An integro-differential equation

by Sepideh Bakhoda   Last Updated September 23, 2018 12:20 PM

I want to solve this integro-differential equation

$(f(x,y,z)-g(x,y,z)\partial_z^{-1}\partial_x)u(x,y,z)=h(x,y,z)$

where $f, g ,h$ are known and $u$ is unknown. In addition

$\partial_z^{-1}(\vec{p_1},\vec{p_2})=\theta(z_1-z_2)\delta(x_1-x_2)\delta(y_1-y_2)$

where $\theta$ is Heaviside step function, $\delta$ is Dirac's delta function and $\vec{p_i}=(x_i,y_i,z_i)$.

Can someone point me in the right direction? How should I start to attach this problem? Is there a general way to solve this kind of equation? I would also be very grateful if you introduce me some references on solving these equations.



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