Nonlinear Ordinary Differential Equations

This is the undergraduate project work that I performed as an intern at the SINP Labs under the guidance of Dr. A.N.Sekar Iyengar.  The work includes MATLAB codes and Simulink models of ODE's like Jerk and Duffing equation showing chaos. Simscape model is also included that can hint towards possible circuit design implenetation of the chaotic circuits.

Autonomous nonlinear differential equations constituted a system of ordinary differential equations, which often applied in different areas of mechanics, quantum physics, chemical engineering science, physical science, and applied mathematics. It is assumed that the second-order autonomous nonlinear differential equations have the types u ′′ (x) − u ′ (x) = f [u(x)] and u ′′ (x) + f [u(x)]u ′ (x) + u(x) = 0 on the range [−1, 1] with the boundary values u[−1] and u[1] provided. We use the pseudospectral method based on the Chebyshev differentiation matrix with Chebyshev-Gauss-Lobatto points to solve these problems. Moreover, we build two new iterative procedures to find the approximate solutions. In this paper, we use the programming language Mathematica version 10.4 to represent the algorithms, numerical results and figures. In the numerical results, we apply the well-known Van der Pol oscillator equation and gave good results. Therefore, they will be able to be applied to other nonlinear systems such as the Rayleigh equations, the Lienard equations, and the Emden-Fowler equations.

Another version of the classic Sumudu Transform called the Elzaki Transform, was put forward as closely
related to the Laplace Transform. In the following paper, the Elzaki Transform Algorithm, which has been built
on the Decomposition Method, is presented to be applied to find approximate solution of a class of non-linear, initial value problems. This method gives an approximate solution in a Convergent-Series form with easily computable components necessitating no linearization or a low perturbation criterion. The most important part of this paper is the error analysis conducted between exact solutions and pade approximate solutions; it proves that our approximate solutions narrow in rapidly to the exact solutions. Moreover, as we will discuss after the results
are resented, this algorithm can also be applicable to more general classes of linear and nonlinear differential
equations.

In the paper we solved a nonlinear 4  equation numerically by Galerkin method. And we compare this results with the results finite difference method as Laya [5], we found that Galerkin finite elements method faster than finite difference method to reach equilibrium state where the density for the 4  equation, also we found that Galerkin method converges towards the steady state solutions faster than finite difference method with less steps in time.

We construct quantum algorithms to compute physical observables of nonlinear PDEs with M initial data. Based on an exact mapping between nonlinear and linear PDEs using the level set method, these new quantum algorithms for nonlinear Hamilton-Jacobi and scalar hyperbolic PDEs can be performed with a computational cost that is independent of M, for arbitrary nonlinearity. Depending on the details of the initial data, it can also display up to exponential advantage in both the dimension of the PDE and the error in computing its observables. For general nonlinear PDEs, quantum advantage with respect to M is possible in the large M limit.

A new algorithm used the Chebyshev pseudospectral method to solve the nonlinear second-order Lienard differential equations Abstract. This article presents a numerical method to determine the approximate solutions of the Lienard equations. It is assumed that the second-order nonlinear Linard differential equations on the range [-1, 1] with the given boundary values. We have to build a new algorithm to find approximate solutions to this problem. This algorithm based on the pseudospectral method using the Chebyshev differentiation matrix (CPM). In this paper, we used the Mathematica version 10.4 to represent the algorithm, numerical results and graphics. In the numerical results, we made a comparison between the CPMs numerical results and the Mathematica's numerical results. The biggest odds were very small. Therefore, they will be able to be applied to other nonlinear systems such as the Rayleigh equations and Emden-fowler equations.

Particle-field interactions are at the heart of modern astrophysics, and computational models are the optimal way to analyze the astronomical phenomena driven by these interactions. In this project, I develop a tool to investigate the behavior of charged particles in electromagnetic fields. I adopt a numerical approach to solve the force equations, in an effort to ensure generalizability in complex time-varying and space-varying fields. Various numerical schemes viz. Euler's method, Runge-Kutta, Leapfrog, Boris pusher are tested against analytical solutions to judge their accuracy and energy-conserving characteristics. On evaluation of these metrics, the symplectic Boris pusher is incorporated into the code as the primary algorithm for computation of results. The applications of the PATIENCE code by students or researchers include the ability to model a variety of physical phenomena such as electromagnetism, planetary orbits, heat diffusion, harmonic oscillators et cetera; all of these situations require the solution of second-order Ordinary Differential Equations using methods which are accurate, computationally efficient, and energy-conserving.