A MATHEMATICAL APPROACH FOR THE SOLUTION OF SINGULARLY PERTURBED DIFFERENTIAL EQUATIONS

Abstract

In the natural sciences and engineering, a range of phenomena are modelled using mathematical equations. These equations in mathematics have different parameters. Minor adjustments to these parameters have an impact on the answers of these equa tions. The perturbation parameter corresponds to this slight modification, which is referred to as a perturbation. Finding these mathematical equations’ exact solutions is challenging. Finding their approximations is therefore the alternate method. The approximation techniques are used to arrive at these solutions. These perturbation methods pave the door for per turbation theory even more. This paper mainly focuses on the derivation of analytic solutions that accurately capture the physical relevance of the nonlinear phenomena involved, which can be difficult to solve explicitly using numerical schemes, especially when the equations are stiff. To solve this problem, we provide an iterative analytical strategy based on the La grange multiplier method. The Lagrange multiplier can be obtained more accurately and efficiently using variational theory and Liouville-Green transforms in a general setting. This method has been demonstrated to be highly accurate and efficient through illustrative examples. The suggested method provides a clear and succinct answer to the problems with numerical methods and is applicable to various nonlinear evolution equations in mathematical physics.