Numerical Solutions of ODE and PDE
Lesson 1
Introduction
Ordinary Differential Equation
An ordinary differential equation (ODE) is an equation stating a relationship between a function of a single independent variable and the derivatives of this functions with respect to the independent variable. For example:
ψ(t, y, y 0 . . . , y m )
The order of an ODE is the order of the highest order derivative in the differential
equation.
If no product of the dependent variable y(t) with itself or any of its derivatives
occurs, then the equation is called linear, otherwise it is non-linear.
Examples are
y 00 + y = 0
linear
y0 + y2 = 0
nonlinear
y 0 + t2 y = 0
linear
y 00 + sin(y) = 0
nonlinear
The general first order ODE is of the form
dy
= f (t, y)
dt
A general solution of an ODE of order m contains m arbitrary constants that can
be determined by prescribing m conditions. There are two different classes of of
ODE, depending on the type of auxiliary conditions specified.
1-1
Introduction
Initial and Boundary Value Problem (IVP & BVP)
If all the auxiliary conditions are specified at the same value of the independent
variable and the solution is to be marched forward from that initial point, the differential equation together with the specified condition is called an IVP.
If the auxiliary conditions are specified at two different values of the independent
variable, the end point or at the boundaries of the domain of interest, the differential
equation is called boundary value problem. For example:
y 00 + P (t, y)y 0 + Q(t, y)y = F (t) y(t0 ) = c1 and y 0 (t0 ) = c2
(IV P )
y 00 + P (t, y)y 0 + Q(t, y)y = F (t) y(t1 ) = d1 and y(t2 ) = d1
(BV P )
Reduction of higher order equations to the system of first order
differential equations
Suppose that an n-th order equation can be solved for the n-th derivative, i.e., it can
be written in the form:
y (n) = f (t, y 0 , y 00 , . . . , y n−1 )
This equation can now be written in a system of first order differential equations by
a standard change of variables:
y1 = y
y2 = y 0
y3 = y 00
..
.
yn = y n−1 .
1-2
Introduction
Then, the resulting first-order system is the following:
y10 = y 0 = y2
y20 = y 00 = y3
y30 = y 000 = y4
..
.
yn0 = y n = f (t, y2 , y3 , . . . , yn ).
In vector form this can simply be written as
y0 = f (t, y)
where y = [y1 , y2 , . . . , yn ]T and f = [y2 , y3 , . . . yn , f ]T .
Let us assume that the initial values for the nth order problem are given as
y(t0 ) = y0 , y 0 (t0 ) = y1 , . . . , y n−1 (t0 ) = yn−1
Clearly, it follows
y(t0 ) = [y0 , y1 , . . . , yn−1 ]T .
Example 1.1 Convert the second order IVP into a system of first order IVP
2y 00 − 5y 0 + y = 0
y(0) = 6; y 0 (0) = −1;
Sol: Let
y2 = y 0 .
y1 = y
It follows then
y10 = y2
1
5
y20 = − y1 + y2
2
2
and
y1 (0) = 6;
y2 (0) = −1
1-3
Introduction
Remark 1.2 The methods of solution of first order initial value problem may be
used to solve the system of first order initial value problems and the nth order
initial value problem.
Suggested Readings
A. Quarteroni, R. Sacco, F. Saleri (2007). Numerical Mathematics. Second Edition.
Springer Berlin.
M.K. Jain, S.R.K. Iyengar, R.K. Jain (2009). Numerical Mathematics. Fifth Edition. New age international publishers, New Delhi.
1-4
Numerical Solutions of ODE and PDE
Lesson 2
Numerical Solutions of IVP
Let us consider the following IVP and discuss its existence and uniqueness of solution:
dy
= f (t, y), y(t0 ) = y0 t ∈ [t0 , b]
dt
(2.1)
Existence and Uniqueness of the Solution of IVP
The IVP (2.1) admits a unique solution y(t) if f (t, y) is uniformly Lipschitz continuous, that is,
|f (t, y1 ) − f (t, y2 )| ≤ L|y1 − y2 | for any t ∈ [t0 , b] and any y1 and y2 .
Here L is called Lipschitz constant.
Finding exact solution of a practical problem is hardly possible, therefore numerical
solutions are required.
Numerical Solutions of IVP
The numerical methods produce approximate values of y(t) at certain points along
the t coordinate called grids or mesh points. In case of equally spaced grid points
we have
ti+1 = ti + h, i = 0, 1, . . . , N − 1;
where h is called the step size.
2-1
tN = b
Numerical Solutions of IVP
Figure 2.1: Grid points
We shall use the following notation for the the approximation of solution of IVPs
un ≈ y(tn ) =: yn .
Single or Multi Step Methods: If the method advances the solution from
one grid point to the next grid point using only data at the single grid point, that is,
un+1 depends only on un , it is called one-step or single step method otherwise it is
called multistep method.
Explicit and Implicit Methods: A method is called explicit method if
un+1 can be computed directly in terms of the previous values uk , k ≤ n, implicit
if un+1 depends implicitly on itself through f .
Single Step Method
A single step method can be written as
un+1 = un + hφ(tn , un , fn , h)
2-2
Numerical Solutions of IVP
where φ is called an increment function.
Figure 2.2: Increment function
2-3
are