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I'd like to know if there is some litterature about the cylindrical Mason-Weaver equation.

The basic Mason-Weaver equation is treated in Wikipedia :

$$\partial_t f(t,x)=\partial_x f(t,x)+\partial_{xx}^2 f(t,x)=\nabla\mathbf{v} $$

and the boundary conditions at $x=x_b$ : $$f(t,x)+\partial_{x} f(t,x)=0 $$

It is treated with the separation of variables : $f(x,t)=F(t)G(x)$

But I'm wondering how to proceed with the cylindrical symmetry case :

$$\partial_t f(t,z,r)=\partial_z f(t,z,r)+\partial_{zz}^2 f(t,z,r) +\frac{1}{r}\partial_r(r\partial_r f(t,z,r)) =\nabla\mathbf{v}$$

and with boundary conditions and the boundary $s(x,y)=0$ :

$$\cos(\theta)(f(t,z,r)+\partial_{z} f(t,z,r) )-\sin(\theta)\partial_r f(t,z,r)=0$$

where the vector $\mathbf{n}=(\sin(\theta),-\cos(\theta))$ is perpendicular to the boundary so that $\mathbf{v}.\mathbf{n}=0$

One can also attack the problem with the separation of variables : $f(t,z,r)=F(t)G(r)H(z)$ and solve each equation.

We then obtain :

$$f(t,z,r)=\sum C_{\beta,k,\lambda} e^{-\beta t} e^{-\frac{1}{2}z}(a\cos(\omega_k z)+b\sin(\omega_k z))\operatorname{J}_0(r/\lambda)$$

where $\omega_k^2=1+4(\kappa-\beta)$ and $\lambda=1/\sqrt{\kappa}$ and $\operatorname{J}_0$ the cylindrical Bessel function (Bessel Function of the First Kind) - whose derivative at $r=0$ is zero.

But I'm then stuck how to find the good constants in order to satisfy the boundary conditions.

Would you have an advice to tackle this problem or recommend articles if this case has already been treated in the litterature please ?

EDIT :

Here is what I thought for a geometry of a well where the boundary is parametrized by : $y=R-\sqrt{R^2-x^2}$ for $0<x<R$.

First at $x=R$, the boundary condition is : $G'(r)=0$, and at $r=0$, $H(z)+H'(z)=0$.

Hence, $R/\lambda$ is a zero of $\operatorname{J}_0'(x)\sim\operatorname{J}_1(x)$.

Moreover, the boundary condition on $r=0$ yields the same result as in $1d$, meaning $\omega_k=\frac{k\pi}{z_{max}-z_{min}}$ (see there)

Now I'm working on how to deal with the other boundary conditions.

I'd like to know if there is some litterature about the cylindrical Mason-Weaver equation.

The basic Mason-Weaver equation is treated in Wikipedia :

$$\partial_t f(t,x)=\partial_x f(t,x)+\partial_{xx}^2 f(t,x)=\nabla\mathbf{v} $$

and the boundary conditions at $x=x_b$ : $$f(t,x)+\partial_{x} f(t,x)=0 $$

It is treated with the separation of variables : $f(x,t)=F(t)G(x)$

But I'm wondering how to proceed with the cylindrical symmetry case :

$$\partial_t f(t,z,r)=\partial_z f(t,z,r)+\partial_{zz}^2 f(t,z,r) +\frac{1}{r}\partial_r(r\partial_r f(t,z,r)) =\nabla\mathbf{v}$$

and with boundary conditions and the boundary $s(x,y)=0$ :

$$\cos(\theta)(f(t,z,r)+\partial_{z} f(t,z,r) )-\sin(\theta)\partial_r f(t,z,r)=0$$

where the vector $\mathbf{n}=(\sin(\theta),-\cos(\theta))$ is perpendicular to the boundary so that $\mathbf{v}.\mathbf{n}=0$

One can also attack the problem with the separation of variables : $f(t,z,r)=F(t)G(r)H(z)$ and solve each equation.

We then obtain :

$$f(t,z,r)=\sum C_{\beta,k,\lambda} e^{-\beta t} e^{-\frac{1}{2}z}(a\cos(\omega_k z)+b\sin(\omega_k z))\operatorname{J}_0(r/\lambda)$$

where $\omega_k^2=1+4(\kappa-\beta)$ and $\lambda=1/\sqrt{\kappa}$ and $\operatorname{J}_0$ the cylindrical Bessel function (Bessel Function of the First Kind) - whose derivative at $r=0$ is zero.

But I'm then stuck how to find the good constants in order to satisfy the boundary conditions.

Would you have an advice to tackle this problem or recommend articles if this case has already been treated in the litterature please ?

EDIT :

Here is what I thought for a geometry of a well where the boundary is parametrized by : $z=R-\sqrt{R^2-r^2}$ for $0<x<R$.

First at $r=R$, the boundary condition is : $G'(r)=0$, and at $r=0$, $H(z)+H'(z)=0$.

Hence, $R/\lambda$ is a zero of $\operatorname{J}_0'(x)\sim\operatorname{J}_1(x)$.

Moreover, the boundary condition on $r=0$ yields the same result as in $1d$, meaning $\omega_k=\frac{k\pi}{z_{max}-z_{min}}$ (see there)

Now I'm working on how to deal with the other boundary conditions.

I'd like to know if there is some litterature about the cylindrical Mason-Weaver equation.

The basic Mason-Weaver equation is treated in Wikipedia :

$$\partial_t f(t,x)=\partial_x f(t,x)+\partial_{xx}^2 f(t,x)=\nabla\mathbf{v} $$

and the boundary conditions at $x=x_b$ : $$f(t,x)+\partial_{x} f(t,x)=0 $$

It is treated with the separation of variables : $f(x,t)=F(t)G(x)$

But I'm wondering how to proceed with the cylindrical symmetry case :

$$\partial_t f(t,z,r)=\partial_z f(t,z,r)+\partial_{zz}^2 f(t,z,r) +\frac{1}{r}\partial_r(r\partial_r f(t,z,r)) =\nabla\mathbf{v}$$

and with boundary conditions and the boundary $s(x,y)=0$ :

$$\cos(\theta)(f(t,z,r)+\partial_{z} f(t,z,r) )-\sin(\theta)\partial_r f(t,z,r)=0$$

where the vector $\mathbf{n}=(\sin(\theta),-\cos(\theta))$ is perpendicular to the boundary so that $\mathbf{v}.\mathbf{n}=0$

One can also attack the problem with the separation of variables : $f(t,z,r)=F(t)G(r)H(z)$ and solve each equation.

We then obtain :

$$f(t,z,r)=\sum C_{\beta,k,\lambda} e^{-\beta t} e^{-\frac{1}{2}z}(a\cos(\omega_k z)+b\sin(\omega_k z))\operatorname{J}_0(r/\lambda)$$

where $\omega_k^2=1+4(\kappa-\beta)$ and $\lambda=1/\sqrt{\kappa}$ and $\operatorname{J}_0$ the cylindrical Bessel function (Bessel Function of the First Kind) - whose derivative at $r=0$ is zero.

But I'm then stuck how to find the good constants in order to satisfy the boundary conditions.

Would you have an advice to tackle this problem or recommend articles if this case has already been treated in the litterature please ?

I'd like to know if there is some litterature about the cylindrical Mason-Weaver equation.

The basic Mason-Weaver equation is treated in Wikipedia :

$$\partial_t f(t,x)=\partial_x f(t,x)+\partial_{xx}^2 f(t,x)=\nabla\mathbf{v} $$

and the boundary conditions at $x=x_b$ : $$f(t,x)+\partial_{x} f(t,x)=0 $$

It is treated with the separation of variables : $f(x,t)=F(t)G(x)$

But I'm wondering how to proceed with the cylindrical symmetry case :

$$\partial_t f(t,z,r)=\partial_z f(t,z,r)+\partial_{zz}^2 f(t,z,r) +\frac{1}{r}\partial_r(r\partial_r f(t,z,r)) =\nabla\mathbf{v}$$

and with boundary conditions and the boundary $s(x,y)=0$ :

$$\cos(\theta)(f(t,z,r)+\partial_{z} f(t,z,r) )-\sin(\theta)\partial_r f(t,z,r)=0$$

where the vector $\mathbf{n}=(\sin(\theta),-\cos(\theta))$ is perpendicular to the boundary so that $\mathbf{v}.\mathbf{n}=0$

One can also attack the problem with the separation of variables : $f(t,z,r)=F(t)G(r)H(z)$ and solve each equation.

We then obtain :

$$f(t,z,r)=\sum C_{\beta,k,\lambda} e^{-\beta t} e^{-\frac{1}{2}z}(a\cos(\omega_k z)+b\sin(\omega_k z))\operatorname{J}_0(r/\lambda)$$

where $\omega_k^2=1+4(\kappa-\beta)$ and $\lambda=1/\sqrt{\kappa}$ and $\operatorname{J}_0$ the cylindrical Bessel function (Bessel Function of the First Kind) - whose derivative at $r=0$ is zero.

But I'm then stuck how to find the good constants in order to satisfy the boundary conditions.

Would you have an advice to tackle this problem or recommend articles if this case has already been treated in the litterature please ?

EDIT :

Here is what I thought for a geometry of a well where the boundary is parametrized by : $y=R-\sqrt{R^2-x^2}$ for $0<x<R$.

First at $x=R$, the boundary condition is : $G'(r)=0$, and at $r=0$, $H(z)+H'(z)=0$.

Hence, $R/\lambda$ is a zero of $\operatorname{J}_0'(x)\sim\operatorname{J}_1(x)$.

Moreover, the boundary condition on $r=0$ yields the same result as in $1d$, meaning $\omega_k=\frac{k\pi}{z_{max}-z_{min}}$ (see there)

Now I'm working on how to deal with the other boundary conditions.

I'd like to know if there is some litterature about the cylindrical Mason-Weaver equation.

The basic Mason-Weaver equation is treated in Wikipedia :

$$\partial_t f(t,x)=\partial_x f(t,x)+\partial_{xx}^2 f(t,x)=\nabla\mathbf{v} $$

and the boundary conditions at $x=x_b$ : $$f(t,x)+\partial_{x} f(t,x)=0 $$

It is treated with the separation of variables : $f(x,t)=F(t)G(x)$

But I'm wondering how to proceed with the cylindrical case :

$$\partial_t f(t,z,r)=\partial_z f(t,z,r)+\partial_{zz}^2 f(t,z,r) +\frac{1}{r}\partial_r(r\partial_r f(t,z,r)) =\nabla\mathbf{v}$$

and with boundary conditions and the boundary $s(x,y)=0$ :

$$\cos(\theta)(f(t,z,r)+\partial_{z} f(t,z,r) )-\sin(\theta)\partial_r f(t,z,r)=0$$

where the vector $\mathbf{n}=(\sin(\theta),-\cos(\theta))$ is perpendicular to the boundary so that $\mathbf{v}.\mathbf{n}=0$

One can also attack the problem with the separation of variables : $f(t,z,r)=F(t)G(r)H(z)$ and solve each equation. But I'm then stuck how to find the good constants in order to satisfy the boundary conditions.

Would you have an advice to tackle this problem or recommend articles if this case has already been treated in the litterature please ?

I'd like to know if there is some litterature about the cylindrical Mason-Weaver equation.

The basic Mason-Weaver equation is treated in Wikipedia :

$$\partial_t f(t,x)=\partial_x f(t,x)+\partial_{xx}^2 f(t,x)=\nabla\mathbf{v} $$

and the boundary conditions at $x=x_b$ : $$f(t,x)+\partial_{x} f(t,x)=0 $$

It is treated with the separation of variables : $f(x,t)=F(t)G(x)$

But I'm wondering how to proceed with the cylindrical symmetry case :

$$\partial_t f(t,z,r)=\partial_z f(t,z,r)+\partial_{zz}^2 f(t,z,r) +\frac{1}{r}\partial_r(r\partial_r f(t,z,r)) =\nabla\mathbf{v}$$

and with boundary conditions and the boundary $s(x,y)=0$ :

$$\cos(\theta)(f(t,z,r)+\partial_{z} f(t,z,r) )-\sin(\theta)\partial_r f(t,z,r)=0$$

where the vector $\mathbf{n}=(\sin(\theta),-\cos(\theta))$ is perpendicular to the boundary so that $\mathbf{v}.\mathbf{n}=0$

One can also attack the problem with the separation of variables : $f(t,z,r)=F(t)G(r)H(z)$ and solve each equation.

We then obtain :

$$f(t,z,r)=\sum C_{\beta,k,\lambda} e^{-\beta t} e^{-\frac{1}{2}z}(a\cos(\omega_k z)+b\sin(\omega_k z))\operatorname{J}_0(r/\lambda)$$

where $\omega_k^2=1+4(\kappa-\beta)$ and $\lambda=1/\sqrt{\kappa}$ and $\operatorname{J}_0$ the cylindrical Bessel function (Bessel Function of the First Kind) - whose derivative at $r=0$ is zero.

But I'm then stuck how to find the good constants in order to satisfy the boundary conditions.

Would you have an advice to tackle this problem or recommend articles if this case has already been treated in the litterature please ?