naginterfaces.library.inteq.fredholm2_​split

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

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_29.3/flhtml/d05/d05aaf.html

Parameters

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.

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

$k1$ must evaluate the kernel $k(x,s)=k_1(x,s)$ of the integral equation for $a\le s\le x$.

Parameters

xfloat

The values of $x$ and $s$ at which $k_1(x,s)$ is to be evaluated.

sfloat

The values of $x$ and $s$ at which $k_1(x,s)$ is to be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

The value of the kernel $k(x,s)=k_1(x,s)$ evaluated at $x$ and $s$.

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

$k2$ must evaluate the kernel $k(x,s)=k_2(x,s)$ of the integral equation for $x<s\le b$.

Parameters

xfloat

The values of $x$ and $s$ at which $k_2(x,s)$ is to be evaluated.

sfloat

The values of $x$ and $s$ at which $k_2(x,s)$ is to be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

The value of the kernel $k(x,s)=k_2(x,s)$ evaluated at $x$ and $s$.

gcallable retval = g(x, data=None)

$g$ must evaluate the function $g\left(x\right)$ for $a\le x\le b$.

Parameters

xfloat

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

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

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

nint

The number of terms in the Chebyshev series required to approximate $f\left(x\right)$.

indint

Determines the forms of the kernel, $k(x,s)$, and the function $g\left(x\right)$.

$ind=0$

$k(x,s)$ is not centro-symmetric (or no account is to be taken of centro-symmetry).

$ind=1$

$k(x,s)$ is centro-symmetric and $g\left(x\right)$ is odd.

$ind=2$

$k(x,s)$ is centro-symmetric and $g\left(x\right)$ is even.

$ind=3$

$k(x,s)$ is centro-symmetric but $g\left(x\right)$ is neither odd nor even.

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 $f\left(x\right)$ evaluated at the first $n$ of $m+1$ Chebyshev points $x_i$, (see Notes).

If $ind=0$ or $3$, $m=n-1$.

If $ind=1$, $m=2\times n$.

If $ind=2$, $m=2\times n-1$.

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

The coefficients $c_{\text{i}}$, for $\text{i}=1,2,\dots ,n$, of the Chebyshev series approximation to $f\left(x\right)$.

If $ind=1$ this series contains polynomials of odd order only and if $ind=2$ the series contains even order polynomials only.

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_split solves 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 $a\le x\le b$, when the kernel $k$ is defined in two parts: $k=k_1$ for $a\le s\le x$ and $k=k_2$ for $x<s\le b$. The method used is that of El–Gendi (1969) for which, it is important to note, each of the functions $k_1$ and $k_2$ must be defined, smooth and nonsingular, for all $x$ and $s$ in the interval $[a,b]$.

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:

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

The values $f_i$ 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 $m=n-1$. However, if the kernel $k$ is centro-symmetric in the interval $[a,b]$, i.e., if $k(x,s)=k(a+b-x,a+b-s)$, 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 $g\left(x\right)$ implies symmetry in the function $f\left(x\right)$. In particular, if $g\left(x\right)$ is even about the mid-point of the range of integration, then so also is $f\left(x\right)$, which may be approximated by an even Chebyshev series with $m=2n-1$. Similarly, if $g\left(x\right)$ is odd about the mid-point then $f\left(x\right)$ may be approximated by an odd 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