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The Poisson-Boltzmann (PB) model is an effective approach for the electrostatics analysis of solvated biomolecules. The nonlinearity associated with the PB equation is critical when the underlying electrostatic potential is strong, but is extremely difficult to solve numerically. In this paper, we construct two operator splitting alternating direction implicit (ADI) schemes to efficiently and stably solve the nonlinear PB equation in a pseudo-transient continuation approach. The operator splitting...

We establish two truncations of Gauss’ square exponent theorem and a finite extension of Euler’s identity. For instance, we prove that for any positive integer $n$, $$\sum _{k=0}^n{(-1)}^k\left[\begin{array}{c}2n-k\\ k\end{array}\right]{(q;q^2)}_{n-k}{q}^{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)}=\sum _{k=-n}^n{(-1)}^k{q}^{k^2},$$ where $$\left[\begin{array}{c}n\\ m\end{array}\right]=\prod _{k=1}^m\frac{1-q^{n-k+1}}{1-q^k}\text{and}{(a;q)}_n=\prod _{k=0}^{n-1}(1-a{q}^k).$$