Meromorphic solutions of Bi-Fermat type partial differential and difference equations

Fermat type functional equation with four terms \(\begin{aligned} f(z)^{n}+g(z)^{n}+h(z)^{n}+k(z)^{n}=1 \end{aligned}\) $$\begin{array}{c}\hfill f{(z)}^n+g{(z)}^n+h{(z)}^n+k{(z)}^n=1\end{array}$$ is difficult to solve completely even if \(n=2,3\) $n=2,3$ , in which the certain type of the above equation is also interesting and significant. In this paper, we first to consider the Bi-Fermat type quadratic partial differential equation \(\begin{aligned} f(z_{1},z_{2})^{2}+\left( \frac{\partial f(z_{1},z_{2})}{\partial z_{1}}\right) ^{2}+g(z_{1},z_{2})^{2}+\left( \frac{\partial g(z_{1},z_{2})}{\partial z_{1}}\right) ^{2}=1 \end{aligned}\) $$\begin{array}{c}\hfill f{(z_1,z_2)}^2+{\left(\frac{\partial f(z_1,z_2)}{\partial z_1}\right)}^2+g{(z_1,z_2)}^2+{\left(\frac{\partial g(z_1,z_2)}{\partial z_1}\right)}^2=1\end{array}$$ in \(\mathbb {C}^{2}\) ${\mathbb{C}}^2$ . In addition, we consider the Bi-Fermat type cubic difference equation \(\begin{aligned} f(z)^{3}+g(z)^{3}+f(z+c)^{3}+g(z+c)^{3}=1 \end{aligned}\) $$\begin{array}{c}\hfill f{(z)}^3+g{(z)}^3+f{(z+c)}^3+g{(z+c)}^3=1\end{array}$$ in \(\mathbb {C}\) $\mathbb{C}$ and obtain partial meromorphic solutions on the above equation.