Journal of Prime Research in Mathematics

# A new parallel numerical algorithm for solving singular Perturbation problems in partial differential equations

Khalid Mindeel Mohammed Al-Abrahemee$$^{a,*}$$, Madeha Shaltagh Yousif$$^a$$
$$^a$$Department of Mathematics, College of Education, Department of Applied Sciences, University of AL, Qadisiyhah, Iraq University of Technology, Baghdad, Iraq.
Correspondence should be addressed to: Khalid Mindeel Mohammed Al-Abrahemee at khalid.mohammed@qu.edu.iq

### Abstract

In this study, a new method based on a neural network has been used as a solution for singular perturbation problems in partial differential equations (SPPDEs). Specifically, a modified neural network with a parallel numerical algorithm was used to train the Levenberg-Marquardt (L-M) algorithm with new data and hypotheses. This method is generally applicable to SPPDEs. We consider the convergence under $$\vartheta_{k}=\min (\|E_{k}\|,\|J_{k}^{T}E_{k}\|)$$ of the L-M algorithm, where $$\|E_{k}\|$$ provides a local error bound, and $$J(\varpi _{k})=E^{‘}(\varpi _{k})$$ is the Jacobian. The sequence generated by the L-M algorithm converges to the solution set quadratically. We use some examples to prove that the proposed algorithm, when implemented in MATHEMATICA 11.2, is more efficient and accurate than the standard algorithm.

#### Keywords:

Nonlinear equations, Local error bound, Levenberg-Marquardt algorithm.