Peace be upon you,
I have the following system of partial differential equations \begin{align*} \begin{cases} \frac{\partial}{\partial a}S(a,b,c,d)=f_1(a)\\ \frac{\partial}{\partial b}S(a,b,c,d)=f_2(b)\\ \frac{\partial}{\partial c}S(a,b,c,d)=f_3(c)\\ \frac{\partial}{\partial d}S(a,b,c,d)=f_4(d)\\ \end{cases} \end{align*} where $f_i()$s are some nonlinear functions.
Does the above system have a unique answer(?) and if has can any one introduce a reference, explaining the techniques for analytic solutions?
Note: The usual PDE references (books, articles, webpages, etc.) speak about the systems for which the number of unknown functions and the number of system equations are equal.
Your problem can be reformulated as follows (upon the change of notation $a=x, b=y, c=z, d=t$).
You are assigning the differential form $$ \omega=f_1(x)dx +f_2(y) dy + f_3(z)dz+f_4(t)dt, $$ which is closed, hence exact on $\mathbb{R}^4$. You want to find a potential function, that is, a function $F=F(x, y, z, t)$ such that $dF=\omega$. One of them is given by the following integral, the others differ by an additive constant: $$ F(x,y,z,t)=\int_\gamma \omega,\qquad \gamma\colon[0, 1]\to \mathbb{R}^4,\ \gamma(0)=0,\ \gamma(1)=(x,y,z,t)$$ You can choose any curve $\gamma$, only its endpoints matter. Taking for instance the line segment $$\gamma(s)=(sx,sy,sz,st),$$ you obtain $$ F(x,y,z,t)=\int_0^1 f_1(sx)x\, ds + \int_0^1 f_2(sy)y\, ds+\int_0^1 f_3(sz)z\, ds+\int_0^1 f_4(st)t\, ds.$$