A matrix version of Fredholm integral equation of the second kind

Fredholm integral equation are often encountered in physics. I have to deal with the following Fredholm matrix equation $$\big[ L (x,y) \big]_{ij} = \big[ A (x,y) \big]_{ij} + \sum_{k} \!\! \int \!\!\mathrm{d} z \, \big[ K(z) \big]_{k} \, \big[ L(z,x) \big]_{k j} \, \big[ A(y,z) \big]_{i k} \, , \;\;\;\;\; (1)$$ where $[L (x,y)]_{i j}$ is the unknown matrix function, $[A(x,y)]_{ij}$ is a known matrix function, and $[K(z)]_{k}$ a known vector kernel. Here ${i,\,j,\,k}$ are index for matrix coefficients, and ${x,\,y,\,z \in \mathbb{R}^{d}}$. Unfortunately, equation $(1)$ is not exactly under the usual form of the Fredholm integral equation.

Let me now rewrite it slightly differently, so as to be able to use the usual formalism. We place ourself for a fixed value of $x\in\mathbb{R}^{d}$. One can then rewrite equation $(1)$ under the form $$\boxed{ \mathbb{L}^{x} (y) = \mathbb{A}^{x} (y) + \int \!\! \mathrm{d} z \,\, \mathbb{K}(y,z) \cdot \mathbb{L}^{x} (z) \, , } \;\;\;\;\; (2)$$ where $\mathbb{L}^{x}$, $\mathbb{A}^{x}$ and $\mathbb{K}$ are three matrices. One can note that $\mathbb{L}^{x}$ and $\mathbb{A}^{x}$ depend on the mute variable $x$, whereas the kernel $[\mathbb{K} (y,z)]_{ik} = [K(z)]_{k} \, [A(y,z)]_{ik}$ is independent of $x$.

Expect for the presence of a matrix multiplication, equation $(2)$ has the same form than the Fredholm equations of the second kind usually considered.

My questions are then as follows :

• How can I solve equation $(2)$ for $\mathbb{L}^{x}$ in this matrix context ? As I am coming from a physics background, you may do any sensitive assumptions for nice properties of the kernel $\mathbb{K}$. What should be the best general way to deal with such equations ?

Thanks to other calculations, I know what I should expect as a result for $\mathbb{L}^{x}(y)$. Indeed, I know I can write the source function $\mathbb{A}^{x} (y)$ under the form $$\big[ \mathbb{A}^{x} (y) \big]_{ij} = \sum_{pq} \psi_{i}^{(p)} (x) \, \mathbb{I}_{pq} \, \psi^{(q) *}_{j} (y) \, , \;\;\;\;\; (3)$$ where $\mathbb{I}_{pq}$ is the identity matrix, and $\psi^{p} (x)$ are basis functions. If I then assume that ${\mathbb{L}^{x} (y)}$ is of the form $$\big[ \mathbb{L}^{x} (y) \big]_{ij} = \sum_{pq} \psi^{(p)}_{i} (x) \, \mathbb{D}_{pq} \, \psi^{(q) *}_{j}(y) \, , \;\;\;\;\; (4)$$ where $\mathbb{D}_{pq}$ is a yet unknown matrix, I can then directly obtain from equation $(2)$ that the matrix $\mathbb{D}$ is given by $$\mathbb{D} = \big[ \mathbb{I} - \mathbb{M} \big]^{-1} \, , \;\;\;\;\; (5)$$ where the matrix $\mathbb{M}$ is defined as $$\mathbb{M}_{pq} = \sum_{k} \int \!\! \mathrm{d} z \, [K (z)]_{k} \, \psi^{(p) *}_{k} (z) \, \psi^{(q)}_{k} (z) \, . \;\;\;\;\; (6)$$

• Assuming only equation (3), can we recover the expression (5), without having to assume (4) ? Does equation $(2)$ can lead only to other solutions ?

Any help/advice on how to effectively deal with such matricial Fredholm integral equation will be much appreciated ! (The problem I am considerering is more complex than equation $(1)$, but this is already a good start !)

• Hi jibe,I am also looking at knowing how to solve matrix Fremhold integral equation of second kind. Do you have any ideas or solutions how to address it? – ems Oct 4 '17 at 15:22
• @ems In practice, the effective resolution of a Fredholm depends quite strongly on its form, and its precise kernel. I think you might to ask a different question here on SE, detailing exactly what equation you want to invert. I would be happy to have a look at your question then (be sure to tag me!). One usual method to solve these equations is the basis method, i.e. using a set of basis `functions' to represent the source terms as I used above. But again, the details of this method depend on the exact form of your equation. – jibe Oct 5 '17 at 16:15
• Thanks @jibe. I follow your suggestion. Btw, have you worked on linear delay Volterra integral equations with limits $t-\tau$ and $t$? I am struggling to know how to solve and obtain bounds for them, knowing that my kernel is bounded. Any references or suggestions are welcome. – ems Oct 5 '17 at 18:50