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.

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

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) [

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. [

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

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

where

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

Also if

Followed from

if the matrix

satisfies

Taking into account of

is the analytical solution to

or

here

It is worth noting that if

which satisfies the corresponding equation of truncated series as follows:

where the unknown parameters can be obtained by enforcing the

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

In this section, the solution scheme for the

in which

in which

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

with the vector

The same algorithm can be applied for solving problems of

In this case, the modified RBF takes the form

where the coefficients

which leads to the four conditions

So, from 3)

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

which can transform problems of

where

The solution

and the updated boundary conditions, as follows:

where the operator in the last equation is described by the formulae of

1) We transform

where

As discussed in the (1+1)-dimensional problems, by considering the

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

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) [

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

Let us consider the multi-term problem discussed in [

with the first kind boundary condition and initial condition conform to

CO |
CPU, sec. | ||||
---|---|---|---|---|---|

4 | 8.77E-3 | 2.96E-2 | 1.96E-2 | 2.8 | 0.5 |

8 | 7.45E-5 | 1.94E-4 | 4.45E-4 | 5.9 | 0.6 |

12 | 3.05E-6 | 7.21E-6 | 2.76E-5 | 8.3 | 0.9 |

16 | 1.81E-7 | 3.88E-7 | 2.07E-6 | 10.4 | 1.3 |

20 | 1.24E-8 | 2.52E-8 | 1.71E-7 | 12.6 | 1.9 |

24 | 9.49E-10 | 1.87E-9 | 1.52E-8 | 14.5 | 2.7 |

26 | 5.80E-10 | 1.22E-9 | 4.72E-9 | 6.1 | 2.9 |

CO | CPU, sec. | ||||
---|---|---|---|---|---|

2 | 5.43E-3 | 2.18E-2 | 1.33E-2 | 2.4 | 0.5 |

4 | 1.16E-5 | 3.10E-5 | 5.34E-5 | 13.0 | 0.8 |

6 | 7.85E-9 | 2.17E-8 | 4.13E-8 | 17.1 | 1.2 |

8 | 1.25E-9 | 3.12E-9 | 5.41E-9 | 7.5 | 1.8 |

10 | 5.80E-10 | 1.22E-9 | 4.72E-9 | 0.3 | 2.3 |

Finally, we show the behaviour of the errors

5 | 10 | 15 | 20 | 25 | |
---|---|---|---|---|---|

1.79E-3 | 1.37E-5 | 1.41E-7 | 1.24E-8 | 3.55E-10 | |

1.79E-3 | 1.37E-5 | 1.41E-7 | 1.24E-8 | 3.91E-10 | |

1.79E-3 | 1.37E-5 | 1.41E-7 | 1.24E-8 | 4.20E-10 | |

1.79E-3 | 1.38E-5 | 1.40E-7 | 1.88E-8 | 6.97E-9 | |

1.79E-3 | 1.38E-5 | 1.83E-7 | 1.10E-7 | 5.42E-7 |

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

CO |
CPU, sec. | ||||
---|---|---|---|---|---|

4 | 1.48E-2 | 2.98E-3 | 3.38E-2 | 2.0 | 0.7 |

8 | 2.03E-4 | 2.91E-5 | 1.01E-3 | 5.1 | 1.3 |

12 | 8.30E-6 | 1.02E-6 | 6.35E-5 | 8.6 | 1.5 |

16 | 4.55E-7 | 5.49E-8 | 4.86E-6 | 10.6 | 1.9 |

20 | 2.96E-8 | 3.64E-9 | 4.11E-7 | 13.4 | 4.2 |

24 | 3.82E-9 | 5.15E-10 | 3.74E-8 | 13.9 | 5.4 |

26 | 5.16E-9 | 8.34E-10 | 1.57E-8 | – | 6.1 |

CO |
CPU, sec. | ||||
---|---|---|---|---|---|

2 | 1.92E-2 | 4.43E-3 | 4.62E-2 | 1.9 | 0.8 |

4 | 2.41E-3 | 5.46E-4 | 5.91E-3 | 3.8 | 1.5 |

6 | 4.69E-5 | 9.59E-6 | 1.18E-4 | 10.1 | 2.4 |

8 | 2.56E-6 | 4.38E-7 | 8.78E-6 | 10.5 | 3.4 |

10 | 4.36E-8 | 6.28E-9 | 2.30E-7 | 31.5 | 4.6 |

12 | 5.16E-9 | 8.34E-10 | 1.57E-8 | 11.7 | 6.6 |

In

5 | 10 | 15 | 20 | 25 | |
---|---|---|---|---|---|

8.86E-4 | 5.29E-6 | 1.39E-7 | 8.20E-9 | 6.33E-9 | |

8.86E-4 | 5.28E-6 | 1.48E-7 | 2.58E-9 | 2.36E-10 | |

8.86E-4 | 2.58E-6 | 1.48E-7 | 2.61E-9 | 2.44E-10 | |

8.86E-4 | 4.01E-6 | 3.24E-6 | 4.29E-6 | 3.67E-6 | |

6.11E-4 | 7.32E-4 | 7.40E-4 | 9.04E-4 | 1.10E-3 |

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

with

CO |
CPU, sec. | ||||
---|---|---|---|---|---|

4 | 1.48E-2 | 2.46E-2 | 3.40E-2 | 3.7 | 0.6 |

8 | 2.43E-4 | 2.98E-4 | 1.04E-3 | 5.8 | 1.6 |

12 | 9.09E-6 | 8.43E-6 | 5.83E-5 | 9.2 | 3.3 |

16 | 4.50E-7 | 3.99E-7 | 4.15E-6 | 11.1 | 5.6 |

20 | 2.80E-8 | 2.44E-8 | 3.22E-7 | 9.7 | 7.9 |

24 | 2.24E-9 | 1.79E-9 | 2.86E-8 | 0.03 | 9.7 |

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

with the same analytical solution. Here

Then, we discuss the solution accuracy on the

5 | 10 | 15 | 20 | 25 | |
---|---|---|---|---|---|

6.69E-3 | 4.72E-5 | 1.28E-6 | 2.74E-8 | 2.25E-9 | |

6.69E-3 | 4.72E-5 | 1.28E-6 | 2.74E-8 | 2.18E-9 | |

6.69E-3 | 4.72E-5 | 1.28E-6 | 2.73E-8 | 2.15E-9 | |

6.69E-3 | 4.71E-5 | 1.40E-6 | 2.43E-7 | 1.05E-6 | |

6.68E-3 | 3.93E-5 | 6.69E-6 | 5.02E-5 | 6.78E-5 |

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

CO |
CPU, sec. | CO |
CPU, sec. | |||||

10 | 8.19E-5 | 5.66E-5 | 14.1 | 1.7 | 7.36E-5 | 5.04E-5 | 14.5 | 6.4 |

12 | 1.09E-5 | 7.76E-6 | 11 | 2.3 | 2.68E-6 | 1.88E-6 | 18.2 | 9.3 |

14 | 8.38E-6 | 5.88E-6 | 1.70 | 3.5 | 6.20E-8 | 2.84E-8 | 24.2 | 13 |

16 | 8.31E-6 | 5.89E-6 | – | 4.2 | 2.90E-8 | 1.15E-8 | 5.7 | 17 |

18 | 8.32E-6 | 5.88E-6 | – | 5.1 | 6.08E-9 | 2.44E-9 | 13.89 | 22 |

20 | 8.32E-6 | 5.88E-6 | – | 6.2 | 1.77E-9 | 6.65E-10 | 11.7 | 27 |

22 | 8.29E-6 | 5.86E-6 | – | 7.6 | 5.08E-10 | 1.88E-10 | 13 | 33 |

24 | 8.26E-6 | 5.84E-6 | – | 9.1 | 1.73E-10 | 5.29E-11 | 12 | 39 |

26 | 8.10E-6 | 5.73E-6 | – | 10.1 | 2.23E-10 | 1.08E-10 | – | 46 |

In

5 | 10 | 15 | 20 | 25 | |
---|---|---|---|---|---|

1.14E-1 | 7.36E-5 | 4.11E-8 | 4.65E-9 | 3.41E-9 | |

1.14E-1 | 7.36E-5 | 3.82E-8 | 1.78E-9 | 6.02E-11 | |

1.14E-1 | 7.36E-5 | 3.80E-8 | 1.96E-9 | 2.23E-10 | |

1.14E-1 | 6.51E-5 | 8.74E-6 | 4.33E-6 | 2.04E-6 | |

1.15E-1 | 7.08E-5 | 4.81E-4 | 2.49E-5 | 2.34E-4 |

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

in the domain governed by

as shown in

Here

where the matrix of diffusivity is

The boundary condition of the Robin type:

and IC conform the exact solution

CPU, sec. | ||||
---|---|---|---|---|

10 | 6.99E-7 | 4.41e-8 | 9.32E-7 | 27 |

20 | 5.82E-7 | 4.86E-8 | 8.23E-7 | 66 |

40 | 1.41E-7 | 1.01E-8 | 1.94E-7 | 249 |

60 | 1.66E-8 | 9.61E-10 | 1.87E-8 | 360 |

80 | 7.44E-9 | 3.78E-10 | 1.45E-8 | 756 |

In the last example, we consider a nonlinear problem

in the star-like domain shown in

where the matrix of diffusivity is

The Dirichlet boundary conditions and IC can be obtained from

CPU, sec. | ||||
---|---|---|---|---|

10 | 5.88E-9 | 2.53E-9 | 6.59E-9 | 30 |

15 | 5.30E-9 | 2.34E-9 | 7.07E-9 | 58 |

20 | 3.17E-8 | 1.40E-8 | 2.82E-8 | 94 |

25 | 2.57E-9 | 1.03E-9 | 3.49E-9 | 138 |

30 | 3.97E-9 | 9.81E-10 | 2.92E-9 | 281 |

35 | 7.08E-10 | 3.86E-10 | 1.06E-9 | 334 |

40 | 4.83E-10 | 1.73E-10 | 5.42E-10 | 380 |

50 | 2.37E-10 | 8.95E-11 | 2.45E-10 | 505 |

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

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

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).

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.

None.

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