On Solutions of the Nonlinear Heat Equation Using Modified Lie Symmetries and Differentiable Topological Manifolds

Abstract

In this paper, we present three simple analytical techniques for obtaining solutions of the nonlinear heat equations. The heat equations, both linear and nonlinear, are very important to the mathematical sciences. This is because they are reduced forms of many models, hard to solve directly. The techniques are based on Lie’ symmetry group theoretical methods. The first is the pure Lie approach, followed by our modified Lie approach. The third is our differentiable topological manifolds approach. As an application, we determine the separation distance, in the quantum superposition principle, relevant to nanoscience.

An analysis of the potential Korteweg-DeVries equation through regular symmetries and topological manifolds

“…This approach, requires setting numerical results, as can be seen by the plot. In this paper we have illustrated that the method of differentiable manifolds, as introduced by [14], yields solutions that are not restricted by any boundaries or assumptions. Figure 1 illustrates that our results compare very well with the numerical results generated by a computer.…”

“…The technique we use to determine exact solutions to the ODE is built on differentiable topological manifolds. We follow it as presented [14]. It is precursor is in [15].…”