Contents
Which method is used to solve ordinary differential equations?
Approximation of initial value problems for ordinary differential equations: one-step methods including the explicit and implicit Euler methods, the trapezium rule method, and Runge–Kutta methods. Linear multi-step methods: consistency, zero- stability and convergence; absolute stability.
What does solved numerically mean?
Solving something numerically means you use approximations that make a problem easier to solve.
How are algorithms used to compute ordinary differential equations?
The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution. Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics.
How to approximate the solution of ordinary differential equations?
There are many ways to solve ordinary differential equations (ordinary differential equations are those with one independent variable; we will assume this variable is time, t ). The techniques discussed in these pages approximate the solution of first order ordinary differential equations (with initial conditions) of the form
How to find the next approximation of a differential equation?
In general, we find the next approximation (at t=t₀+ (n+1)h) from the previous approximation (at t=t₀+n·h) This process is repeated indefinitely to get our approximate solution. This method is called Euler’s method and is covered in detail (with examples) on the next page.
When to use an approximate method for solving a problem?
In other words, problems where the derivative of our solution at time t, y (t), is dependent on that solution and t (i.e., y’ (t)=f (y (t),t) ). The usefulness of approximate techniques are described below, but we will jump right into a discussion of how such a technique can work.