Numerical Solutions of ODE
and PDE
Introduction
• For simple differential equations, it is possible to find closed form
solutions
• Example:
• The more general equation is approached in a similar spirit, in the
sense that usually there is a general solution dependent on a
constant.
• For a first-order linear equation Method of Integrating factors can
be used.
• If a(t)=λ
• Multiplying by 𝑒 −𝜆𝑡
• The general solution of the first-order equation normally depends
on an arbitrary integration constant.
• To single out a particular solution, we need to specify an additional
condition.
• Usually such a condition is taken to be of the form
• For a general right-side function it is usually not possible to solve
the initial value problem analytically.
• One such example is for the equation
• In such a case, numerical methods are the only plausible way to
compute solutions.
• Moreover, even when a differential equation can be solved
analytically, the solution formula, usually involves integrations of
general functions.
• The integrals mostly have to be evaluated numerically.
Multistep Methods
• Reformulate the differential equation by integrating over [tn, tn+1]
• To evaluate the integral we approximate g(t) by using polynomial
interpolation and then integrate the interpolating polynomial.
• By using a polynomial of degree q at points (tn+1,tn,tn-1, …,tn-q+1)
• From the theory of polynomial interpolation and using first order
interpolator
• Incorporating this into the ODE problem equation
• Dropping the final term
• Example: Solve the following ODE
Euler’s Method
• As before, Y (t) denotes the true solution of the initial value
problem with the initial value Y0:
• Numerical methods for solving this will find an approximate
solution y(t) at a discrete set of nodes,
• For simplicity the points are equally spaced
• To derive Euler’s method, consider the standard derivative
approximation from beginning calculus,
• Applying this to the problem at tn
• Euler’s method is defined by taking this to be exact:
• Example: The true solution of the problem
• is
• Euler’s method is given by
• Example: solve
• whose true solution is
• Euler’s method for this differential equation is
• Error analysis of Euler’s method
• We begin by applying Taylor’s theorem to approximating Y(tn+1),
• Since
• The final term is called the truncation error for Euler’s method
• By subtracting the Euler’s approximation formulation
• The error has two parts
• The truncation error that is introduced at each tn+1
• The propagated error
• Reading assignment
• Stability
• Round off error accumulation
SYSTEMS OF DIFFERENTIAL EQUATIONS
• A general form of m first-order differential equation is
• We seek the functions Y1(t), . . . , Ym(t) on some interval t0 ≤ t ≤ b.
• An efficient representation is
• Example: the following problem can be written as
Higher order Differential Equations
• In practice most problems involve higher order equations.
• An example is solving the second-order equation that results from
Newton’s second law of motion
• This can be reformulated as a system of first-order equations, by
setting
• Then
• A general differential equation of order m can be written as
• It is reformulated as a system of m first-order equations by
introducing
• Then the equivalent initial value problem for a system of first-order
equations is
• Example: The initial value problem
• Is reformulated as
NUMERICAL METHODS FOR SYSTEMS
• The derivation of the Euler’s method for a system of equations is
done similarly as the case of a first order differential equation by
using the Taylor’s expansion
• In this formula, the matrix-vector notation is used that was introduced for
the case of system of differential equations.
• Higher order differential equations are also solved in a similar
manner by converting them into a system of first order differential
equations.