MATLAB Mathematics - SERC - Index of

5 Differential Equations
odephas3
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Introduction to Initial Value ODE Problems
What Is an Ordinary Differential Equation?
The ODE solvers are designed to handle ordinary differential equations. An
ordinary differential equation contains one or more derivatives of a dependent
variable y with respect to a single independent variable t, usually referred to
as time. The derivative of y with respect to t is denoted as y′ , the second
derivative as y′′ , and so on. Often yt () is a vector, having elements
y 1 , y 2 ,…,
y n .
Types of Problems Handled by the ODE Solvers
The ODE solvers handle the following types of first-order ODEs:
• Explicit ODEs of the form y′ = fty ( , )
• Linearly implicit ODEs of the form
matrix
• Fully implicit ODEs of the form
Mt ( , y) ⋅ y′
ftyy′ ( , , ) = 0
= fty ( , ), where M(t,y) is a
(ode15i only)
Using Initial Conditions to Specify the Solution of Interest
Generally there are many functions yt () that satisfy a given ODE, and
additional information is necessary to specify the solution of interest. In an
initial value problem, the solution of interest satisfies a specific initial
condition, that is, y is equal to y 0 at a given initial time t 0 . An initial value
problem for an ODE is then
y′ = fty ( , )
yt ( 0 ) = y 0
(5-1)
If the function ft ( , y)
is sufficiently smooth, this problem has one and only one
solution. Generally there is no analytic expression for the solution, so it is
necessary to approximate yt () by numerical means, such as using one of the
ODE solvers.
5-4