How to determine the integration constants in solving the Klein Gordon equation?

[Safinaz]

Homework Statement

How to solve the following wave equation for the scalar $\phi(t,x)$ :

Relevant Equations

$\partial_i \dot{\phi}=0$

Where $\partial_i= \partial/\partial x$. And (.) is the derivative with respect for time $\partial/\partial t$

I solved by

$\int d \dot{\phi} = \int d x \to \dot{\phi} = x+ c_1 \to \int d \phi = \int d t ( x+c_1) \to \phi = x t + c_1 t + c_2$

Is this way correct? To determine $c_2$ use initial condition: $\phi(0,x)=0$ that yields $c_2=0$, but how to get $c_1$ ?

 

[pasmith]

The kernel of the operator $\partial_x$ consists of all functions $f$ for which $\partial_x f = 0$; this includes constants, but also includes functions which depend only on $t$. Therefore the most general element of this kernel is $A(t)$. Hence $$\int \partial_x \dot \phi\,dx = \int 0\,dx \Rightarrow \dot\phi = A(t).$$ Now integrate with respect to $t$. This time, the most general element of the kernel of $\partial/\partial t$ is a function of $x$ alone.

 

[Orodruin]

Homework Statement: How to solve the following wave equation for the scalar $\phi(t,x)$ :
Relevant Equations: $\partial_i \dot{\phi}=0$

Where $\partial_i= \partial/\partial x$. And (.) is the derivative with respect for time $\partial/\partial t$

I solved by

$\int d \dot{\phi} = \int d x \to \dot{\phi} = x+ c_1 \to \int d \phi = \int d t ( x+c_1) \to \phi = x t + c_1 t + c_2$

Is this way correct? To determine $c_2$ use initial condition: $\phi(0,x)=0$ that yields $c_2=0$, but how to get $c_1$ ?

Your function provably does not satisfy $\partial_i\partial_t \phi = 0$. Just try to differentiate it!

(apart from what was already said)

 

[Safinaz]

The kernel of the operator $\partial_x$ consists of all functions $f$ for which $\partial_x f = 0$; this includes constants, but also includes functions which depend only on $t$. Therefore the most general element of this kernel is $A(t)$. Hence $$\int \partial_x \dot \phi\,dx = \int 0\,dx \Rightarrow \dot\phi = A(t).$$ Now integrate with respect to $t$. This time, the most general element of the kernel of $\partial/\partial t$ is a function of $x$ alone.

You mean

$\dot{\phi} =A(t) \to \int \partial_t \phi = \int A(t) dt \to \phi = A(t) t + c ?$
but what is A(t) ?

 

[Orodruin]

You mean

$\dot{\phi} =A(t) \to \int \partial_t \phi = \int A(t) dt \to \phi = A(t) t + c ?$
but what is A(t) ?

First of all, you cannot integrate a general function A(t) with respect to t and obtain A(t) t. Not even in single variable calculus.

Second, you are still missing what was said. If $\partial_t \phi = A(t)$, then $\phi = a(t) + f(x)$, where $a’(t) = A(t)$. The “integration constant” when integrating a partial derivative is generally a function of all of the other variables.

 

[Safinaz]

First of all, you cannot integrate a general function A(t) with respect to t and obtain A(t) t. Not even in single variable calculus.

Second, you are still missing what was said. If $\partial_t \phi = A(t)$, then $\phi = a(t) + f(x)$, where $a’(t) = A(t)$. The “integration constant” when integrating a partial derivative is generally a function of all of the other variables.

Okay. But now how to get the definition of $f(x)$ and $a(t)$ ?

 

[Orodruin]

Okay. But now how to get the definition of $f(x)$ and $a(t)$ ?

Just as you need boundary or initial conditions to fix integration constants for ODEs, you will need boundary/initial conditions to fix those functions.

 

[Safinaz]

Just as you need boundary or initial conditions to fix integration constants for ODEs, you will need boundary/initial conditions to fix those functions.

Hello. Thanks so much for your answer. I was trying to find proper IC and BC to find $\phi(t,x)$ . Assuming:

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

Also $\phi(t,x)$ obays the Klein Gordon’s equation :
$\left( \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} \right) \phi = 0$

in which solution:
$\phi(t,x) = ( c_1 e^{kt} + c_2 e^{-kt} ) ( c_3 e^{kx} + c_4 e^{-kx} ) …………(1)$

To find the constants in Eq. (1) , BC leads to $c_3 = c_4= 1/2$ and the IC leads to to $c_1=- c_2= 1/2$ is that correct? But how to know $k$ ?