A set of equation is derived from an elastic problem with axisymmetry. The solution is assumed in a serial form, whose coeffecients are almost found but still remain elusive because of the expansion of the non-homogenious term. Could someone help me out?
The equations about $u(r,z)$ and $w(r)$ are $$\partial_{rr}u+\frac{\partial_ru}{r}-\frac{u}{r^2}+a^2\partial_{zz}u=0 ……(a)$$ $$\int_{-\frac{h}{2}}^{\frac{h}{2}} (b(\partial_r+\frac{1}{r})(\partial_zu+\partial_rw)-c\nabla^2\nabla^2w)dz=f……(b)$$ where $a,b$ and $c$ are known constant, $f$ is a known function of $r$, $\nabla^2=(\partial_r+\frac{1}{r})\partial_r$ is the 2D Laplacian operator under cylinderical coordinates, and $r\in[0,R], z\in[-\frac{h}{2},\frac{h}{2}]$.
The BCs (1~7) are $$u(0,z)=0, u(R,z)=0$$ $$w(R)=0, \partial_rw(0)=0, \partial_rw(R)=0$$ $$\partial_zu(r,-\frac{h}{2})+\partial_rw(r)=\partial_zu(r,\frac{h}{2})+\partial_rw(r)=0$$
First, I try to gain knowledge about the probable form of $u(r,z)$ through eq.(a). Combined with the boundary conditions, $u(0,z)=u(R,z)=0$, and the physcial fact that $u(r,z)$ is anti-symmetric with respect to $z$, eq.(a) is partly solve by saperation of variables, i.e. $$u(r,z)=\sum_{k=1}^{\infty}u_k \sinh(\frac{a_k}{a}z) J_1(a_kr) ……(c)$$
where, $u_k$ is the coefficients dependent on $k$ and to be determined, $a_k=j_k^{(1)}/R$. $J_{\nu}(r)$ is the $\nu$th ordered Bessel function and $j_{k}^{(\nu)}$ is the $k$th zero of $J_{\nu}(r)$.
Besides, ${\{J_0(a_kr)\}}$ for $k=0,1,2,...$ is also a set of complete orthogonal bases in $(0,R)$ with weight $r$ (e.g.,see this lecture note). Therefore, $w(r)$ may be represented by $$ w(r)=\sum_{k=0}^{\infty}w_k J_0(a_kr)……(d) $$ where, $w_k$ is another coefficient to be determined and dependent on $k$.
Plug eq.(d) into the boundary condition that $\partial_ru(r,\frac{h}{2})+\partial_rw(r)=0$, one obtains $$w_k=0, (k=0)$$and $$w_k=\frac{u_k}{a}\cosh(\frac{a_kh}{2a}),(k=1,2,3,...)……(e)$$ which suggests that the term $J_0(a_0r)=1$ is not included in the expansion of $w(r)$.
Plug eqs.(c~e) into eq.(b), we arrive at $$f=\sum_{k=1}^{\infty}u_kp_kJ_0(a_kr)……(f)$$ where $p_k$ is a known parameter dependent on $k$.
So far, seemingly so good. If we could expand $f$ as a series of $J_0(a_kr)$, the problem would be solved.
Calculate this expansion, we get $$ f=\sum_{k=0}^{\infty}f_kJ_0(a_kr)……(g)$$ where $f_k$ is a known coefficients dependent on $k$.
However, eq.(g) has one extra term at $k=0$ compared with eq.(f), which hinders me and $u_k$ couldn't be finalized especially when $f$ is a constant.
It seems to be so close to the solution. Could someone help me out?
Any other suggestion on solving the equations would also be appriciated