11 Numerical Evaluation of Integrals in Laplace Domain Anisotropic …
159
For such change of variables, we need to be able to obtain inverse function q
−1
(y),
and phase function must not have any stationary points, so q
(x) = 0 in interval
of integration. In our case in integrals I 1 [ϕ, τ ] and I 2 [ϕ, τ ], amplitude and phase
function are not available in explicit closed form, and we need to evaluate them
numerically for every value of integration variable. So to apply Filon-type method,
we need to numerically calculate inverse of the phase function. Also, we do not know
beforehand if phase function has any stationary points.
Levin’s method (Levin 1982) is entirely different method suitable for solving
highly oscillatory integrals with more complicated phase functions and does not
require computation of moments. Levin’s method transforms integration problem
into the problem of finding a certain function F(x) to satisfy the following ordinary
differential equation (ODE)
F
(x) + iγ q
(x)F(x) = f (x).
(11.24)
If we solve this ODE, then we can evaluate the integral (11.22) trivially by
substituting (11.24) into (11.22)
I =
1
−1
F
(x) + iγ q
(x)F(x)
e
iγ q(x) dx
=
1
−1
d
dx
F(x)e
iγ q(x)
dx = F(1)e
iγ q(1)
− F(−1)e
iγ q(−1)
.
(11.25)
To obtain F(x), it is approximated by F n (x) that is a combination of some linearly
independent basis functions
F(x) ≈ F n (x) =
n
k=1
β k ν k (x).
(11.26)
To determine the coefficients β k , the following system of collocation equations is
solved on a sequence of collocation points
x j
, j = 1, n:
n
k=1
β k ν
k (x j ) + iγ q
(x j )
n
k=1
β k ν k (x j ) = f (x j ).
(11.27)
Equation (11.27) is a complex-valued linear algebraic system for coefficients β k
After solving it, we can approximate the integral (11.22) as
I ≈ I n =
n
k=1
β k ν k (1)e
iγ q(1)
−
n
k=1
β k ν k (−1)e
iγ q(−1)
.
(11.28)
159
For such change of variables, we need to be able to obtain inverse function q
−1
(y),
and phase function must not have any stationary points, so q
(x) = 0 in interval
of integration. In our case in integrals I 1 [ϕ, τ ] and I 2 [ϕ, τ ], amplitude and phase
function are not available in explicit closed form, and we need to evaluate them
numerically for every value of integration variable. So to apply Filon-type method,
we need to numerically calculate inverse of the phase function. Also, we do not know
beforehand if phase function has any stationary points.
Levin’s method (Levin 1982) is entirely different method suitable for solving
highly oscillatory integrals with more complicated phase functions and does not
require computation of moments. Levin’s method transforms integration problem
into the problem of finding a certain function F(x) to satisfy the following ordinary
differential equation (ODE)
F
(x) + iγ q
(x)F(x) = f (x).
(11.24)
If we solve this ODE, then we can evaluate the integral (11.22) trivially by
substituting (11.24) into (11.22)
I =
1
−1
F
(x) + iγ q
(x)F(x)
e
iγ q(x) dx
=
1
−1
d
dx
F(x)e
iγ q(x)
dx = F(1)e
iγ q(1)
− F(−1)e
iγ q(−1)
.
(11.25)
To obtain F(x), it is approximated by F n (x) that is a combination of some linearly
independent basis functions
F(x) ≈ F n (x) =
n
k=1
β k ν k (x).
(11.26)
To determine the coefficients β k , the following system of collocation equations is
solved on a sequence of collocation points
x j
, j = 1, n:
n
k=1
β k ν
k (x j ) + iγ q
(x j )
n
k=1
β k ν k (x j ) = f (x j ).
(11.27)
Equation (11.27) is a complex-valued linear algebraic system for coefficients β k
After solving it, we can approximate the integral (11.22) as
I ≈ I n =
n
k=1
β k ν k (1)e
iγ q(1)
−
n
k=1
β k ν k (−1)e
iγ q(−1)
.
(11.28)