How to solve the following (integro-differential eq.)

[Hortensius]

Hi all,

I'm struggeling with an equation of the following form:

$$\frac{\partial A\left(x,t\right)}{\partial t} = \int_{x_0}^{x_1} f\left(x,x',t\right)A\left(x',t\right) dx'$$

The problem is defined on the interval $$x_0 \leq x \leq x_1$$, where $$f\left(x,x',t\right)$$ is a known function. We have the initial condition $$A\left(x,0\right) = g\left(x\right)$$.

For my specific problem I found the qualitative behavior of the solution $$A\left(x,t\right)$$ by straightforward numerical integration. But I would like to be able to find analytic solutions... Any suggestions on how to proceed here?
Any help is very much appreciated!

 

[jvicens]

Hi there,

It seems like you are dealing with a partial differential equation (PDE) with an integral term. This type of equation can be challenging to solve analytically, but there are some techniques that may be helpful.

One approach is to use separation of variables, where you assume that the solution can be written as a product of two functions, one depending only on x and the other depending only on t. This can sometimes lead to a simpler equation that can be solved analytically.

Another technique is to use a series expansion, where you approximate the solution as a sum of simpler functions. This can be useful when the integral term is difficult to handle analytically.

You can also try to transform the equation into a different form, such as using a change of variables or applying a Fourier transform. This can sometimes simplify the equation and make it easier to solve.

If all else fails, you can also try to find numerical methods that can give you an accurate solution. These methods can often handle complex equations and provide a good approximation to the solution.

I hope these suggestions are helpful. Good luck with your research!