Simulation of Singularly Perturbed Problems in the age AI

Around 1940's with the advent of computer simulations, numerical methods came into existence. Scientists came to conclusion that analytical methods work efficiently for well-defined problems but for differential equations defined on complicated domains, it is not always feasible to apply these techniques. It has been observed that transformation of continuum models into discretized systems may not effectively capture the whole information embodied in the models. As a result of which, at that time, one big question arose that how the approximation error can be effectively measured and minimized during computer simulations. Applaud to the efforts of researchers, scientists and mathematicians that today we have reached at a stage where most of classical mathematical models of applied science and engineering can be numerically solved within the desired accuracy.In the present work, analysis of various singularly perturbed differential equations has been presented using finite difference method and finite element method. Numerical validation has been carried to validate the theoretical findings. It has been shown that parameter uniform convergence can be achieved numerically.