A special introduction about myself is given in Lesson 01.
L22 - ODEs, IVPs, Multi-Step Methods, Adams-Bashforth, Adams-Moulton, Explicit, Implicit
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[ Ссылка ]+(Dr.+Ab+Rahman+Ahmad).docx/file
This is Video No 22 in the series. Lesson 22 - Ordinary Differential Equations, IVP, Multi-Step Methods, Adams-Bashforth, Adams-Moulton, Explicit, Implicit
An Ordinary Differential Equation (ODE) is an equation that consists of a function of its derivative and one variable. The term ordinary is used in contrast with the term partial with respect to more than one variable. Some examples of ODEs are given in the video.
The examples shown in the video are first order ODEs (on the left) where the highest derivative in the equations is of the first order, whilst the equations on the right are the examples of second order ODEs. The task is to solve an ODE which involves integration at a point and yields a function . However, for some ODEs it is not easy to perform integration hence some initial values will be introduced.
INITIAL VALUE PROBLEMS (IVPs)
An ODE together with a specified value where this value is referred as an initial condition, is called an Initial Value Problem (IVP). See the examples of IVPs in the video.
It is necessary to determine whether the unique solution exists or otherwise, prior to solving an IVP. Furthermore, for small changes made in a problem, will this too yield small changes in a solution? This is important since rounding errors might affect the solution when it is solved numerically. Some definitions and theorems for the ODEs are required for discussion but since these are quite complicated, these will not be discussed. The selected problems are assumed well-posed based on the definition stated in the video.
An approximation for y is generated at various points in an interval [a,b] where they are called as grid points. When an approximate solution at this point is obtained, the solution at other points in the interval can also be found using interpolation methods discussed previously. In this lesson, we consider the grid points in the interval [a.b] are equidistant. This can be done by selecting a positive integer N, and the grid points {x0, x1, x2, ..., xN} where xi=a+ih for i = 0, 1, 2,... N. The distance between one point to another, h =(b-a)/N is called step size.
MULTI-STEP METHODS
More details will be furnished
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