Difficulties in solving following PDE

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Discussion Overview

The discussion revolves around the challenges of solving a partial differential equation (PDE) of the form Uzz + f(t)*Uz = g(t)*Ut analytically, with specified boundary and initial conditions. Participants seek assistance in addressing the complexities involved in finding a solution that satisfies these conditions.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant requests clarification on the functions f(t) and g(t) to proceed with the solution.
  • Another participant suggests that if f(y) and g(y) are constants, the characteristics of the PDE can be analyzed using the discriminant b²-4ac.
  • A different participant asserts that while a formal expression for the general solution can be derived, the challenge lies in determining the arbitrary functions alpha() and beta() that meet the boundary conditions, especially since f(t) and g(t) are unspecified.
  • There is acknowledgment that even with specified functions, finding a solution that fits the boundary conditions may still be very difficult.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the path forward, as there are differing views on the implications of the unspecified functions f(t) and g(t) and the feasibility of finding a solution that meets the boundary conditions.

Contextual Notes

The discussion highlights limitations related to the unspecified nature of f(t) and g(t), which affects the ability to derive specific solutions. Additionally, the complexity of satisfying boundary conditions with the general solution remains unresolved.

mohammad449
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Dear Friends,
I encountered with some difficulties in solving following PDE (off course, analytically not numerically), so I would really appreciate it if you help me in this matter.
The PDE is: Uzz+f(t)*Uz=g(t)*Ut
where U(z,t), f(t), and g(t)

B.Cs and I.C are:
U(0,t)=b;
U(infinity,t)=a;
U(z,0)=a;
where a & b are constant.
I am looking forward to hearing from you,
Many Thanks,
Best Regards,
 
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It would help if you specify f(t) and g(t)
 


U zz+ f(y) ux = g(y)Uy
Uzz+ f(y)Ux-g(y) Uy=0
Since f(y) and g(y) are constants we can use b^2-4ac to determine the characteristics

don know whether am on right path
 


It isn't difficult to find a formal expression for the general solution of the PDE. (see attachment). As usual, the main difficulty is to find the solution fitting with the boundary conditions, among the infinity of solutions provided by the general formula.
As expected, the formal expression includes two arbitrary functions, namely alpha() and beta() with our natations.
The main problem will be to find what are thoses functions alpha() and beta() in order to fulfil the given boundary conditions. As far as the formula contains some functions, f(t) and g(t) which aren't specified, this is impossible. And even if they were specified, this would be probably very difficult, except by chance in some particular cases of f(t) and g(t).
 

Attachments

  • PDE.JPG
    PDE.JPG
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thanks for your consideration about this matter
 

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