naginterfaces.library.inteq.fredholm2_​smooth

naginterfaces.library.inteq.fredholm2_smooth(k, g, lamda, a, b, odorev, ev, n, data=None)

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_30.3/flhtml/d05/d05abf.html

Parameters

kcallable retval = k(x, s, data=None)

$k$ must compute the value of the kernel $k(x,s)$ of the integral equation over the square $a\le x\le b$, $a\le s\le b$.

Parameters

xfloat

The values of $x$ and $s$ at which $k(x,s)$ is to be calculated.

sfloat

The values of $x$ and $s$ at which $k(x,s)$ is to be calculated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

The value of the kernel $k(x,s)$ evaluated at $x$ and $s$.

gcallable retval = g(x, data=None)

$g$ must compute the value of the function $g\left(x\right)$ of the integral equation in the interval $a\le x\le b$.

Parameters

xfloat

The value of $x$ at which $g\left(x\right)$ is to be calculated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

The value of $g\left(x\right)$ evaluated at $x$.

lamdafloat

The value of the parameter $\lambda$ of the integral equation.

afloat

$a$, the lower limit of integration.

bfloat

$b$, the upper limit of integration.

odorevbool

Indicates whether it is known that the solution $f\left(x\right)$ is odd or even about the mid-point of the range of integration. If $odorev$ is $True$ then an odd or even solution is sought depending upon the value of $ev$.

evbool

Is ignored if $odorev$ is $False$. Otherwise, if $ev$ is $True$, an even solution is sought, whilst if $ev$ is $False$, an odd solution is sought.

nint

The number of terms in the Chebyshev series which approximates the solution $f\left(x\right)$.

dataarbitrary, optional

User-communication data for callback functions.

Returns

ffloat, ndarray, shape $\left(n\right)$

The approximate values $f_{\text{i}}$, for $\text{i}=1,2,\dots ,n$, of the function $f\left(x\right)$ at the first $n$ of $m+1$ Chebyshev points (see Notes), where

$m=2n-1$	if $odorev=True$ and $ev=True$.
$m=2n$	if $odorev=True$ and $ev=False$.
$m=n-1$	if $odorev=False$.

cfloat, ndarray, shape $\left(n\right)$

The coefficients $c_{\text{i}}$, for $\text{i}=1,2,\dots ,n$, of the Chebyshev series approximation to $f\left(x\right)$. When $odorev$ is $True$, this series contains polynomials of even order only or of odd order only, according to $ev$ being $True$ or $False$ respectively.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $1$)

On entry, $a=⟨value⟩$ and $b=⟨value⟩$.

Constraint: $b>a$.

(errno $2$)

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 form

$$f\left(x\right)-\lambda {\int }_a^{b}k(x,s)f\left(s\right)ds=g\left(x\right)$$

for the function $f\left(x\right)$ in the range $a\le x\le b$.

An approximation to the solution $f\left(x\right)$ is found in the form of an $n$ term Chebyshev series ${{\sum }^{\prime }}_{i=1}^n{c}_i{T}_i\left(x\right)$, where ${}^{\prime }$ indicates that the first term is halved in the sum. The coefficients $c_{\text{i}}$, for $\text{i}=1,2,\dots ,n$, of this series are determined directly from approximate values $f_{\text{i}}$, for $\text{i}=1,2,\dots ,n$, of the function $f\left(x\right)$ at the first $n$ of a set of $m+1$ Chebyshev points

$$x_i=\frac{1}{2}(a+b+(b-a)\times \cos [(i-1)\times \pi /m])\text{,}i=1,2,\dots ,m+1\text{.}$$

The values $f_i$ 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 $m=n-1$. However, advantage may be taken of any prior knowledge of the symmetry of $f\left(x\right)$. Thus if $f\left(x\right)$ is symmetric (i.e., even) about the mid-point of the range $(a,b)$, it may be approximated by an even Chebyshev series with $m=2n-1$. Similarly, if $f\left(x\right)$ 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 $m=2n$.

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