naginterfaces.library.inteq.fredholm2_smooth¶
- naginterfaces.library.inteq.fredholm2_smooth(k, g, lamda, a, b, odorev, ev, n, data=None)[source]¶
fredholm2_smooth
solves any linear nonsingular Fredholm integral equation of the second kind with a smooth kernel.For full information please refer to the NAG Library document for d05ab
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d05/d05abf.html
- Parameters
- kcallable retval = k(x, s, data=None)
must compute the value of the kernel of the integral equation over the square , .
- Parameters
- xfloat
The values of and at which is to be calculated.
- sfloat
The values of and at which is to be calculated.
- 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 compute the value of the function of the integral equation in the interval .
- Parameters
- xfloat
The value of at which is to be calculated.
- dataarbitrary, optional, modifiable in place
User-communication data for callback functions.
- Returns
- retvalfloat
The value of evaluated at .
- lamdafloat
The value of the parameter of the integral equation.
- afloat
, the lower limit of integration.
- bfloat
, the upper limit of integration.
- odorevbool
Indicates whether it is known that the solution is odd or even about the mid-point of the range of integration. If is then an odd or even solution is sought depending upon the value of .
- evbool
Is ignored if is . Otherwise, if is , an even solution is sought, whilst if is , an odd solution is sought.
- nint
The number of terms in the Chebyshev series which approximates the solution .
- dataarbitrary, optional
User-communication data for callback functions.
- Returns
- ffloat, ndarray, shape
The approximate values , for , of the function at the first of Chebyshev points (see Notes), where
if and .
if and .
if .
- cfloat, ndarray, shape
The coefficients , for , of the Chebyshev series approximation to . When is , this series contains polynomials of even order only or of odd order only, according to being or respectively.
- Raises
- NagValueError
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, and .
Constraint: .
- (errno )
A failure has occurred due to proximity of an eigenvalue.
- Notes
fredholm2_smooth
uses the method of El–Gendi (1969) to solve an integral equation of the formfor the function in the range .
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 a set of simultaneous linear algebraic equations formed by applying a quadrature formula (equivalent to the scheme of Clenshaw and Curtis (1960)) to the integral equation at each of the above points.
In general . However, advantage may be taken of any prior knowledge of the symmetry of . Thus if is symmetric (i.e., even) about the mid-point of the range , it may be approximated by an even Chebyshev series with . Similarly, if is anti-symmetric (i.e., odd) about the mid-point of the range of integration, it may be approximated by an odd Chebyshev 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