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# A Novel Accurate Method for Multi-Term Time-Fractional Nonlinear Diffusion Equations in Arbitrary Domains

1
Ningbo Yongxin Engineering Consulting Co., Ltd., Ningbo, 315042, China

2
Taizhou Huanfeng Water Conservancy Engineering Co., Ltd., Taizhou, 318050, China

3
A. Pidhornyi Institute of Mechanical Engineering Problems, National Academy of Sciences of Ukraine, Kharkiv, 61046, Ukraine

4
Material and Structure Department, Nanjing Hydraulic Research Institute, Nanjing, 210029, China

5
College of Mechanics and Materials, Hohai University, Nanjing, 211100, China

* Corresponding Authors: Sergiy Reutskiy. Email: ; Ji Lin. Email:

(This article belongs to the Special Issue: Advances on Mesh and Dimension Reduction Methods)

*Computer Modeling in Engineering & Sciences* **2024**, *138*(2), 1521-1548. https://doi.org/10.32604/cmes.2023.030449

**Received** 06 April 2023; **Accepted** 20 June 2023; **Issue published** 17 November 2023

## Abstract

A novel accurate method is proposed to solve a broad variety of linear and nonlinear (1+1)-dimensional and (2+1)- dimensional multi-term time-fractional partial differential equations with spatial operators of anisotropic diffusivity. For (1+1)-dimensional problems, analytical solutions that satisfy the boundary requirements are derived. Such solutions are numerically calculated using the trigonometric basis approximation for (2+1)-dimensional problems. With the aid of these analytical or numerical approximations, the original problems can be converted into the fractional ordinary differential equations, and solutions to the fractional ordinary differential equations are approximated by modified radial basis functions with time-dependent coefficients. An efficient backward substitution strategy that was previously provided for a single fractional ordinary differential equation is then used to solve the corresponding systems. The straightforward quasilinearization technique is applied to handle nonlinear issues. Numerical experiments demonstrate the suggested algorithm’s superior accuracy and efficiency.## Keywords

In this study, we consider an efficient and accurate numerical technique for the problem of the following form:

where

is the operator of the anisotropic diffusivity.

which are time differential operators of integer or fractional orders. And

Eq. (1) can be used to describe a wide variety of mathematical models in engineering and science as particular cases. First of all, these are the models of the heat equations obeying the classical Fourier law. At the same time, there are many models of heat exchange based on the non-Fourier laws which also can be represented by this equation. Besides, it can also be used for simulating the unsteady flow of the non-Newtonian fractional Maxwell fluid [1] and the Oldroyd-B fluid [2]. The general equation given in Eq. (1) also includes the Benjamin-Bona-Mahony and the Benjamin-Bona-Mahony-Burgers equations [3,4], which have been used for modeling long waves of small amplitudes. Besides, the Sobolev equation also falls into this group [5,6].

So, the multi-term systems of equations are introduced for modeling many real-life applications which involved some complicated processes that cannot be accurately described by systems of the single-term. It is evident that only a small number of problems can be analyzed analytically under some idealized conditions which are useful for parametric research. However, the derivation of analytical solutions for the nonlinear fractional equations with general spatial differential operators is not a trivial task. Applying numerical methods to deal with these equations would be more desirable. The finite difference method (FDM) [7–9] and the finite element method (FEM) [10–12] have been proposed widely as available techniques. Jin et al. [13] investigated the multi-term time-fractional diffusion equations using the Galerkin FEM. Recently an alternating direction implicit (ADI) scheme has been presented by Huang et al. in [14]. The weighted meshless spectral method was proposed by Hussain and Haq for problems arising in mass and heat transfer [15]. A method based on the Haar wavelets and the finite difference scheme was proposed by Oruç, Esen, and Bulut for two-dimensional time fractional reaction-sub-diffusion equation in [16]. An accurate computational method based on the linear barycentric interpolation method has been proposed by Oruç [17] recently for solving fractional Rayleigh-Stokes problems. The numerical method which combines the Laplace transform with the local radial basis functions method was presented by Li et al. in [18] for fractional diffusion models. A Galerkin method based on the second kind Chebyshev wavelets was presented by Soltani Sarvestani et al. in [19] for solving the multi-term time fractional diffusion-wave equation. Very recently an alternating direction implicit Legendre spectral method has been developed by Liu et al. in [20]. A mixed meshless method was proposed by Nikan et al. in [21]. The weak Galerkin finite element method for the space discretization combined with the discretization of the Caputo time derivative by L1 method was proposed by Zhou et al. in [22] for time-fractional quasi-linear diffusion equation. A method that combines the L1 interpolation of the fractional derivative with a high-order compact finite volume scheme was applied by Su et al. in [23] for the 2D multi-term time fractional sub-diffusion equation. The technique which combines approximation of the time-fractional derivative via the L1 formula and approximation of the integer order space derivatives by truncated one and two-dimensional wavelet series was presented by Ghafoor et al. for one and two dimensional higher order multi-term time-fractional partial differential equations in [24].

Inspired by the advantages of meshless methods which do not need the requirement of domain mesh, a variety of meshless methods have been proposed in the literature for solving fractional equations in both theoretical and numerical aspects. There exist several types of meshless methods, from which the radial basis function (RBF)-based meshless methods are the most popular type. With the merits of Euclidian distances as variables, RBF-based methods are flexible and usual for high-dimension problems under irregular complicated domains. Piret et al. [25] made the first attempt to use the RBF discretization for the fractional diffusion. Hosseini et al. [26] solved the time-fractional telegraph equations using the RBF discretization with the finite difference method for temporal discretization. Liu et al. [27] provided the implicit RBF approach with error analysis. The radial basis function-finite difference method (RBF-FD) [28] was also used to solve the fractional equations. In this paper, an accurate and efficient method based on the key idea of the backward substitution method has been proposed [29–34]. In this method, the general approximation based on the pure radial basis functions and their correcting terms are formed. Applying the collocation method, we can reduce the considered problems into the system of fractional ordinary differential equations. Then the corresponding problems are solved efficiently with the help of the Müntz polynomial basis since the fractional derivative of a Müntz polynomial is again a Müntz polynomial.

The remainder is organized as follows. These preliminaries including the definition of fractional derivative, the backward substitution method for the linear system of fractional ordinary differential equations, and the quasilinearization techniques are described in Section 2. The main method of application to (1+1)-dimensional and (2+1)-dimensional problems are described in Section 3 where the description of the method in application to the problems with the spatial operator of the fourth order has also been discussed. The numerical examples which illustrate the method presented are placed with the comparisons by some well-known numerical methods in Section 4. Finally, in Section 5, a short conclusion and discussion is provided.

2.1 Solutions for Linear System of Fractional Ordinary Differential Equations

The method presented in this study is based on the effective algorithm for the solution of linear systems of the fractional ordinary differential equations (FODEs). This algorithm is described detailed in this subsection. Let us consider the linear system of fractional ordinary differential equation

with zero initial conditions

where

as the basis function where the right hand side of Eq. (5) can be approximated by the linear combination of these functions

where

Some properties of Müntz polynomials should be discussed and noted here. Let {

Also if

Followed from Eq. (10), we have the following equation:

if the matrix

satisfies

Taking into account of Eq. (6), the following series:

is the analytical solution to Eq. (11) and the unknown vectors

or

here

It is worth noting that if Eq. (16) is fulfilled for any

which satisfies the corresponding equation of truncated series as follows:

where the unknown parameters can be obtained by enforcing the Eq. (16) at several selected time steps

2.2 Quasilinearization Procedure for Nonlinear Problem

The nonlinear term

Let us denote

Assuming that

As a result the original equation is transformed to a sequence of linear equations with the initial approximations of the form

and the system of iteration will stop under the control of the error given by two successive evaluations

3 The Solution Scheme for the Considered Problem

In this section, the solution scheme for the Eq. (22) with the known boundary conditions (BC) and initial conditions (IC) will be discussed in details, as follows:

in which

in which

3.1 Algorithm for (1+1)-Dimensional Problem

In this case the governing equation takes the form

or in the explicit form

with the IC

and the BCs

at the endpoints of the interval

which satisfies the equation

the BCs

and zero IC

Here

Let us define the following two functions where the parameters are determined from the initial condition and boundary condition:

satisfying the conditions

Then, it is easy to prove that the following combination:

satisfies the boundary conditions

Suppose that the

where

where

and the following initial condition and boundary condition:

In order to approximate

where

where coefficients

As a result we get the linear system

for each pair of the coefficients

in which the unknown

Let

or simplified in the matrix form as shown below:

Here the derivatives can be obtained in the analytical form

and

then the system Eq. (57) coincides with the linear system of FODEs shown in Eq. (5) and can be solved using the algorithm described there. Using M Müntz polynomials in solving of the system of the FODEs shown in Eq. (5), we get the approximate solution

with the vector

The same algorithm can be applied for solving problems of Eq. (27) with the spatial operator of the fourth order

In this case, the modified RBF takes the form

where the coefficients

which leads to the four conditions

So, from 3)

3.2 Algorithm for (2+1)-Dimensional Problem

Following the decomposition for the (1+1)-dimensional problem, we have

which can transform problems of Eq. (22) into the form

where

The solution

and the updated boundary conditions, as follows:

where the operator in the last equation is described by the formulae of Eqs. (25) and (26). To transform the time-dependent fractional equations into a system of fractional ordinary equations and apply the algorithm described in the previous subsection for (1+1)-dimensional problems, we should perform the following steps for this goal:

1) We transform Eq. (71) to the homogeneous BCs by introducing the primary approximation

where

As discussed in the (1+1)-dimensional problems, by considering the Eq. (75) and the governing equation, we have the equation to determine the correcting solution

where the new source term is

which can now be solved using the method as described in the (1+1)-dimensional problems. The main difference for the solution process of the (1+1)-dimensional problems and the (2+1)-dimensional problems is that the initial approximation of the solution

3.3 The Procedure for Obtaining

Since the artificially designed

for the (1+1)-dimensional problems and the (2+1)-dimensional problems, respectively, where the parameter

where the weighted parameters are determined using the collocation approach

and the

It is important to note that we need approximation of the function

To make this possible, we use the point interpolation method (PIM) [38,39]. Let

and the coefficients

where

is the so-called square moment matrix. Assuming that

Using vectors

As a result, we get the approximate solution

where the approximate solution also depends on the parameters

In this section, the accuracy and efficiency of the proposed method is verified. The maximum absolute error (MAE,

where

Furthermore, for the 1D problems, we let

4.1 (1+1)-Dimensional Problems

Let us consider the multi-term problem discussed in [13]

with the first kind boundary condition and initial condition conform to

Fig. 1 and Table 1 show the behaviour of the errors with respect to the number of the modified RBFs

Fig. 2 and Table 2 show the computed

Finally, we show the behaviour of the errors

Let us consider the multi-term problem of the following form:

with the boundary condition of third kind (Robin BCs)

The BCs and IC conform the exact solution

Some results of the calculations are presented in Tables 4 and 5 and displayed in the Figs. 4 and 5. From Tables 4 and 5, we can see that for the problem of the Robin boundary conditions, the proposed method can yield accurate results with small number of basis functions and collocation nodes. And the overall convergence order of the propose scheme is large than two. The same conclusions can be made from Figs. 4 and 5, where the computed maximum absolute errors vs. the M and N are displayed. From these figures, we can see that with the increasing of N for the case

In Table 6, we show the effect of the

Consider the nonlinear time-fractional Huxley equation of the following form:

with

Table 7 and Fig. 6 show the trends of the computed errors with the growth of N with fixed M. The predicted solutions are obtained after 3 iteration of the quasilinearization procedure. The initial approximations of the solution and its derivatives are provided randomly. From the results, we can verify that for the randomly provided initial approximations, we can obtained the accurate solutions in three iterations. And with the increasing of the N, the proposed algorithm converges fast. The same problem was considered by Hadhoud et al. in [41] using a numerical technique based on the cubic B-spline collocation method and the mean value theorem for integrals. The maximal absolute errors obtained there are near

Now let us move onto the multi-term nonlinear time-fractional equation with the Huxley nonlinear term,

with the same analytical solution. Here

Fig. 7 demonstrates the behaviour of the errors with the grows of the total amount of basis function

Then, we discuss the solution accuracy on the

Let us consider the nonlinear problem with the spatial differential operator of the fourth order

where the spatial operator

and the nonlinear term is

The boundary condition and initial condition conform to the exact

In Table 10, we display the effect of the parameter

4.2 (2+1)-Dimensional Problems

Let us consider the linear fractional equation of the following form:

in the domain governed by

as shown in Fig. 9.

Here

where the matrix of diffusivity is

The boundary condition of the Robin type:

and IC conform the exact solution

Table 11 shows the errors as the functions of N where M is fixed at 25, and the maximum error is about

In the last example, we consider a nonlinear problem

in the star-like domain shown in Fig. 10. Here

where the matrix of diffusivity is

The Dirichlet boundary conditions and IC can be obtained from

In order to solve (1+1)-dimensional and (2+1)-dimensional time-fractional nonlinear diffusion equations of multi-term accurately, we propose a simple technique in this paper. First, we convert the original problem into a new one under homogeneous initial conditions by subtracting the initial conditions from the unknown desired solutions. Then the key issue is to solve the problem with zero initial conditions. After that, we try to seek the function

Acknowledgement: The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.

Funding Statement: This research was funded by the National Key Research and Development Program of China (No. 2021YFB2600704), the National Natural Science Foundation of China (No. 52171272), and the Significant Science and Technology Project of the Ministry of Water Resources of China (No. SKS-2022112).

Author Contributions: The authors confirm contribution to the paper as follows: study conception and design: T. Hu, C. Huang, and S. Reutskiy; data collection: J. Lu; analysis and interpretation of results: J. Lin and S. Reutskiy; draft manuscript preparation: J. Lu and J. Lin. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: None.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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*Computer Modeling in Engineering & Sciences*,

*138*

*(2)*, 1521-1548. https://doi.org/10.32604/cmes.2023.030449

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*Comput. Model. Eng. Sci.*, vol. 138, no. 2, pp. 1521-1548. 2024. https://doi.org/10.32604/cmes.2023.030449

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