Solve analytically the following coupled PDEs: \begin{cases} \partial_t F(x,t) = -2\alpha_0 x \left[ \partial_x G(x,t)-\partial_x^2 G(x,t) \right], \[6pt] \partial_t G(x,t) = -2\alpha_0 x \left[ \partial_x F(x,t)+\partial_x^2 F(x,t) \right]. \tag{1} \end{cases} with a domain x∈[x0,∞),x0≡c+1,α0=constant,c>12.x\in[x_0,\infty), \qquad x_0\equiv c+1, \qquad \alpha_0=\text{constant}, \qquad c>\frac{1}{2}. The initial conditions are F(x,0)= \begin{cases} 1, & x=x_0,\\ 0, & x>x_0, \end{cases} \qquad \text{and} \qquad \left.G(x,0)\right|_{x>x_0}=0. \tag{2} The boundary conditions are: F(x_0,t)=1, \tag{3} G(x_0,t)=0, \tag{5} \lim_{x\to\infty}F(x,t)=0, \tag{7} \lim_{x\to\infty}G(x,t)=0. \tag{8} Goal: Find explicit analytical solutions of the PDEs (1) satisfying the appropriate initial and boundary conditions, i.e., determine the expressions for F(x,t)F(x,t) and G(x,t)G(x,t) .