# Integral equation with endpoint singularity

• rplcs
In summary, the conversation discusses solving an integral equation with a singularity at the endpoint 1. The speaker is using numerical methods, specifically looking at Nystrom method, but is unsure if it is applicable for this type of integral equation. They are seeking suggestions for a suitable method.
rplcs
I am trying to solve this integral equation numerically. The kernel has a singularity at the endpoint 1. Any suggestions??

f(s) = \int_0^1 \frac{1+st}{(1-st)^3} f(t) dt

rplcs said:
I am trying to solve this integral equation numerically. The kernel has a singularity at the endpoint 1. Any suggestions??

f(s) = \int_0^1 \frac{1+st}{(1-st)^3} f(t) dt
Which numerical algorithm are you using? Most will quite happily handle end-point singularities.

Hootenanny said:
Which numerical algorithm are you using? Most will quite happily handle end-point singularities.

I am looking at methods based on quadrature rules like Nystrom method. But the question is are these method applicable to this integral equation. Nystrom like methods are suitable for very specific singular kernels.

please suggest which method is suitable for this integral equation.

## 1. What is an integral equation with endpoint singularity?

An integral equation with endpoint singularity is an equation that involves an integral where the integrand has a singularity at one or both of the endpoints of integration. This type of equation is often encountered in physics and engineering problems where the solution is a function that is not smooth at the endpoints.

## 2. How is an integral equation with endpoint singularity solved?

Solving an integral equation with endpoint singularity can be a challenging task. It typically involves using specialized techniques such as contour integration, singular value decomposition, or Green's functions. These methods allow for the removal of the singularity and the solution of the equation using standard integral calculus techniques.

## 3. What are some applications of integral equations with endpoint singularity?

Integral equations with endpoint singularity have a wide range of applications in physics, engineering, and mathematics. They are commonly used in solving boundary value problems in electromagnetics, fluid mechanics, and quantum mechanics. They also have applications in signal processing, image reconstruction, and inverse problems.

## 4. How do integral equations with endpoint singularity differ from regular integral equations?

The main difference between integral equations with endpoint singularity and regular integral equations is the presence of a singularity in the integrand. This singularity can make the solution of the equation more challenging and often requires the use of specialized techniques. Regular integral equations, on the other hand, do not have these singularities and can be solved using standard integral calculus methods.

## 5. What are some common challenges when working with integral equations with endpoint singularity?

One of the main challenges when working with integral equations with endpoint singularity is the removal of the singularity in the integrand. This can be a complex process and may require the use of advanced mathematical techniques. Additionally, the solution of these equations may not always be unique, and the presence of multiple solutions can complicate the interpretation of the results.

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