Combination of variables is the technique that reducing the PDE to one independent variables (i.e. become ODE) by introducing a suitable change of variables.
But the general solution of a PDE should contain arbitrary functions as the arbitrary parts, while the general solution of an ODE only have arbitrary constants as the arbitrary parts, does it mean that solving PDE by combination of variables always cannot find the general solutions?
Saying typically, combination of variables is the technique described in the following references:
- Combination of Variable Solutions
- Solution of Partial Differential Equations - Combination of Variables
- Second-order PDE
For the example in Combination of Variable Solutions, it claims that solving $\dfrac{\partial\theta}{\partial t}=\gamma\dfrac{\partial^2\theta}{\partial x^2}$ with $t>0$ , $0<x<\infty$ , $\theta(x,0)=\theta_0$ , $\theta(0,t)=\theta_1$ , $\theta(\infty,t)=\theta_0$ by combination of variables is $\theta(x,t)=\theta_1+(\theta_0-\theta_1)\text{erf}\left(\dfrac{x}{\sqrt{4\gamma t}}\right)$ .
I discover that when solving the same problem by separation of variables, i.e. Random diffusion coefficient in the Fourier equation with replacing $u$ by $\theta$ , $u_0$ by $\theta_1$ , $u_L$ by $\theta_0$ , $k(t)$ by $\gamma$ , the solution is $\theta(x,t)=\theta_1+\int_0^\infty\dfrac{2(\theta_0-\theta_1)e^{-\gamma ts^2}\sin xs}{\pi s}ds=\theta_1+(\theta_0-\theta_1)\text{erf}\left(\dfrac{x}{\sqrt{4\gamma t}}\right)$ . This seems show that solving PDE by combination of variables takes the same effect to solving PDE by separation of variables.
But when the problem is replacing to $\dfrac{\partial\theta}{\partial t}=\gamma\dfrac{\partial^2\theta}{\partial x^2}$ with $t>0$ , $0<x<\infty$ , solve it by combination of variables Combination of Variable Solutions claims that $\theta(x,t)=I_1\text{erf}\left(\dfrac{x}{\sqrt{4\gamma t}}\right)+I_2$ , but solve it by separation of variables Random diffusion coefficient in the Fourier equation claims that $\theta(x,t)=C_1x+C_2+\int_0^\infty C_3(s)e^{-\gamma ts^2}\sin xs~ds+\int_0^\infty C_4(s)e^{-\gamma ts^2}\cos xs~ds$ . This seems that knowing whether solving PDE by combination of variables and solving PDE by separation of variables take the same effect or not are difficult.
So how should I recognize those issues? In fact $I_1$ and $I_2$ are arbitrary constants or arbitrary functions? If they are arbitrary functions, i.e. $I_1=I_1(f(x,t))$ and $I_2=I_2(g(x,t))$ , $f(x,t)$ and $g(x,t)$ are equal to what?