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Why is the method of separation of variables is the only method used in physics for solving partial differential equations despite that they are not the most general solutions? Do these solutions form some kind of complete set? Is there a proof?

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    $\begingroup$ It is not the only method, it is elementary, for linear equations general solutions can often be written as (infinite) sums of those with separated variables, it is a generalized version of Fourier series. $\endgroup$ – Conifold Mar 30 '19 at 6:01
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    $\begingroup$ Physicists and engineers love mathematical models where the assumption of linear superposition is possible as they are simple and easy to understand. $\endgroup$ – James Arathoon Mar 30 '19 at 10:22
  • $\begingroup$ @Conifold "it is a generalized version of Fourier series" Do the product solutions form a complete set? In Fourier series, we know that functions satisfying the Dirichlet conditions can be expanded into functions of sines and cosines which form the basis. Do product solutions form a basis? $\endgroup$ – mithusengupta123 May 22 '19 at 4:43
  • $\begingroup$ They typically do in an appropriate space, technical conditions that ensure that depend on a problem. $\endgroup$ – Conifold May 22 '19 at 7:49
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Because this method will work. It reduces many partial differential equations down to ordinary differential equations and if we can solve those then we’re in business and the method will allow us to get a solution to the partial differential equations.

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It's definitely not the only method. Separation of variables is convenient to use if the domain is finite, i.e. $x\in[0,1]$.

If the domain is inifnite, i.e. $x\in[0,\infty)$, the fundamental solution is often used instead.

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  • $\begingroup$ But such product solutions are not general solutions. For example, $F(x,y)=c_1x+c_2y$ is a solution of 2D Laplace equation and it is not of the product form as assumed in separation of variables technique $\endgroup$ – mithusengupta123 Mar 30 '19 at 8:23
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    $\begingroup$ But they are! Any continuous bounded function in a finite domain can be expressed as a sum of Fourier series, including the one you listed. Separation of variables doesn't just give you a product solution, it gives you a series solution. $\endgroup$ – Dylan Mar 30 '19 at 14:13
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    $\begingroup$ @Dylan You say $[0,1]$ is "finite", $[0,+\infty)$ is "infinite". This is not the convenient adjectives : $[0,1]$ is bounded, $[0,+\infty)$ is not bounded. Besides $\{0,1\}$ is a finite set with 2 elements. $\endgroup$ – Jean Marie Apr 27 '19 at 8:55
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Fourier's separation of variables technique was one of the earliest methods for solving linear partial differential equations. A good number of classical problems such as the Laplace equation, the Heat equation, and the wave equation can be solved this way, and the solutions are explicit. (By the way, $x+y = x\cdot 1 + 1\cdot y$ is a separated function.) There are not a lot of other techniques that can be used to find explicit solutions; so the technique remains useful for gaining theoretical and practical information about these classical equations of Engineering and Physics.

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It is also, in connection with PDFs (Probability Density Functions) a reason of independence for example in the simplest case where the variables are $x$ (space) and $t$ (time) when we know that there is no connection between space and time.

Indeed a PDF $f(x,t)$ can be written $f_1(x)f_2(t)$ if and only if the associated Random Variables $X$ and $T$ are independent. And this is frequently the case.

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