Journal of Function Spaces

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Fractional Delay Differential Equations and their Numerical Solutions

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Research Article | Open Access

Volume 2021 |Article ID 5581102 | https://doi.org/10.1155/2021/5581102

Jian Chen, Yong Huang, Taishan Zeng, "-Multiscale Galerkin’s Scheme with Multilevel Augmentation Algorithm for Solving Time Fractional Burgers’ Equation", Journal of Function Spaces, vol. 2021, Article ID 5581102, 11 pages, 2021. https://doi.org/10.1155/2021/5581102

-Multiscale Galerkin’s Scheme with Multilevel Augmentation Algorithm for Solving Time Fractional Burgers’ Equation

Academic Editor: Qifeng Zhang
Received23 Jan 2021
Accepted11 Feb 2021
Published02 Mar 2021

Abstract

In this paper, we consider the initial boundary value problem of the time fractional Burgers equation. A fully discrete scheme is proposed for the time fractional nonlinear Burgers equation with time discretized by -type formula and space discretized by the multiscale Galerkin method. The optimal convergence orders reach in the norm and in the norm, respectively, in which is the time step size, is the space step size, and is the order of piecewise polynomial space. Then, a fast multilevel augmentation method (MAM) is developed for solving the nonlinear algebraic equations resulting from the fully discrete scheme at each time step. We show that the MAM preserves the optimal convergence orders, and the computational cost is greatly reduced. Numerical experiments are presented to verify the theoretical analysis, and comparisons between MAM and Newton’s method show the efficiency of our algorithm.

1. Introduction

In this paper, we consider the following time fractional Burgers equation [17]: with the initial and boundary conditions, given by where and are given functions, and the notation denotes the Caputo fractional partial derivative of order defined by in which represents the Gamma fuction.

The time fractional Burgers equation is a kind of nonlinear subdiffusion convection equation occurring in several physical problems such as unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe, propagation of weak shock, compressible turbulence shallow-water waves, shock waves in a viscous medium, waves in bubbly liquids, and electromagnetic waves [1, 4, 8]. Till now, there have been several analytical techniques developed to solve the time fractional Burgers equation. These methods include the Cole-Hopf transformation, Laplace transform, variable separation method [8], Adomian decomposition method [4], homotopy analysis method [6], and so on.

However, even if some fractional differential equations can be solved, the expressions of their exact solutions are often expressed by special functions, which are difficult to apply in practice. Moreover, due to the nonlocality of the fractional operators, analytical methods do not always work well on most fractional differential equations in real applications. Hence, it is of great importance to develop reliable and efficient numerical methods for solving fractional differential equations. Nowadays, the numerical methods cover the quadratic B-spline Galerkin method [2], cubic B-spline finite element method [3], finite difference methods [1, 913], and Fourier pseudospectral schemes [14]. There are also some other numerical methods (see, for example, [8, 1517]).

In this paper, we first present a fully discrete scheme for solving the time fractional Burgers equation with the time approximated by the -type formula and the space discretization based on the multiscale Galerkin method. We give rigorous convergence analysis for the fully discrete scheme, which shows that the scheme enjoys the optimal convergence order in the norm and in the norm, respectively, where , and are the time step size, space step size, and the order of piecewise polynomial space, respectively. Since the time fractional Burgers equation is a nonlinear differential equation, the fully discrete scheme results in a system of nonlinear algebraic equation at each time step. Iteration methods such as the Newton iteration method and the quasi-Newton iteration method are often employed to solve these nonlinear equations. In this case, a large amount of computational effort is demanded to compute and update the Jacobian matrix in each iteration process. The higher accuracy of the approximate solution is required, the larger dimension of the subspace is needed, and the longer computational time is consumed. To overcome this problem, we develop the multilevel augmentation method for solving the fully discrete scheme. The MAM solves a nonlinear equation at a high level consisting of two parts: solving the nonlinear equation only in a fixed initial subspace with the dimension much lower than that of the whole approximate subspace; compensating the error by matrix-vector multiplications at the high level. The MAM reduces the computational costs significantly and leads to a fast solution for the fully discrete scheme. We prove that the MAM preserves the same optimal convergence order as the original fully discrete scheme. The idea of MAM was first introduced in [18] for solving the linear Fredholm integral equations of the second kind. The theoretical setting of MAM was established by Chen et al. in [19] for solving operator equations covering both first kind and second kind equations; they further develop MAM for solving the nonlinear Hammerstein integral equation in [20]. We modified the framework and extended the idea of MAM to solve general nonlinear operator equations of the second kind and applied it to the Sine-Gordon equation in [21]. Readers are referred to [2227] and the references therein for more applications of MAM.

This paper is organized in seven sections. In “Preliminaries,” some necessary notations, multiscale orthonormal bases in Sobolev space, and useful lemmas are introduced. In “ Scheme for Discretization of Caputo Derivative in Time,” we introduce the -formula for time discretization. In “Fully Discrete Scheme and Convergence,” a fully discrete scheme for time fractional Burgers equation is established, and the convergence analysis are given. The MAM and its convergence analysis are developed in “Multilevel Augmentation Method for Solving the Fully Discrete Scheme.” The numerical experiments are provided in “Numerical Experiments” to verify the theoretical estimates. Finally, a conclusion is included in “Conclusion.”

2. Preliminaries

Denote . Let stand for the inner product on the space with the norm . We denote by the Sobolev space of elements satisfying the homogeneous boundary conditions that . The inner product and norm of are defined by respectively. Let be a positive integer, we denote by the subspace of whose elements are the piecewise polynomials of order with knots where the notation Obviously, the sequence of is nested, that is which yields the following decomposition: where is the orthogonal complement of in

It is easily concluded from the definition of and that the dimensions of and are given by

and respectively.

Define two affine mappings on the interval by and which map the interval into and respectively. Associated with the two mappings, we introduce two linear operators as follows:

Lemma 1 (see [26]). Let be an orthonormal basis of Then the functions form an orthonormal basis for

Lemma 1 shows that the space can be recursively constructed by the linear operators and once has been given. Therefore, the basis of the space can be constructed by Lemma 1 step by step. For the details of the construction and more, the readers can refer to [26].

Let be an orthogonal projection operator from into with respect to the inner product that is, for all or

The following approximation results on the operator will be used later. Throughout this paper, unless stated otherwise, denotes a generic positive constant whose value may differ in different occurrences.

Lemma 2 (see [28]). If then where

3. Scheme for Discretization of Caputo Derivative in Time

For a positive integer let be the time step size and for Let be the solution of on

Define

For the approximation of fractional derivative we use the following scheme [29, 30]: where and

Lemma 3 (see [30]). If and then

Lemma 4 (see [30]). Suppose Let

Then

4. Fully Discrete Scheme and Convergence

In this section, we present a fully discrete scheme for the time fractional Burgers equation (1), and we derive the error estimates and convergence of the proposed fully discrete scheme. The Galerkin method associated with the multiscale basis introduced in “Preliminaries” is employed to discretize the spatial variable. The fully discrete scheme in weak formulation for (1) reads as follows: for each find such that where and denotes the interpolation operator.

We present an optimal error estimate of the fully discrete scheme (16) in the following theorem.

Theorem 5. Suppose that the problem (1)–(2) has a unique solution Then where

Proof. Denote We conclude from (1) and (16) that satisfies Taking in (18), we have We estimate the terms of the right-hand side of (19) one by one. For the first term in right-hand side of (19), using the Cauchy-Schwarz inequality and Lemma 2, we have To estimate the second term in the right-hand side of (19), we conclude from Lemma 4 that For the last term in the right-hand side of (19), using integration by parts and the Cauchy-Schwartz inequality, we have where and due to the smoothness of and the approximation
On the other hand Substituting (20)–(23) into (19) and noting that we deduce that Choose such that and denote ; then, we have By Gronwall’s inequality, we have which, together with Lemma 2 and the initial error estimate, yields that This completes the proof.

Remark 6. If we choose in (19) and make a similar analysis as the above Theorem 5, we can obtain the optimal convergence order in norm

5. Multilevel Augmentation Method for Solving the Fully Discrete Scheme

At each time step, the fully discrete scheme (16) leads to a nonlinear system, which makes the computational cost expensive. We present a fast multilevel augmentation method in this section to solve these nonlinear systems. To this end, we rewrite (16) into where and

Define a nonlinear operator as follows:

Similar to the proof of Lemma 3 in [23], we applied the Riesz representation theorem to the right-hand side of (29); there exists a element such that

Then, Equation (29) can be reformulated as or equivalently

Since Equation (16) has been reformulated as a nonlinear operator equation of the second kind (33), and has the properties (P1) and (P2) described in [23], then MAM developed in [21] is applicable.

We now briefly describe the MAM for solving (33). As we presented in “Preliminaries,” the approximation subspace sequence is nested, for a fixed positive integer is any nonnegative integer, and we have the following decomposition:

Now, we are in a position to solve (33) with and is fixed and smaller than Firstly, we solve (33) with exactly and obtain Next, we obtain an approximation of of (33) with To this end, we decompose

with

With the help of (34), Equation (33) with can be rewritten as an equivalent form as

Note that

Equation (35) becomes

The in the right-hand side can be approximated by the previous level solution We compute

Replace in (36) by , and solve from

Let which is an approximation to the solution

This procedure is repeated times to obtain an approximation of the solution of (33) with . The solution is called a multilevel augmentation solution. Since at any step we only need to invert the same nonlinear operator with a fixed small instead of the nonlinear operator This means the algorithm has a high computational efficiency. At every time step, the fully discrete scheme (33) is solved by the MAM, and the whole process can be summarized as the following algorithm:

Let be two fixed positive integers and .
Step 1: obtain the approximation of initial value function
   
where
Step 2: for (), do the following:
 (I): Solve from (33) with Set
 (II): Compute that is
   
where with
   
 (III): solve from the following equation
   
that is
   
 (IV): Let Set and go back to (III) until

Theorem 7. Let be the exact solution of (1) and be the approximation solution obtained by Algorithm 1. Suppose that the solution of Equation (33) belongs to for Then, there exist a positive integer such that for all and

Proof. As stated in [20, 21, 23], is the solution of the equation: which is equivalent to the following discrete form: where Rearranging the terms, we have Noting that the exact solution at satisfies Subtracting (45) from (46), we obtain that for all Denote and then Using these notations and noting that we derive the error equation as follows: Let and where is the positive constant appearing in (22); then, We take in (48) and estimate the terms in the right-hand side of (48).
For the first three terms in the right-hand side of (48), similar to the analysis of (20)–(22), we have By the Cauchy-Schwartz inequality, Young’s inequality, and noting that we have For the last term in the right-hand side of (48), it follows from integration by parts, the Cauchy-Schwartz inequality, and the Young inequality that On the other hand side, as presented in (23), we have Combining (49)–(54) and we have Choose such that and denote Then, we have When the exact solution of (33) belongs to there exists a positive integer for all and any (see [21, 23]): Combining (56) and (57), we conclude from Gronwall’s inequality that Noting that then