Heat eqn with a variable coeff: given u_t=u_(xx)-2xu_x+sinh 3x

[armez]

Given an initial value problem for the eqn u_t=u_xx-2xu_x+sinh 3x with the zero initial value.
Is it possible to manage the term 2xu_x ?

 

[stapel]

Given an initial value problem for the eqn u_t=u_xx-2xu_x+sinh 3x with the zero initial value.
Is it possible to manage the term 2xu_x ?

What, specifically, do you mean by "managing" this term?

When you reply, please provide the full and exact text of the exercise, the complete instructions, and a clear listing of your efforts so far. Thank you!

 

[armez]

Dear stapel, thank you for your attention to my question!
I've no exact text for this task other than to solve the Cauchi problem I've formulated.
Could you please specify your question regarding the problem?
Should I explain what is the zero initial value?
Well, it means that the solution u(x,t) satisfy the condition u(x,0)=0.
As regards "to manage" the term 2xu_x, I just wanted to emphasize that
this is the only term making the problem non-trivial. For example, the
equation like u_t=u_xx+sinh(3x) is a standard nonhomogenous heat
conduction equation and there is no problem to find a solution for the
Cauchi problem with the zero initial value for this equation. Moreover, it would
be quite easy to solve the same problem for the equation with a constant
coefficient like u_t=u_xx-2u_x+sinh(3x) by reducing it to the equation above.
So, my question is which method is applicable in the case when the
coefficient is variable.
I've no instructions regarding the problem. That is the reason why I've
posted my question. Please, note that I don't need an exact or complete
solution of this problem. Just a hint would be fine.

 

[Deleted member 4993]

Dear stapel, thank you for your attention to my question!
I've no exact text for this task other than to solve the Cauchi problem I've formulated.
Could you please specify your question regarding the problem?
Should I explain what is the zero initial value?
Well, it means that the solution u(x,t) satisfy the condition u(x,0)=0.
As regards "to manage" the term 2xu_x, I just wanted to emphasize that
this is the only term making the problem non-trivial. For example, the
equation like u_t=u_xx+sinh(3x) is a standard nonhomogenous heat
conduction equation and there is no problem to find a solution for the
Cauchi problem with the zero initial value for this equation. Moreover, it would
be quite easy to solve the same problem for the equation with a constant
coefficient like u_t=u_xx-2u_x+sinh(3x) by reducing it to the equation above.
So, my question is which method is applicable in the case when the
coefficient is variable.
I've no instructions regarding the problem. That is the reason why I've
posted my question. Please, note that I don't need an exact or complete
solution of this problem. Just a hint would be fine.

Did you try series solution?

 

[armez]

Did you try series solution?

Of course. After the separation of the variables, I've got a Sturm-Liouville problem for the Hermite equation. It is well known that this problem has the Hermite functions as the eigenfunctions satisfying an additional condition at the infinity - a special exponent-power asymptotic estimate.

However, there is no such condition in the main problem. The uniqueness theorem says that no additional condition is required. Moreover, I think, to calculate the expansion coefficients for the function sinh(3x) would be a problem.