations are derived wherever possible. It is also explained how conservation laws can be derived to time fractional partial differential equations.

A method is presented to derive the Lie point symmetries of time fractional partial differential equations in the sense of Riemann-Liouville fractional derivative. The applicability of the method has been illustrated through time fractional Burgers-Korteweg-de Vries with time dependent variable coefficients, time fractional dissipative Zabolotskaya-Khokhlov equation, time fractional generalised Benjamin equation and time fractional diffusion equation with variable coefficients. Using the obtained Lie point symmetries, it is shown that each of the above mentioned time fractional partial differential equations can be transformed into a ordinary differential equations of fractional order. Exact solutions of the above mentioned time fractional equations are derived wherever possible. It is also explained how conservation laws can be derived to time fractional partial differential equations.

PurposeThe purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain.Design/methodology/approachThe proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations.FindingsThe fractional derivative of Lagrange polynomial is a big hurdle in classical DQM. To overcome this problem, the authors represent the Lagrange polynomial in terms of shifted Legendre polynomial. They construct a transformation matrix which transforms the Lagrange polynomial into shifted Legendre polynomial of arbitrary order. Then, they obtain the new weighting coefficients matrices for space fractional derivatives by shifted Legendre polynomials and use these in conversion of a non-linear fractional partial differential equation into a system of fractional ordinary differential equations. Convergence analysis for the proposed method is also discussed.Originality/valueMany engineers can use the presented method for solving their time and space fractional non-linear partial differential equation models. To the best of the authors’ knowledge, the differential quadrature method has never been extended or implemented for non-linear time and space fractional partial differential equations.

The main objective of this paper is to improve the optimal homotopy analysis method to find the approximate solutions for the linear and nonlinear partial fractional differential equations. The fractional derivatives are described in the Caputo sense. The optimal homotopy analysis method in applied mathematics can be used for obtaining the analytic approximate solutions for some nonlinear partial fractional differential equations such as the time and space fractional nonlinear Schrödinger partial differential equation and the time and space fractional telegraph partial differential equation. The optimal homotopy analysis method contains the h parameter which controls the convergence of the approximate solution series. Also, this method determines the optimal value of h as the best convergence of the series of solutions.

A new semi-analytical method for solving a class of time fractional partial differential equations with variable coefficients

ARA-residual power series method for solving partial fractional differential equations

A higher order numerical method for time fractional partial differential equations with nonsmooth data

Motivated by the widely used ansätz method and starting from the modified Riemann—Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.

Fundamental solutions and conservation laws for conformable time fractional partial differential equation

Error estimates of some high-order numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a high-order numerical method proposed recently by Li et al. [21] for solving time fractional partial differential equation. We prove that this method has the convergence order O(τ 3−α ) for all α ∈ (0, 1) when the first and second derivatives of the solution are vanish at t = 0, where τ is the time step size and α is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. [21]. We show that this new method also has the convergence order O(τ 3−α ) for all α ∈ (0, 1). The proofs of the error estimates are based on the energy method developed recently by Lv and Xu [26]. We also consider the space discretization by using the finite element method. Error estimates with convergence order O(τ 3−α + h 2) are proved in the fully discrete case, where h is the space step size. Numerical examples in both one- and two-dimensional cases are given to show that the numerical results are consistent with the theoretical results.

In this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich’s fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

Fractional order has the characteristics of memory and non-locality and it is different with integer order. Therefore, fractional differential equations can be used to describe some abnormal natural phenomena. At the same time, how to solve the fractional order partial differential equation and differential equations with fractional order has become a very important research field. Besides analytic solution, it is also important to investigate the numerical methods for fractional differential equations. In the paper, fundamental solution of the time fractional partial differential equation has been deduced, which is derived by Furrier transform and Laplace transform. According to the simulation, there is little difference between numerical solution and the exact solution when the solution is the time variable function. The results show the validity of the method.

In this work, we have presented analysis of time fractional order linear and non-linear partial differential equations with initial value and boundary conditions by applying Riemann - Leivoulli fractional integral. As the explained differential equations are related to natural phenomenon, it may be observed under various circumstances for which the possible outcome may vary. The properties and nature of physical states of these equations have been emphasised more precisely by taking fractional order. Fractional order homotopy perturbation method has tackled the approximate solutions in the series form of well known time fractional order linear and non linear differential equations. Numerical simulations are demonstrated prominently in graphical format by using Matlab.