Checking the representation formula of differential operator $square -b$.
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I am trying to do a very simple check of the representation formula of the
equation $square u = bu$. I am setting $b=1$.
I set $u(x,t) =cos x$, a stationary solution. The representation of $u(x,t)$, according to Polyanin (this book) when $b<0$ is
$$ u(x,t) =frac12[f(x+t) + f(x-t)] + fract2int_x-t^x+tfracI_1(sqrtt^2 - (x- xi)^2)sqrtt^2 - (x- xi)^2f(xi)dxi$$
Here $f(x) = u(x,0)$, and $I_1$ is the modified Bessel function. I omit the further terms expressing the initial velocity and the inhomogeneous forcing because both of these are zero, if $u(x,t) = cos(x)$.
My problem is that I cannot see how this works. I have done numerical simulations and cannot get $cos(x)$ out of the RHS (it diverges). However, if I use the representation formula for the inhomogeneous wave equation (d'Alembert's formula), everything obviously works:
$$ cos(x) =frac12[cos(x+t) + cos(x-t)] + frac12int_0^tint_x-s+t^x+s-tcos(xi)dxi ds$$
This integral is easy and is true.
So where am I going wrong?
integration pde wave-equation
asked Aug 9 at 15:28
Wapiti
23318
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up vote
0
down vote
favorite
I am trying to do a very simple check of the representation formula of the
equation $square u = bu$. I am setting $b=1$.
I set $u(x,t) =cos x$, a stationary solution. The representation of $u(x,t)$, according to Polyanin (this book) when $b<0$ is
$$ u(x,t) =frac12[f(x+t) + f(x-t)] + fract2int_x-t^x+tfracI_1(sqrtt^2 - (x- xi)^2)sqrtt^2 - (x- xi)^2f(xi)dxi$$
Here $f(x) = u(x,0)$, and $I_1$ is the modified Bessel function. I omit the further terms expressing the initial velocity and the inhomogeneous forcing because both of these are zero, if $u(x,t) = cos(x)$.
My problem is that I cannot see how this works. I have done numerical simulations and cannot get $cos(x)$ out of the RHS (it diverges). However, if I use the representation formula for the inhomogeneous wave equation (d'Alembert's formula), everything obviously works:
$$ cos(x) =frac12[cos(x+t) + cos(x-t)] + frac12int_0^tint_x-s+t^x+s-tcos(xi)dxi ds$$
This integral is easy and is true.
So where am I going wrong?
integration pde wave-equation
asked Aug 9 at 15:28
Wapiti
23318
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to do a very simple check of the representation formula of the
equation $square u = bu$. I am setting $b=1$.
I set $u(x,t) =cos x$, a stationary solution. The representation of $u(x,t)$, according to Polyanin (this book) when $b<0$ is
$$ u(x,t) =frac12[f(x+t) + f(x-t)] + fract2int_x-t^x+tfracI_1(sqrtt^2 - (x- xi)^2)sqrtt^2 - (x- xi)^2f(xi)dxi$$
Here $f(x) = u(x,0)$, and $I_1$ is the modified Bessel function. I omit the further terms expressing the initial velocity and the inhomogeneous forcing because both of these are zero, if $u(x,t) = cos(x)$.
My problem is that I cannot see how this works. I have done numerical simulations and cannot get $cos(x)$ out of the RHS (it diverges). However, if I use the representation formula for the inhomogeneous wave equation (d'Alembert's formula), everything obviously works:
$$ cos(x) =frac12[cos(x+t) + cos(x-t)] + frac12int_0^tint_x-s+t^x+s-tcos(xi)dxi ds$$
This integral is easy and is true.
So where am I going wrong?
integration pde wave-equation
asked Aug 9 at 15:28
Wapiti
23318
I am trying to do a very simple check of the representation formula of the
equation $square u = bu$. I am setting $b=1$.
I set $u(x,t) =cos x$, a stationary solution. The representation of $u(x,t)$, according to Polyanin (this book) when $b<0$ is
$$ u(x,t) =frac12[f(x+t) + f(x-t)] + fract2int_x-t^x+tfracI_1(sqrtt^2 - (x- xi)^2)sqrtt^2 - (x- xi)^2f(xi)dxi$$
Here $f(x) = u(x,0)$, and $I_1$ is the modified Bessel function. I omit the further terms expressing the initial velocity and the inhomogeneous forcing because both of these are zero, if $u(x,t) = cos(x)$.
My problem is that I cannot see how this works. I have done numerical simulations and cannot get $cos(x)$ out of the RHS (it diverges). However, if I use the representation formula for the inhomogeneous wave equation (d'Alembert's formula), everything obviously works:
$$ cos(x) =frac12[cos(x+t) + cos(x-t)] + frac12int_0^tint_x-s+t^x+s-tcos(xi)dxi ds$$
This integral is easy and is true.
So where am I going wrong?
integration pde wave-equation
asked Aug 9 at 15:28
Wapiti
23318
edited Aug 9 at 19:27
asked Aug 9 at 15:28
Wapiti
23318
asked Aug 9 at 15:28
Wapiti
23318
asked Aug 9 at 15:28
Wapiti
23318
23318
add a comment |Â
add a comment |Â
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