We proposed a method by utilizing method of steps and Hermite wavelet method, for solving the fractional delay differential equations. This technique first converts the fractional delay differential equation to a fractional nondelay differential equation and then applies the Hermite wavelet method on the obtained fractional nondelay differential equation to find the solution. Several numerical examples are solved to show the applicability of the proposed method.
The future state of a physical system depends not only on the present state but also on its past history. Functional differential equations provide a mathematical model for such physical systems in which the rate of change of the system may depend on the influence of its hereditary effects. Delay differential equations have numerous applications in mathematical modeling [
Delay differential equation is a generalization of the ordinary differential equation, which is suitable for physical system that also depends on the past data. During the last decade, several papers have been devoted to the study of the numerical solution of delay differential equations. Therefore different numerical methods [
Method of steps is easy to understand and implement. In the method of steps [
Hermite wavelet method [
In the present work, we established a technique by combining both the method of steps and the Hermite wavelets method for solving the fractional delay differential equation. We also implemented the Hermite wavelet method for solving fractional delay differential equation, as described in Example
The Hermite polynomials
We can expand any function
Let
By following the procedure in [
Since
The method of steps [
Proposed method consists of two methods, method of steps and Hermite wavelet method. We first implement the method of steps to the fractional delay differential equation (
In the fractional delay differential equation the solution
It is a fractional nondelay differential equation because
We solve the obtained fractional nondelay differential equation (
Approximate the unknown function
Substitute (
Set the residual (
Continue the procedure for the subsequent interval; delay differential equation on
In this section, we utilize the proposed scheme for finding the numerical solution of linear and nonlinear fractional delay differential equations. The notations
Consider the fractional delay differential equation
The results obtained by the proposed method, by taking
Comparison of the solution by proposed method at













5 

0.0062 

0.0010 

0.0074 
10 

0.0134 



0.0082 
50 

0.0690 



0.0052 
Solutions by the proposed method,
Consider the following fractional delay differential equation:
According to Table
Comparison of the solution by proposed method,






0.0  1.0000  1.0000 


0.2  0.8187  0.8187 


0.4  0.6703  0.6703 


0.6  0.5488  0.5488 


0.8  0.4493  0.4493 


1.0  0.3679  0.3679 


Solutions by the proposed method at different
Consider the fractional pantograph equation
By fixing
Comparison of proposed solution
Solution by proposed method for









0.2  1.22140  1.22140 

0.00 

0.4  1.49182  1.49182 



0.6  1.82212  1.82212 



0.8  2.22554  2.22554 



1.0  2.71828  2.71828 



Solutions by the proposed method at different
Consider the following fractional neutral functional differential equation with proportional delay:
We implement the proposed method by fixing
Absolute errors using proposed method at

SLC method [ 
RKHSM [ 
RKT method [ 



 
0.1 




0.2 




0.3 




0.4 




0.5 




0.6 




0.7 




0.8 




0.9 




1.0 




Solution by the proposed method at different
Consider the fractional nonlinear delay differential equation
Solution by proposed method at different values of
Absolute errors by using proposed method at different











0.1 




0.2 




0.3 




0.4 




0.5 




0.6 




0.7 




0.8 




0.9 




1.0 




Solution by the proposed method at different
Consider the fractional nonlinear neutral delay differential equation
We can approximate the solution of (
We fix
Comparison of proposed and Hermite wavelet methods









0.0  1.0000000000  1.0000000000  1.0000000000 


0.1  1.1051709181  1.1051709181  1.1051709181 


0.2  1.2214027582  1.2214027581  1.2214027582 


0.3  1.3498588076  1.3498588076  1.3498588076 


0.4  1.4918246977  1.4918246976  1.4918246976 


0.5  1.6487212707  1.6487212707  1.6487212707 


0.6  1.8221188004  1.8221188004  1.8221188004 


0.7  2.0137527076  2.0137527075  2.0137527075 


0.8  2.2255409285  2.2255409284  2.2255409285 


0.9  2.4596031112  2.4596031112  2.4596031112 


1.0  2.7182818276  2.7182818276  2.7182818285 


For the problem (
For this purpose, we use Maple 13 in system with Core Duo CPU 2.00 GHz and RAM 2.50 GB.
It is shown that proposed method gives excellent results when applied to different fractional linear and nonlinear delay differential equations. The results obtained from the proposed method are more accurate and better than the results obtained from other methods, as shown in Tables
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the anonymous reviewers for their valuable comments which led to the improvement of the paper.