Recursion-transform method on computing the complex resistor network with three arbitrary boundaries

Abstract

We perfect the recursion-transform method to be a complete theory, which can derive the general exact resistance between any two nodes in a resistor network with several arbitrary boundaries. As application of the method, we give a profound example to illuminate the usefulness on calculating resistance of a nearly $m\times n$ resistor network with a null resistor and three arbitrary boundaries, which has never been solved before since the Greens function technique and the Laplacian matrix approach are invalid in this case. Looking for the exact solutions of resistance is important but difficult in the case of the arbitrary boundary since the boundary is a wall or trap which affects the behavior of finite network. For the first time, seven general formulae of resistance between any two nodes in a nearly $m\times n$ resistor network in both finite and infinite cases are given by our theory. In particular, we give eight special cases by reducing one of general formulae to understand its application and meaning