Two numerical algorithms are derived to compute the fractional diffusionwave equation with a reaction term. Firstly, using the relations between Caputo and RiemannLiouville derivatives, we get two equivalent forms of the original equation, where we approximate RiemannLiouville derivative by a secondorder difference scheme. Secondly, for secondorder derivative in space dimension, we construct a fourthorder difference scheme in terms of the hyperbolictrigonometric spline function. The stability analysis of the derived numerical methods is given by means of the fractional Fourier method. Finally, an illustrative example is presented to show that the numerical results are in good agreement with the theoretical analysis.
The realm of fractional differential equations has drawn increasing attention and interest due to their important applications in biology, physics, chemistry, biochemistry, medicine, and engineering [
As far as we know, there are some numerical methods for the subdiffusion equations [
In this paper, we study the fractional diffusionwave equation with a reaction term in the following form:
Roughly speaking, the previous fractional differential equation is obtained from the classical diffusionwave equation by replacing the secondorder time derivative by a fractional derivative of order
The remainder of this paper is organized as follows. In Section
In the present section, we introduce some important definitions and lemmas which are used later on. The following definitions and lemmas can be found in [
The left RiemannLiouville integral operator
The left Caputo derivative operator
The left RiemannLiouville derivative operator
Suppose
The eigenvalues of the tridiagonal matrix
Let
We consider a uniform mesh
where
To obtain the expressions of the coefficients in (
Through the straightforward calculations, we obtain the following expressions:
in which
From (
In view of the continuity of the firstorder derivative at the common nodes
and
Obviously, from the expressions of
When
If we use
where
Using Taylor’s series for (
From the previous expression, we find that we can obtain different difference schemes by choosing different parameters
In the present section, we introduce two schemes for the fractional diffusionwave equation (
Based on Lemma
The RiemannLiouville derivative (with homogeneous initial value conditions) can be approximated by the following scheme:
There exist various choices of the generating functions which can lead to different approximation order
Let
If we take the generating function in the form
If we take the generating function as
Here we introduce two methods to determine the corresponding coefficients
In the following, we give another simple but interesting method to obtain the coefficients
By using
and (
Comparing the previous equation with (
which has the following recursive relation:
The coefficients
Finally, let
where
Next, we study the solvability and the local truncation error.
For convenience, denote
Then we can give the following matrix form of the difference scheme (
The difference equation (
From Lemma
The local truncation error of the difference scheme (
We now define the local truncation error as
This finishes the proof.
In the following, we discuss the stability of Numerical Algorithm
Let
So, we can easily obtain the following roundoff error equation:
We suppose that the solution of (
When
Note that
Differentiating (
Combing (
where
Next, for the secondorder time derivative
Let
where
Similar to Numerical Algorithm I, we discuss the solvability and local truncation error of Numerical Algorithm II.
Denote
The difference equation (
Obviously, the eigenvalues of the tridiagonal matrix
Hence,
Therefore, the solution of (
In order to get the local truncation error analysis, we need the following lemma.
Consider
Let
The local truncation error of difference scheme (
We now define
By using (
Noticing that
Next, we study the stability of Numerical Algorithm II (see (
As before, we can easily obtain the following roundoff error equation:
Considering the extreme value
For large enough
It follows that
Because the sine function is bounded by
Furthermore, stability condition (
where
In addition, the numerical check of the validity of the stability condition (
Moreover, from Theorems
In this section, we give a numerical example to demonstrate the efficiency of the derived numerical algorithms.
Consider the following equation:
The analytical solution of this equation is
On one hand, we use Numerical Algorithm I (see (
The maximum error and temporal and spatial convergence orders by Numerical Algorithm I (see (

The maximum error  Temporal convergence order  Spatial convergence order  

1.1 


—  — 


1.5366  3.0733  


1.9857  3.9714  


1.9976  3.9951  


1.9992  3.9984  
 
1.3 


—  — 


1.3937  2.7873  


1.9519  3.9038  


1.9813  3.9626  


1.9925  3.9851  
 
1.5 


—  — 


1.2263  2.4526  


1.8859  3.7719  


1.9490  3.8980  


1.9743  3.9487  
 
1.7 


—  — 


1.0657  2.1314  


1.7833  3.5665  


1.8956  3.7912  


1.9358  3.8715  
 
1.9 


—  — 


1.0310  2.0621  


1.5901  3.1802  


1.7954  3.5909  


1.8588  3.7176 
Dependence of the numerical solution for different values of
Dependence of the numerical solution for different values of
The error of the analytical solution and the numerical solution for
Next, we use Numerical Algorithm II (
Figures
The maximum error and temporal convergence order by Numerical Scheme II (see (

The maximum error  The temporal convergence order  

1.1 


— 


1.8470  


1.6287  


1.0846  
 
1.2 


— 


1.9210  


1.7879  


1.3919  
 
1.3 


— 


1.9454  


1.8686  


1.5943  
 
1.4 


— 


1.9572  


1.9111  


1.7193  
 
1.5 


— 


1.9678  


1.9526  


1.8581  
 
1.6 


— 


1.9740  


1.9800  


1.9612  
 
1.7 


— 


1.9741  


1.9845  


1.9809  
1.8 


— 


1.9724  


1.9824  


1.9754  
 
1.9 


— 


1.9699  


1.9777  


1.9595 
Dependence of the numerical solution for different values of
Dependence of the numerical solution for different values of
The error of the analytical solution and the numerical solution for
Finally, we checked the stability condition given in (
Comparison of the analytical solution and numerical solution for
Comparison of the analytical solution and numerical solution for
In this paper, we construct two difference schemes for solving the fractional diffusionwave equation with reaction term. It is proved that Numerical Algorithm I (
The work was partially supported by the Key Program of Shanghai Municipal Education Commission under Grant no. 12ZZ084 and the grant of “The FirstClass Discipline of Universities in Shanghai.”