- #1

Safinaz

- 259

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- Homework Statement
- I try to solve Klein Gordon’s equation for specific boundary and initial conditions

- Relevant Equations
- The Klein Gordon’s equation for a masses scalar is given by :

## \left( \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} \right) \phi (x, t) = 0 ##………(1)

My solution:

Let ## \phi (x, t) = F(x) A(t) ##, then Eq. (1) becomes

##

\frac{1}{A(t)} \frac{\partial^2}{\partial t^2} - \frac{1}{F(x)} \frac{\partial^2}{\partial x^2} = 0

##

So that : ## \frac{\partial^2}{\partial t^2} = k^2 ~A (t)##, and ## \frac{\partial^2}{\partial x^2} = k^2 ~F (x)##.

Leads to :

##

\phi(t,x) = ( c_1 e^{kt} + c_2 e^{-kt} ) ( c_3 e^{kx} + c_4 e^{-kx} )

##

Assuming BC and IC :

##

bc={\phi[t,0]==1,(D[\phi[t,x],x]/.x->Pi)==0}

##

##

ic={\phi[0,x]==0,(D[\phi[t,x],t]/.t->0)==1}

##

BC leads to ##c_3 = c_4= 1/2 ## and the IC leads to to ##c_1=- c_2= 1/2 ##.

Ending up by :

##

\phi(t,x) = \frac{1}{4} e^{k(t-x)} - \frac{1}{4} e^{-k(t-x)}……… (2)

##

So any help are these steps correct till Eq. (2) ? And how to determine ##k##?

Let ## \phi (x, t) = F(x) A(t) ##, then Eq. (1) becomes

##

\frac{1}{A(t)} \frac{\partial^2}{\partial t^2} - \frac{1}{F(x)} \frac{\partial^2}{\partial x^2} = 0

##

So that : ## \frac{\partial^2}{\partial t^2} = k^2 ~A (t)##, and ## \frac{\partial^2}{\partial x^2} = k^2 ~F (x)##.

Leads to :

##

\phi(t,x) = ( c_1 e^{kt} + c_2 e^{-kt} ) ( c_3 e^{kx} + c_4 e^{-kx} )

##

Assuming BC and IC :

##

bc={\phi[t,0]==1,(D[\phi[t,x],x]/.x->Pi)==0}

##

##

ic={\phi[0,x]==0,(D[\phi[t,x],t]/.t->0)==1}

##

BC leads to ##c_3 = c_4= 1/2 ## and the IC leads to to ##c_1=- c_2= 1/2 ##.

Ending up by :

##

\phi(t,x) = \frac{1}{4} e^{k(t-x)} - \frac{1}{4} e^{-k(t-x)}……… (2)

##

So any help are these steps correct till Eq. (2) ? And how to determine ##k##?

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