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