The integral equation derived in the previous section can be solved numerically by discretizing the computational domain using a proper meshing scheme. The original functional equation is reduced to a set of discretized linear algebraic equations over elementary cells. The unknown quantities are found by solving this system of linear equations, and many other parameters can be computed thereafter. This method of numerical solution of integral equations is known as the Method of Moments (MoM). In this method, the unknown electric current is represented by an expansion of basis functions as follows:
:<math> \mathbf{J(r)} = \sum_{n=1}^N {I_n}^{(J)} \mathbf{ {f_n}^{(J)}(r) }</math><!--[[File:07_numerical-solutions_tn.gif]]-->
where [[File:08_numerical-solution_tn.gif]] <math>\mathbf{ {f_n}^{(J)} }</math> are the generalized vector basis functions for the expansion of electric currents, and [[File:09_numerical-solution_tn.gif]] <math>{I_n}^{(J)}</math> are the unknown complex amplitudes of these basis functions, which have to be determined. Substituting these expansions yields the following discretized integral equation:
:<math>\mathcal{L}_E \left( \mathbf{E^i} +\iiint_V \mathbf{ \overline{\overline{G}}_{EJ}(r|r') } \cdot \sum_{n=1}^N {I_n}^{(J)} \mathbf{ {f_n}^{(J)}(r') } \, d\nu' \right) = 0</math><!--[[File:10_numerical-solution_tn.gif]]-->
In order to solve the above equation, the method of moments uses Galerkin's technique to turn it into a set of linear algebraic equations. This is accomplished by testing the above equations using the basis functions, leading to the following linear system:
:<math>\mathbf{[Z] \cdot [I] = [V]}</math><!--[[File:11_numerical-solution_tn.gif]]-->
where
:<math>Z_{ij} = \iiint_{V_i} \mathbf{ {f_i}^{(J)}(r) } \, d\nu \cdot \iiint_{V_j} \mathbf{ \overline{\overline{G}}_{EJ}(r|r') \cdot {f_j}^{(J)}(r') } \, d\nu'</math><!--[[File:12_numerical-solution_tn.gif]]-->
and
:<math> V_i = \iiint_{V_i} \mathbf{ {f_i}^{(J)}(r) \cdot E^i(r) } \, d\nu </math><!--[[File:13_numerical-solution_tn.gif]]-->
Using a rooftop expansion of the currents on the wires, we can discretize the Pocklington integral equation. In order to convert the discretized integral equation into a system of linear system of algebraic equations, we use Galerkin’s testing process, in which the testing functions are chosen to be identical to the expansion basis functions. However, to avoid the source singularity at r=r’, the expansion functions are placed at the center of the wires, while the test functions are evaluated on the surface of the wires, assuming a finite non-zero radius for all wires. The solution vector [I] is then found as:
:<math>\mathbf{[I] = [Z]^{-1} \cdot [V] } </math><!--[[File:24_galerkin_tn.gif]]-->
where [Z]<sup>-1 </sup> is the inverse of the impedance matrix and [V] is the excitation vector.
=== Pocklington’s Integral Equations for Wire Structures ===