I seen two variations of the Black-Scholes PDE with either $+{\frac {\partial V}{\partial t}}$ or $-{\frac {\partial V}{\partial t}}$, and wanted to ask why that is?
a) https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_equation#Solving_the_PDE $${\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0$$
b) https://www.quantstart.com/articles/C-Explicit-Euler-Finite-Difference-Method-for-Black-Scholes $$-{\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0$$
Please see below derivation of Black-Scholes PDE:
\begin{align*} \Pi &= C(S,t)+\Delta S\\ d\Pi &=dC(S,t)+\Delta dS\\ &= C_S dS+C_t dt + \frac{1}{2}C_{SS}d[S]+ \frac{1}{2}C_{tt}d[t]+ C_{St}d[S,t]+\Delta dS\\ &= C_S(S\mu dt+S\sigma dW)+C_tdt+\frac{1}{2}C_{SS}S^2\sigma^2dt+\Delta (S\mu dt+S\sigma dW)\\ &= (\Delta S\sigma+C_SS\sigma)dW+(C_SS\mu+C_t+\frac{1}{2}C_{SS}S^2\sigma^2+\Delta S\mu)dt \end{align*} Hedge: \begin{align*} d\Pi&\stackrel{!}{=}\Pi rdt\Rightarrow \Delta = -C_S:\\ &\Pi rdt= (C_t+\frac{1}{2}C_{SS}S^2\sigma^2)dt\\ &\Leftrightarrow(C-C_S S) rdt= (C_t+\frac{1}{2}C_{SS}S^2\sigma^2)dt \end{align*} $$\Rightarrow C_t+\frac{1}{2}C_{SS}S^2\sigma^2-r(C-C_S S)= 0$$