Reduction of Ordinary Differential Equations to the Birkhoff Canonical Form

Jordan Canonical Form: Application to Differential Equations

Jordan Canonical Form ( JCF) is one of the most important, and useful, concepts in linear algebra. In this book we develop JCF and show how to apply it to solving systems of differential equations. We first develop JCF, including the concepts involved in it–eigenvalues, eigenvectors, and chains of generalized eigenvectors.We begin with the diagonalizable case and then proceed to the general cas...

Fundamental solutions to Kolmogorov equations via reduction to canonical form

This paper finds fundamental solutions to the backward Kolmogorov equations, usually interpretable as transition density functions for variables x that follow certain stochastic processes of the form dx = A(x, t)dt + cxγdX and dx = A(x, t)dt +α1 +α2x+α3xdX . This is achieved by first reducing the relevant PDEs that the density functions satisfy to their canonical form. These stochastic processe...

APPLICATION NEURAL NETWORK TO SOLVE ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we introduce a hybrid approach based on neural network and optimization teqnique to solve ordinary differential equation. In proposed model we use heyperbolic secont transformation function in hiden layer of neural network part and bfgs teqnique in optimization part. In comparison with existing similar neural networks proposed model provides solutions with high accuracy. Numerica...

Introduction to Ordinary Differential Equations

Reduction of Order for Systems of Ordinary Differential Equations

The classical reduction of order for scalar ordinary differential equations (ODEs) fails for a system of ODEs. We prove a constructive result for the reduction of order for a system of ODEs that admits a solvable Lie algebra of point symmetries. Applications are given for the case of a system of two second-order ODEs which admits a solvable four-dimensional Lie algebra of point symmetries.