Solving non-linear system by linearization

In summary: Otherwise you might need to use a different method like the Runge-Kutta method. However, it is not clear from your summary what the professor is getting at by saying "global solution".
  • #1
nikolafmf
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I have non-linear system of ordinary differential equations to solve by first linearising it. I know it can be linearised by expanding right side of equations by Taylor series and keeping only the linear terms. Then I can solve the linear system of differential equations with given initial conditions.

Here is the problem. The solution will only be acceptable locally, but not globally. But professor of mine told me to solve the system globally and numerically in this way by finding solution step by step and to compare it with Euler and Runge-Kutta methods for non-linear systems of differential equation.

The question is, how can such linearised system be solved numerically step by step, in a way that solution will be "good enough" globally for the given (non-linear) system?
 
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  • #2
Well, the first thing you would have to do is say what is meant by "good enough"!

What you could do is this: given a non-linear equation with a given initial condition, solve it with a numerical, perhaps Runge-Kutta, method. That solution will come with estimates of how large the independent variable can be from the starting value and still be within "good enough". Choose some x within that region and evaluate f at that x to give a new starting point to solve a new numerical problem.

Of course, it is always possible that the feasible range of x values will be smaller and smaller each to you repeat, possibly fast enough that you cannot extend the solution "globally". It is simply true that most (perhaps "almost all") non-linear differential equation problems do not have "global" solutions.
 
  • #3
This might be what your prof is getting at:

X'=F(X). In the Euler approximation you replace F(X) with the constant F(X_0) and follow that vector for small time t1, and then switch to F(X_t1) and repeat. Here you do the same thing except that at point X_0 you replace F(X) with its linearization around X_0, namely

F(X_0)+dF(X-X_0), where dF is calculated at X_0.

That is a linear ODE system, so you solve it up to time t1 until you reach point X(t1). Then you repeat the process at this new point. I don't know numerical stuff, but the benefit is probably that you can do larger time steps since the linear problem is close to the nonlinear problem for longer than is the case with the Euler steps. However, at each step you have to solve a system of ODEs, which is much more complicated than moving along a constant vector. So Euler's method is probably a much faster way to numerically calculate solutions.

Of course, one does have to pay attention to details like global existence, etc. If F(X) is compactly supported or if it doesn't grow too fast, global existence holds.
 

1. What is the purpose of linearization in solving non-linear systems?

Linearization is a mathematical technique used to approximate the behavior of a non-linear system by breaking it down into smaller, simpler linear systems. This allows for easier analysis and solution of the non-linear system.

2. How is linearization different from linear systems?

Linear systems have the property of superposition, meaning that the output is directly proportional to the input. Non-linear systems do not have this property, so linearization is necessary to approximate their behavior.

3. What are the steps involved in solving a non-linear system by linearization?

The first step is to identify the non-linear system and determine the variables and parameters involved. Then, the non-linear equations are linearized by using Taylor series expansion. The linearized equations are then solved using standard linear algebra techniques. Finally, the solution is checked for accuracy and the linearized results are converted back to the original non-linear system.

4. Are there any limitations to using linearization for solving non-linear systems?

Yes, linearization is only effective for approximating the behavior of non-linear systems in a small range around a given point. If the non-linear system has significant changes in behavior at different points, linearization may not accurately represent the system's behavior.

5. Can linearization be used for all types of non-linear systems?

No, linearization is most effective for non-linear systems that are close to being linear. Highly non-linear systems with complex behavior may not be accurately approximated using linearization techniques.

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