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I want to find analytical solutions of the following integro-differential equation:

$\left(A\nabla_{\rho}^2 + B\nabla_z^2\right)f(\vec{r}) = C \int{ g(\vec{r},\vec{r\,'}) f(\vec{r\,'})d\vec{r\,'}}, \qquad g(\vec{r},\vec{r\,'}) = e^{-\alpha|z-z'|-\beta\rho^{\,'2}},$

where $A$, $B$, $C$, $\alpha$, and $\beta$ are real constants and $\nabla_{\rho}$ and $\nabla_z$ are gradients over $\rho$ and z, and $h$ is a well-known integrable function. I considered cylindrical coordinates where the position vector is given by $\vec{r} = \vec{\rho} + \vec{z}$ with $\vec{\rho} = \vec{x} + \vec{y}$.

It will be appreciated if you can leave your comments.

Best regards

Houshyar
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  • Hello and welcome to math.stackexchange. Where does this interesting problem come from? Are there additional conditions for the unknown function $f$? What have you tried? Are you familiar with the Fourier transform? – Hans Engler Jan 13 '22 at 15:44
  • Hi Hans, It is stationary heat transfer. f is temperature.I used variables separation method by considering Gaussian form for ro-dependency of f(r): f(ro,z)=exp(-a.ro^2)sum[(ro^n).f_n(z)] for n=[0, +inf). Then I considered homogeneous equation (without the integral) to find a special solution.Thus, I found a homogeneous equation for f_n(z) for n>= 2 that can be found in terms of f_0(z) and f_1(z) which should be obtained.Using Gauss-Jordan to solve: [A].[f_n(z)]=[0], where [A] is a matrix of coefficients, did not cause a general term to truncate the series on a specific n. Thanks for comments. – Houshyar Jan 13 '22 at 16:14

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