An Analytic Solution for Riccati Matrix Delay Differential Equation using Coupled Homotopy-Adomian Approach

: An efficient modification and a novel technique combining the homotopy concept with Adomian decomposition method (ADM) to obtain an accurate analytical solution for Riccati matrix delay differential equation (RMDDE) is introduced in this paper . Both methods are very efficient and effective. The whole integral part of ADM is used instead of the integral part of homotopy technique. The major feature in current technique gives us a large convergence region of iterative approximate solutions .The results acquired by this technique give better approximations for a larger region as well as previously. Finally, the results conducted via suggesting an efficient and easy technique, and may be addressed to other non-linear problems.


Introduction:
In this study, a homotopy -Adomian approach is presented for addressing the following RMDDE: [ , ], t c T  1 by history condition: 0 ( ) ( ), G t Z t  c t c     , c is non negative integer number.
2 For many years, Liao implemented the initial thoughts of homotopy in topology to suggest a semi analytic approach for non-linear equations, called Homotopy Analysis Method (HAM); 1 , 2 . According to homotopy of topology, the evidence of this approach exists with or without small physical parameters in the considered problem. It is an analytic approach for finding series solution of various kinds of non-linear problems 3 . This technique is a strong technique used to find the solution of non-linear phenomena 3,4 .
ADM since 1980s, is an example of a semi analytic approach which can be implemented to many kinds of equations including non-linear PDEs 5,6,7 . It gives an effective computational manner for equations and can get results as an infinite series converging to exact solution 6,8,9 . Authors in 8 , implemented a new Adomian's technique for finding type of Riccati and compared with original ADM.
Also, Abazari 10 , found the approximate analytical approach for RMDE using DTM. Khalid et al. in 11 , implemented VIM to find the analytic solution to RMDE. A modification of DTM has been made by Shaher et al. 12 , to find the approximate results to FHN neurons mode. Mohammed et al. 13 found that the solution methodology lies in generating an infinite conformable chain with reliable wave pattern by minimizing the residual error functions. Mohammed et al. 14 , applied the DTM to obtain results of the fractional order Lu types. Approximate results for a SEIR epidemic type by homotopy method have been conducted 15 . In this study, a new implementation of homotopy-Adomian approach is implemented to obtain solve RMDDE.
A motivation in these methods provides the solution of DEs in infinite series. The most important advantages of these methods is its ability in providing us a continuous representation of the approximate solution, which allows better information of the results with whole interval. In addition it gives a highly accurate solution.
A feature for this technique as compared with the other methods for obtaining the solution, is that it is efficient not only for a small value of x but also for a larger value. Absolute residual error for each component is conducted. A suggested approach showed that it is so efficient, accurately and gives more extend of the concourse region.

Homotopy-Adomian Approach for RMDDE
Firstly, the standard HAM for RMDDE is reviewed. For this reason, consider the nonlinear mm  of equation 1: where , G , P S and D are matrices mm  ; , Now, by the method of successive for delay,  is a matrix function of homotopy, so: and mth-order deformation matrix equation related to (3) can see: Now, the mth-order formulation matrix equation 11 after addressing the inverse operator with given history condition, for 1 m  will transform to: Q is a matrix mm  and () Gt is a matrix mm  assumed to be bounded, [ , ], t c T  Ft is polynomial Adomian matrices. Where: The implementation of ADM to the matrix equation 13 is: Currently, the whole integral for HAM in equation 12 is replaced by integral of ADM in equation 18. In this case, more accurate solution, efficient, and more extend of the concourse area is obtained compare with the obtained by equation 12, means that:

Numerical Example
Now, the homotopy-Adomian approach presented above to RMDDE is applied. In the following example, the term of delay in the other part. It can be seen that this technique is very strong, effect and reliable. Hence the non-linear 22  RMDDE:

t y t z t x t t w t x t y t y t w t x t z t z t z t t w t w t y t z t
In this example, it can be noted that the delay is in the noise part [equation 21], it can also be observed that the delay part disappears after addressing the method of successive [

Conclusion:
A homoptopy-Adomian approach is discussed to obtain RMDDE. All results from the table above indicate that the suggested approach award highly precise results in extended area. Results conducted by this approach award better approximations for whole interval time as well as the neighborhood of the initial condition.