Can I solve the discreate ODE by considering the continuous case?

In summary, many of the methods used to solve continuous ODEs can also be applied to discrete ODEs, but there are important differences and considerations to keep in mind. A discrete ODE can be represented using a difference equation, and the main differences between solving a discrete ODE and a continuous ODE lie in the methods and techniques used. Discrete ODEs can be approximated from continuous ODEs with a small enough step size, and they have practical applications in fields such as engineering, physics, and biology.
  • #1
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There is a linear version of so-called lattice Schrodinger equation (LSE), it is just a variation form of nonlinear Schrodinger equation. But the LSE is the discrete case on N lattices. I wonder if I can solve the continuous case and then take the solution at specific lattice for the discrete case?
 
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  • #2
Well first of all I'd like to see the equation.
What is discrete in the equation, space? time? both?
Does it involve an approximation of the derivative or is it a "standart" difference equation?
 

1. Can I use the same methods to solve a discrete ODE as I would for a continuous ODE?

Yes, many of the methods used to solve continuous ODEs can also be applied to discrete ODEs. However, there are some important differences and considerations to keep in mind.

2. How do I represent a discrete ODE mathematically?

A discrete ODE can be represented using a difference equation, which relates the values of a function at discrete time steps. For example, the forward difference equation for a first-order ODE is given by yn+1 = yn + h*f(yn, tn), where h is the step size and f(yn, tn) is the derivative of the function y at time tn.

3. What are the main differences between solving a discrete ODE and a continuous ODE?

The main differences lie in the methods and techniques used for solving each type of ODE. For discrete ODEs, numerical methods such as Euler's method and Runge-Kutta methods are commonly used, while for continuous ODEs, analytical and semi-analytical methods such as separation of variables, Laplace transforms, and power series solutions are often employed.

4. Can I approximate a continuous ODE with a discrete ODE?

Yes, in many cases, a continuous ODE can be approximated by a discrete ODE with a small enough step size. This is useful for numerical computations and simulations, as solving a discrete ODE is often computationally faster and more efficient than solving a continuous ODE.

5. What are some practical applications of solving discrete ODEs?

Discrete ODEs are commonly used in various fields such as engineering, physics, and biology to model and analyze systems that change over time. Some examples include population growth models, chemical reaction kinetics, and electrical circuit analysis.

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