Spline-Fourier Approximations of Discontinuous Waves

( ) l
1 X
?even
q l (ik )l (qt+ l +2)! eik x
?even
l=0
The integration over G(x; t) produces splines with larger indexes. The splines sj that are su ciently smooth (e.g. j > p) are rounded as follows
1 j =1 1
k ?1
=
?1
(1)
Let fsj : j = 1; 2; : : :g be a set of periodic splines de ned as polynomials on (?1; 1) such that s (x) = x; x 2 (?1; 1), s0j = sj? , sj 2 Cj? (?1; 1), j = 2; 3; : : :. These splines are closely related to the monosplines, which are wellknown in the spline theory, as well as the Bernoulli polynomials. In fact n!sn and the nth monospline di er by a constant and sn (x) = n Bn ( x ) where Bn is the nth Bernoulli polynomial. From the Fourier series expansions
1 1
dp f (x) dxp
2
dx ?
M X
Therefore we have the inclusion
m
amp ? 2
2
N X k
=1
(k ) p jbk j
2
!1 2
2
=1
(2p ? 1) p N p?
2 2
2
1 2
1
f (x) 2 a +
0
M p XX
=1
m
j
amj sj (x + 1 ? m ) +
+ +2 +1 + +2
ZZ q
G(x;t)
q! sj (y + )dyd =
q X
+1
ZZ q
G(x;t)
l
=0
1 ? 2 (qt+1 ? l)! sj
+1 +1
l
q ?l
+ +1
q sj q (x ? t) (x + t) ? ? 1 l 2
+1 + +2
q! sj (y ? )dyd = ?
q X 1 l
1 2 1 2
3 Numerical Method for the Wave Equation
U (x; t) = A (t) +
0
M p XX X
1 =1
m
j
=1
" ?
=
Amj" (t)sj (x + "t + 1 ? m ) +
1
N X
6 ?
1
using an iterative procedure RR U k = (1 ? )U k + g + G x;t U k where G(x; t) is a triangle with vertices (x; t); (x ? t; 0); (x + t; 0) and ! 1 g (x + t) + g (x ? t) + Z x t g ( )d + Z Z g(x; t) = 2 (y; )dyd x?t G x;t Essential part of each iteration is the evaluation of the integral. The following formulas are used:
2 1 0 1 2
of a function f : ?1; 1] ! R is approximated by a vector of the form 8 :0 k N > ck >
> > > <c : ?N k c(k) = > ?k > p > > (?1)k P a : jkj > N > : jk j
+1
ZZ q
l
=0
tq ?l s (x + t)+ 1 q s j q (x ? t) 2 (q +1 ? l)! j l 2
+ +1
G(x;t)
1 ik y q! e dyd = ik =2
l l=0
q
+2
1 eik x t + ik
( + ) + +2
q
+2
q X tq?l eik x?t ? 2 (q ? l)! eik x
2 Spline-Fourier Series
Let M be the set of all periodical function with period 2 which are p-times di erentiable at every point of (?1; 1] except for a nite number of points ; ; : : : ; M and let d f 2 L ?1; 1]. There is a unique representation of dx every function f 2 M in the form
( +1) ( ) ( ) 2 ( ) + 1 1 2 ( )
k= N k=0
Bk (t)eik x
ZZ q
G(x;t)
q q! sj (y)dyd = sj q (x + t)+(?1) sj q (x ? t) ? 2
+ +2 + +2
q X
?even
l=0
l
tq?l s (q ? l)! j l (x)
1 Introduction
This work is inspired by the ideas for validated approximations of function space problems 3] and essentially it is a further application of the Fourier Hyper Functoid described in 2] where the coe cient vector c = (: : : ; c? ; c? ; c ; c ; c ; : : :) of the Fourier series 1 X f (x) ck eik x
1 2 n 2 ! +1 2
sj (x) = ? (i 1)j
k= k=0
1 X (?1)k ik x kj e ;
6 ?1
j 1
Anguelov R.: Spline-Fourier Approximations of Discontinuous Waves
111
we can see that the approximation of the Fourier coe cients of the form (1)is equivalent to approximating the function f by linear combinations of fsj : j = 1; : : : ; pg feik x : k = 0; 1; : : :; N g For practical reasons that can be seen below we would prefer to use the spline notations rather then the above ansatz (1) for the coe cients.
1 X
k= k=0
=1
6
?1
bk eik x + ?"N ; "N ]
112
Anguelov R.: Spline-Fourier Approximations of Discontinuous Waves
We consider a wave equation of the form u(x; t)tt ? u(x; t)xx = (t)u(x; t) + (x; t)) u(x; 0) = g (x) ; ut (x; 0) = g (x) with periodic boundary conditions at x = ?1 and x = 1, assuming that , g , g or some of their derivatives may be discontinuous but they can be represented as a spline-Fourier series (2). An interval enclosure of the solution is calculated in the form
p 1 2 p 2
f (x) = a +
0
M p XX
=1
where
1 X
k= k=0
m
j
amj sj (x + 1 ? m ) +
1 X
6