Obviously such a decomposition is not possible for an arbitrary function h(x,t). Could you provide some background on how you obtained the function h? Do you have an explicit form for it?

Depending on theoretical and practical considerations for h(x,t), you might be able to use the convolution theorem from Fourier analysis. With g'(x) defined as g(-x) your proposed form becomes a convolution which (perhaps) can be integrated.

Specifically, if we define k(t) = integral(h(x,t), (x,-infinity,infinity)) then the Fourier transform of k is the product of the transforms of f and g'. For this to be computationally tractable you need to be able to compute k, then its FT, then factor that into a product of two functions that have (inverse) FTs, then compute those IFTs.

Before this practical consideration there is of course the theoretical one of whether these various integrals and integral transforms even exist, for your given h(x,t). So, as noted in a prior response, things really (as in "entirely") depend on specifics of that function.

Really all you need to do is to factor h(x,t).

If you are looking for verifiable criteria whether h(x,t) possesses such a factorisation and a procedure to obtain it, the following might be useful:

I. case: a is not given

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(I only realised that you asked for the easier case "a given", when I had already written this; for "a given" s. below)

Under the assumption (A) h(x,t)=g(x) f(x-at) the quotient

(1) q(x,t):=\frac{\partial h(x,t)}{t} / h(x,t) = - a f'(x-at) /f(x-at)

is a function that depends on (x-at) only, i.e.

(2) q(x,t)=p(x-at).

(P) The gradient of q(x,t) therefore would be a vector field p'(x-at) (1, -a) with parallel directions (1,-a).

If the condition (P) is not satisfied, the assumption was wrong,

if it is, the vector field yields the hitherto unknown a.

(C) Again, if the assumption is correct q(x,t) must be constant for x-at=const.

If (C) is satisfied, p(z)=q(z,0) satisfies (2). In the next step (1) yields a differential

equation for f:

(3) -a f'(z)/ f(z)=p(z)

In the last step one would obtain g.

II. case: a is given.

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In this case one can either directly verify (2) and continue as in (3) or rather consider

(4) r(t)=h(0,t)/h(0,0)

Under the assumption (A) we have

r(t)=c f(-at), i.e. f(z)=r(-z/a)

(neglecting the constant, which can be arbitrarily chosen).

Then one can check whether h(x,t)/f(x-at) depends on x only, and, if so, obtain g(x); if not, (A) was wrong.

a very simple test is the following one

suppose the décomposition exists for a given a

then you compute the linear combination of the partial derivatives

L=h'_t+ah'_x =a g'(x)f(x-at)

so you see that if you divide h(x,t)=g(x)f(x-at) by this combination you find :

L(x,t)/h(x,t)= a g'(x)/g(x)

which does not depend of t : so this is a necessary condition (it gives you a test)

more over observe that this quantity gives you the derivative of ln |g(x)|

so you have the possibility to find g and consequently f .

In fact i wanted to say :

if you dividey this combination by h(x,t)=g(x)f(x-at)....

That's not possible in general.

Introduce a new variable, y=x-at. The transition from (x,t) to (x,y) is linear invertible unless a is zero. Let F(x,y)=h(x,t). If F splits to a product g(x)f(y), then its logarithm satisfies the differential equation \partial_x\partial_y log F(x,y)=0. The converse is also true. You can rewrite this criterion back in terms of partial derivatives in (x,t). Of course, you should assume differntiability etc.

Without assumption of differentiability, fix some y_1. If F(x,y)=f(x)g(y), then F(x,y)/F(x,y_1) depends only on y and is equal to g(y)/g(y_1). Conversely, suppose that F(x,y)/F(x,y_1) is independent of x. Denote it h(y). Then F(x,y)= F(x,y_1) h(y) is a factorization. Since the transition from (x,t) to (x,y) is equivalence, this is the complete solution. Again, you can get back to the old coordinates.

Thank you for all your answers and sorry for this very late reply. I have just read your contributions, it is the first time I use this question feature in ResearchGate and I thought I would be informed of your contributions by email.

To better define the problem, the constant a is given and positive (it is a rotational velocity) and all functions are differentiable. This problem comes from electromagnetics of rotating machines. A key quantity is the airgap (space between rotor and stator) flux density distribution which depends on time t and angle x. In some cases the time simulation in FEM is very long, and it does not necessarily "add" some information: all the physical quantities of the rotor simply rotates. The flux density is decomposed in a permeance function that account for slotting effects, and a magnetomotive force function that account for field sources (e.g. rotor magnets). My aim is to build a simulation strategy to obtain the full flux density distribution without making a simulation for every timestep. I was working on a test based on convolution, but I am going to study your other proposals.