This paper focuses on a hybrid multistep and its twin oneleg methods and implementing them on implicit mixed differential algebraic equations. The orders of convergence for the above methods are discussed and numerical tests are solved.
Consider the ordinary differential system:
The author presents hybrid multistep methods that take the form
Applying Newton’s interpolation formula for this data gives the following scheme:
The hybrid multistep method and its twin oneleg depend on two parameters,
Differentialalgebraic equations (DAEs) often take place in highly scientific technology domains, such as automatic control engineering, simulation of electrical networks, and chemical reaction kinetics [
Here LM (
As a modification technique that applies the same arguments of
In the following section, the hybrid multistep (HMS) method in (
In the case of
Method (
Let
The order of convergence of the secondorder hybrid method in (
Let the local truncation errors be defined by
Therefore,
In this case,
Since
In the case of
Method (
The order of convergence of the secondorder oneleg twin in (
Applying the method in (
The residues of (
Expanding (
Substitute
The solution
Here, some numerical results are presented to evaluate the performance of the proposed technique [
The exact solutions are
The exact solution is
Rectifier circuit.
In Figure
The exact and approximate solutions for the current of the diode circuit.
The exact and numerical solutions for the voltage
The exact and approximate solutions of the voltage
The above tests are solved by the hybrid multistep method in (
The errors of solutions for the first test.


Er( 
Er( 


HMS  2  0.01 


4  0.01 



2  0.001 



4  0.001 



2  0.000l 



4  0.0001 





HOL  2  0.01 


4  0.01 



2  0.001 



4  0.001 



2  0.0001 



4  0.0001 





HBDF  2  0.01 


4  0.01 



2  0.001 



4  0.001 



2  0.0001 



4  0.0001 


The errors of solutions for the second test.


Er( 
Er( 
Er( 


HMS  0.5  0.01 



1  0.01 




0.5  0.001 




1  0.001 




0.5  0.0001 




1  0.0001 






HOL  0.5  0.01 



1  0.01 




0.5  0.001 




1  0.001 




0.5  0.0001 




1  0.0001 






HBDF  0.5  0.01 



1  0.01 




0.5  0.001 




1  0.001 




0.5  0.0001 




1  0.0001 



The errors of solutions for the third test.


Er( 
Er( 
Er( 


HMS  1  0.01 



3  0.01 




1  0.001 




3  0.001 




1  0.0001 




3  0.0001 






HOL  1  0.01 



3  0.01 




1  0.001 




3  0.001 




1  0.0001 




3  0.0001 






HBDF  1  0.01 



3  0.01 




1  0.001 




3  0.001 




1  0.0001 




3  0.0001 



This paper focuses on the implementation of hybrid multistep classes and their twin oneleg classes on implicit mixed differential algebraic equations. The orders of convergence for these classes are discussed. Numerical tests are introduced, which show that the introduced methods give better results than HBDF.
The authors declare that there is no conflict of interests regarding the publication of this paper.