naginterfaces.library.inteq.fredholm2_​split

naginterfaces.library.inteq.fredholm2_split(lamda, a, b, k1, k2, g, n, ind, data=None)[source]

fredholm2_split solves a linear, nonsingular Fredholm equation of the second kind with a split kernel.

For full information please refer to the NAG Library document for d05aa

https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/d05/d05aaf.html

Parameters
lamdafloat

The value of the parameter of the integral equation.

afloat

, the lower limit of integration.

bfloat

, the upper limit of integration.

k1callable retval = k1(x, s, data=None)

must evaluate the kernel of the integral equation for .

Parameters
xfloat

The values of and at which is to be evaluated.

sfloat

The values of and at which is to be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
retvalfloat

The value of the kernel evaluated at and .

k2callable retval = k2(x, s, data=None)

must evaluate the kernel of the integral equation for .

Parameters
xfloat

The values of and at which is to be evaluated.

sfloat

The values of and at which is to be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
retvalfloat

The value of the kernel evaluated at and .

gcallable retval = g(x, data=None)

must evaluate the function for .

Parameters
xfloat

The values of at which is to be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
retvalfloat

The value of evaluated at .

nint

The number of terms in the Chebyshev series required to approximate .

indint

Determines the forms of the kernel, , and the function .

is not centro-symmetric (or no account is to be taken of centro-symmetry).

is centro-symmetric and is odd.

is centro-symmetric and is even.

is centro-symmetric but is neither odd nor even.

dataarbitrary, optional

User-communication data for callback functions.

Returns
ffloat, ndarray, shape

The approximate values , for , of evaluated at the first of Chebyshev points , (see Notes).

If or , .

If , .

If , .

cfloat, ndarray, shape

The coefficients , for , of the Chebyshev series approximation to .

If this series contains polynomials of odd order only and if the series contains even order polynomials only.

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, and .

Constraint: .

(errno )

A failure has occurred due to proximity of an eigenvalue.

Notes

fredholm2_split solves an integral equation of the form

for , when the kernel is defined in two parts: for and for . The method used is that of El–Gendi (1969) for which, it is important to note, each of the functions and must be defined, smooth and nonsingular, for all and in the interval .

An approximation to the solution is found in the form of an term Chebyshev series , where indicates that the first term is halved in the sum. The coefficients , for , of this series are determined directly from approximate values , for , of the function at the first of a set of Chebyshev points:

The values are obtained by solving simultaneous linear algebraic equations formed by applying a quadrature formula (equivalent to the scheme of Clenshaw and Curtis (1960)) to the integral equation at the above points.

In general . However, if the kernel is centro-symmetric in the interval , i.e., if , then the function is designed to take advantage of this fact in the formation and solution of the algebraic equations. In this case, symmetry in the function implies symmetry in the function . In particular, if is even about the mid-point of the range of integration, then so also is , which may be approximated by an even Chebyshev series with . Similarly, if is odd about the mid-point then may be approximated by an odd series with .

References

Clenshaw, C W and Curtis, A R, 1960, A method for numerical integration on an automatic computer, Numer. Math. (2), 197–205

El–Gendi, S E, 1969, Chebyshev solution of differential, integral and integro-differential equations, Comput. J. (12), 282–287