Inhomogeneous Partial Differential Equations

[Arindam]

Given,

u _t - u_xx = xt ; 0 < x < Pi
u(x,0) = 1 ;
u_x (0,t) = 0, u_x(pi,t) = 0


How to find the solution u using separation of variables?

 

[Deleted member 4993]

Given,
u _t - u_xx = xt ; 0 < x < Pi
u(x,0) = 1 ; & u_x (0,t) = 0, & u_x(pi,t) = 0
How to find the solution u using separation of variables?

start with assumption u(x,t) = g(x) * h(t) ← this is a standard assumption for PDE.

Please show us what you have tried and exactly where you are stuck.

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Please share your work/thoughts about this problem

 

[Arindam]

I have no problem with the homogeneous part and for the inhomogeneous part, I opt for the change of function such that u(x,t) = 1/2 [t²]x + v(x,t)
and then v_t - v_xx =0 and then there is similar changes in the initial conditions. But what would be my subsequent steps?

 

[Deleted member 4993]

I have no problem with the homogeneous part and for the inhomogeneous part, I opt for the change of function such that u(x,t) = 1/2 [t²]x + v(x,t)
and then v_t - v_xx =0 and then there is similar changes in the initial conditions. But what would be my subsequent steps?

Instead of "using words" - please share your mathematical work.

 

[Arindam]

Kindly check!