# Mathematica Numerical solution of integral equation with parameters

1. Jan 13, 2017

### illuminates

Hello! Could you tell me about how to take the next numerical calculation in mathematica? (perhaps there are special packages).
I have an expression (in reality slightly more complex):

$V=x^2 + \int_a^b x \sqrt{x^2-m^2} \left(\text Log \left(e^{-\left(\beta \left(\sqrt{\left(\sqrt{l^2-m^2}+U\right)^2+(m+x)^2+N}+u\right)\right)}+1\right)\right) \, dl$

Code ( (Unknown Language)):

V=x^2+\!$$\*SubsuperscriptBox[\(\[Integral]$$, $$a$$, $$b$$]$$l \*SqrtBox[\( \*SuperscriptBox[\(l$$, $$2$$] -
\*SuperscriptBox[$$m$$, $$2$$]\)] $$Log(1 + \*SuperscriptBox[\(e$$, $$-\(\[Beta](u + \*SqrtBox[\( \*SuperscriptBox[\(( \*SqrtBox[\( \*SuperscriptBox[\(l$$, $$2$$] -
\*SuperscriptBox[$$m$$, $$2$$]\)] + U)\), $$2$$] +
\*SuperscriptBox[$$(m + x)$$, $$2$$] + N\)])\)\)])\) \[DifferentialD]l\)\)

where $x$ is function of $l$; $m$, $N$ are constants; $\beta$, $u$, $U$ are parameters.
I need to find the dependence $U$ on $u$ and $\beta$ (in order to draw graph) from an equation:
$\frac {\partial V} {\partial x}=0$
($x$ will be needed to set a constant after differentiation; In reality,there is not the derivative, but a variation)

If I have an integral (without parameters) rather than equation, I would try to do the following ones:

1) to define the a region of integration (due to graphical representation of function)
2) to tabulate integrand
3) to calculate the integral that is to get a number.

Nevertheless I have the equation, which probably requires other method. I would appreciate Mathematica literature on this subject, or help.

I do not know how actual it is to calculate integral equation, but the integral can be led to another kind:
$x \sqrt{x^2-m^2} \left(\text {Log} \left(e^{-\left(\beta \left(\sqrt{\left(\sqrt{l^2-m^2}+U\right)^2+(m+x)^2+N}+u\right)\right)}+1\right)\right) \to$
$\left(x^2-m^2\right)^{3/2} \frac{\text{\cosh(\beta u)} +\exp \left(-\beta \sqrt{\left(\sqrt{l^2-m^2}+U\right)^2+(m+x+y)^2+(q+z)^2}\right)}{\text{\cosh (\beta u)} -\cosh \left(\beta \sqrt{\left(\sqrt{l^2-m^2}+U\right)^2+(m+x+y)^2+(q+z)^2}\right)}$

Code ( (Unknown Language)):

(x^2-m^2)^(3/2) (cosh(\[Beta]u)+exp(-\[Beta] Sqrt[(Sqrt[l^2-m^2]+U)^2+(m+x+y)^2+(z+q)^2]))/(cosh(\[Beta]u)-cosh(\[Beta] Sqrt[(Sqrt[l^2-m^2]+U)^2+(m+x+y)^2+(z+q)^2]))

$V=x^2+\int_{a}^{b} x u U \beta l dl$

$\frac {\partial V} {\partial x}=0$

$x$ is assumed a constant after differentiating.

$\int_a^b \beta l u U \, dl+2 x=0$

$U=\frac{4 x}{\beta u \left(a^2+b^2\right)}$

In all I get the dependency $U$, on $u$ and $\beta$

I need to do the same if the integral is not taken analytically.

2. Jan 15, 2017

### the_wolfman

Can you help me understand your simple example. For starters you arrive at the conclusion that U is a function of x. However, when you differentiate the integral term by x you ignore this dependence. Second you state the x is a function of l. If this is the case then you should not ignore this dependence when evaluating the integral.