- #1

chwala

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- Homework Statement
- Kindly see attached;

- Relevant Equations
- separation of variables, in essence breaking down a pde into two ordinary differential equations.

I am reading on this part; and i realize that i get confused with the 'lettering' used... i will use my own approach because in that way i am able to work on the pde's at ease and most importantly i understand the concept on separation of variables and therefore would not want to keep on second guessing on the 'letters used' in the textbook notes ...as attached below;

This is (my approach) steps to the working...

##U_t ##= ##k U_{xx}##

Let ##U(x,t) ##= ##X(x) T(t)##

Therefore,

##XT^{'}##=## kX^{''}T##

##\frac{T^{'}}{kT} ##=##\frac{X^{''}}{X} ##=##-λ##

From here we have one initial condition,i.e ##u(x,0)##=##f(x)## and two spatial boundary conditions, i.e ##u_x(0,t)=0## and ##u_x(L,t)=0##

Therefore using the Boundary conditions, we shall have,

##u_x##= ##X'T##

##0##=##X'(0)T(t)## and also, ##0= X'(L)T(t)##

If ## T(t)=0##, then we shall have a trivial solution and therefore for a non- trivial solution to be realized we need to have,

##X'(0)=0## →##X'(L)=0##

and therefore our problem is reduced to,

##T'+λkT=0## and ##X^{''} + λX=0##

##U_x(0)=0##

##U_x(L)=0##

I intend to use this approach in problems related to pde...regards,

This is (my approach) steps to the working...

##U_t ##= ##k U_{xx}##

Let ##U(x,t) ##= ##X(x) T(t)##

Therefore,

##XT^{'}##=## kX^{''}T##

##\frac{T^{'}}{kT} ##=##\frac{X^{''}}{X} ##=##-λ##

From here we have one initial condition,i.e ##u(x,0)##=##f(x)## and two spatial boundary conditions, i.e ##u_x(0,t)=0## and ##u_x(L,t)=0##

Therefore using the Boundary conditions, we shall have,

##u_x##= ##X'T##

##0##=##X'(0)T(t)## and also, ##0= X'(L)T(t)##

If ## T(t)=0##, then we shall have a trivial solution and therefore for a non- trivial solution to be realized we need to have,

##X'(0)=0## →##X'(L)=0##

and therefore our problem is reduced to,

##T'+λkT=0## and ##X^{''} + λX=0##

##U_x(0)=0##

##U_x(L)=0##

I intend to use this approach in problems related to pde...regards,

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