Let say I have a minimal example that contains most of the mathematics I am looking for. The example is as follows:
\begin{align} &\frac{\partial u^0}{\partial t}(x,t)+\frac{\partial^2 u^0}{\partial x^2}(x,t) + u^0(x,t)\frac{\partial u^0}{\partial x}(x,t)\\ &\qquad\qquad\qquad\quad\;\;+\int_0^1\!\!\!\int_0^t u^1(x,t;x_1,t_1)\frac{\partial u^1}{\partial x}(x,t;x_1,t_1)dt_1 dx=0\\ &\frac{\partial u^1}{\partial t}(x,t;x_1,t_1)+\frac{\partial^2 u^1}{\partial x^2}(x,t;x_1,t_1)+u^0(x,t)\frac{\partial u^1}{\partial x}(x,t;x_1,t_1)=0 \end{align}
wherein, $0\leq x\leq 1$ and $t\geq 0$. The initial and boundary conditions are as follows:
\begin{align} &x=0: && u^0(0,t)=0\,, && u^1(0,t;x_1,t_1)=0\\ &x=1: && u^0(1,t)=1\,, && u^1(1,t;x_1,t_1)=0\\ &t=0: && u^0(x,0)=0\,, && u^1(x,0;x_1,t_1)=0 \end{align}
These are a coupled set of PDE in $(x,t)$, somewhat of first kind Fredholm integral equation WRT $x_1$ and somewhat of first kind Volterra integral equation WRT $t_1$.
Now here are my questions:
a) The variables $x_1$ and $t_1$ never appear outside the integrals, they are like mere parameters in the second equations, can we find a systematic way to find the dependency on $x_1$ and $t_1$ in $u^1$ or such a dependency is arbitrary?
comment: it seems $u^1$ can be considered as constant WRT these variables and also other dependencies are possible, but in some physical applications (not applicable to mine for lack of symmetry here) the researchers have Fourier transformed the equations and in the Fourier space both $k$ and $k^1$ have appeared in the equations even outside the integrals, so not strange to see the answer is non-trivially depending on both $k$ and $k_1$.
b) Is there any efficient and easy method to solve such a system of equations?
The method of Fourier transform in each variable is a powerful one. $u(x,t)\to \hat{u}(\xi, t)\to \tilde{\hat{u}}(\xi, \omega)$ turns integro-differential equations into a set of algebraic equations that can be easily solved.