# Numerical solution of integral equation with parameters

• Mathematica
• illuminates
In summary, the conversation discusses how to take the next numerical calculation in Mathematica, specifically involving an expression for V and an equation for finding the dependence of U on u and β. It also mentions using a simpler example to understand the process.
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:
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:
    (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]))

Perhaps it was necessary to start with something simpler. Let

##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.

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.

## 1. What is an integral equation with parameters?

An integral equation with parameters is a mathematical equation that involves both unknown functions and parameters. The unknown functions are typically represented by an integral, while the parameters are constants that affect the solution of the equation.

## 2. Why do we need to solve integral equations with parameters numerically?

Some integral equations with parameters do not have analytical solutions, meaning they cannot be solved using traditional algebraic methods. Therefore, we need to use numerical methods to approximate the solution and obtain a numerical result.

## 3. What are some common numerical methods used to solve integral equations with parameters?

Some common numerical methods used to solve integral equations with parameters include the Gauss-Legendre quadrature, the trapezoidal rule, and the Simpson's rule. These methods involve breaking down the integral into smaller parts and approximating the solution using numerical calculations.

## 4. How do you choose the appropriate numerical method for solving an integral equation with parameters?

The choice of numerical method depends on the specific properties of the integral equation, such as the type of integral, the number of parameters, and the desired accuracy of the solution. It is important to understand the strengths and limitations of each method in order to select the most appropriate one for a particular problem.

## 5. What are some applications of solving integral equations with parameters?

Integral equations with parameters have a wide range of applications in various fields of science and engineering, such as physics, chemistry, and economics. They can be used to model and analyze complex systems, including biological processes, electrical circuits, and economic systems. Solving these equations numerically allows for a better understanding of these systems and can lead to important insights and discoveries.

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