There are some mistakes in Euler’s works. Some of them form the basis of most of the sciences, including differential equations and complex analysis. We discuss them here.
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.
We outline symmetry analysis on the Hamilton-Bellman-Jacobi equation presented by Heath in [1] as an equation for mean-variance hedging and analyzed by Leach in [2] using classical symmetry analysis. We further apply modified symmetry analysis on this equation and compare our analytic results with numerical solutions. We do this by introducing a new infinitesimal parameter into the group generators [3]. This helps us solve the unintegrable solutions usually appearing in invariant solutions.
We investigate a case of the generalized Korteweg – De Vries Burgers equation. Our aim is to demonstrate the need for the application of further methods in addition to using Lie Symmetries. The solution is found through differential topological manifolds. We apply Lie’s theory to take the PDE to an ODE. However, this ODE is of third order and not easily solvable. It is through differentiable topological manifolds that we are able to arrive at a solution
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