Chebyshev polynomial approximation to approximate partial differential equations

and has been shown to perform well empirically (Rivlin, 1990). Chebyshev nodes are
also known to possess a further convenient property, i.e. equi-oscillation ³(Judd, 1998).

As important as the choice of the nodes interpolants is that of a family of functions from
which the approximant P will be drawn. We suggest using a Chebyshev polynomial. This
is defined as⁴:

Γ_i(s) = cos(ia cos(s)) i = 0,1,..., n

with Γ₀(s)=1,Γ₁(s)=s,and Γ_n+1(s) =2sΓ_n - Γ_n-1 (s)

Therefore:

n
V(s)=∑ciΓi(s)
i=0

A Chebyshev basis polynomial, in conjunction with Chebyshev interpolation nodes,
produces an efficient interpolation equation which is very accurate and stable over n .
However, in our case, to solve the problem in (3), the polynomial we choose should be
able to replicate, not just the function V at s₁,s₂,...,s_n , but also its derivatives
s₁^',s₂^' ,...,s_n^' . Therefore the approximant that solves our problem can be defined as
follows⁵:

n

∑c_iΓ_i(s) =V  (s_i),∀_i = 1,..., n₁

i=1
n

∑ciΓ'i(s')=V  '(si),∀i=1,...,n2

i =1

³ This property states that the maximum error of a cubic function, for example, shall be reached at least five
times, and the sign of this error should alternate between the interpolation points.

⁴ Note that in this application we use the general formula for the Chebyshev basis, however there exists also
a recursive formula.

⁵ Note that, although one can also use Hermite polynomials to approximate the functional and the slopes,
the latter are inefficient (Judd, 1998).

<<   4   5   6   7   8   9   10   >>

More intriguing information