Solution of 1<sup>st</sup> Order Stiff Ordinary Differential Equations Using Feed Forward Neural Network and Bayesian Regularization Algorithm

Abstract

A stiff equation is a differential equation for which certain numerical methods are not stable, unless the step length is taken to be extraordinarily small. The stiff differential equation includes few terms that could result in speedy variation in the solution. When integrating a differential equation numerically, the requisite step length should be incredibly small. In the solution curve, much variation can be observed where the solution curve straightens out to approach a line with slope almost zero. The phenomenon of stiffness is observed when the step-size is unacceptably small in a region where the solution curve is very smooth. A lot of work on solving the stiff ordinary differential equations (ODEs) have been done by researchers with numbers of numerical methods that currently exist. Extensive research has been done to unveil the comparison between their rate of convergence, number of computations, accuracy, and capability to solve certain type of test problems. In the present work, an advanced Feed Forward Neural Network (FFNN) and Bayesian regularization algorithm-based method is implemented to solve first order stiff ordinary differential equations and system of ordinary differential equations. Using proposed method, the problems are solved for various time steps and comparisons are made with available analytical solutions and other existing methods. A problem is simulated using proposed FFNN model and accuracy has been acquired with less calculation efforts and time. The outcome of the work is showing good result to use artificial neural network methods to solve various types of stiff differential equations in near future.