How Do You Solve This Complex Partial Differential Equation?

[femiadeyemi]

Hi All,
Please I need your assistance to solve this PDE below:

$\frac{\partial^2 X}{\partial t^2} - \frac{\partial^2 X}{\partial z^2} + a(z,t) \frac{\partial X}{\partial t} + b(z,t) \frac{\partial X}{\partial z} +c(z,t) X =\Phi(z,t)$

With initial and boundary condition:
$X(z,0)=\frac{\partial X(z,0)}{\partial t}=0$
$X(0,t)=X(L,t)=0$

Thank you in advance.
FM

 

[Chestermiller]

Are you trying to solve an equation like this analytically or numerically? Solving it numerically isn't too difficult, but solving it analytically is.

 

[femiadeyemi]

Thank you for your response. I want to solve it analytically

Are you trying to solve an equation like this analytically or numerically? Solving it numerically isn't too difficult, but solving it analytically is.

 

[berkeman]

Hi All,
Please I need your assistance to solve this PDE below:

$\frac{\partial^2 X}{\partial t^2} - \frac{\partial^2 X}{\partial z^2} + a(z,t) \frac{\partial X}{\partial t} + b(z,t) \frac{\partial X}{\partial z} +c(z,t) X =\Phi(z,t)$

With initial and boundary condition:
$X(z,0)=\frac{\partial X(z,0)}{\partial t}=0$
$X(0,t)=X(L,t)=0$

Thank you in advance.
FM

Are you trying to solve an equation like this analytically or numerically? Solving it numerically isn't too difficult, but solving it analytically is.

Thank you for your response. I want to solve it analytically

What is this equation from? Is this for schoolwork, or research, or other?

 

[femiadeyemi]

It's research

What is this equation from? Is this for schoolwork, or research, or other?

 

[SteamKing]

If it's research, shouldn't you be doing some of the research?

 

[dextercioby]

Doing some (not too in depth) reasearch in the field of solving PDE's doesn't leave with too many options to try to find a solution. The method of characteristics or separating variables should be the first ones you should try. Whether a (preferably closed form and expressible in terms of known special functions) solution can be found is directly dependent on the fact that the 4 coefficient functions have a 'nice', i.e. preferable constant form, so that the PDE would have the smallest possible non-linearity (even though, as written, it's classifield as linear).

Either way, your best research is done with a smart computer software such as Mathematica.