I need to solve the three-dimensional Laplacian $$\nabla^2T(x,y,z)=0 \tag {A}$$
on a cuboidal domain $x\in[0,a], y\in [0,b], z\in [0,c]$.
(A) is subjected to the following boundary conditions
$$T(0,y,z)=T_{w} \tag{1}$$ $$\frac{\partial T}{\partial x} \bigg \vert_{a,y,z}=0 \tag {2}$$ $$T(x,0,z)=T(x,b,z)=T(x,y,0)=T(x,y,c)=0 \tag{3}$$
I need some guidance in attempting this problem. I know that all the homogeneous Dirichlet conditions described by $(3)$ enables us to write a preliminary solution of the form: $T(x,y,z)=T_{nm}(x)\sin\bigg(\frac{n\pi y}{b}\bigg)\sin\bigg(\frac{m\pi z}{c}\bigg)$. Some help in determining the function $T_{nm}(x)$ is needed.
Attempt After applying Mattos's suggestion, I can write the following
$$T(x,y,z)=\bigg[A_{nm}\bigg(-\tanh(\sqrt{\gamma}a)\cosh{(\sqrt{\gamma}x)}+\sinh{(\sqrt{\gamma}x)}\bigg)\bigg]\sin\bigg(\frac{n\pi y}{b}\bigg)\sin\bigg(\frac{m\pi z}{c}\bigg)$$
On applying boundary condition $(1)$ $$T_w=\bigg[A_{nm}\bigg(-\tanh(\sqrt{\gamma}a)\bigg)\bigg]\sin\bigg(\frac{n\pi y}{b}\bigg)\sin\bigg(\frac{m\pi z}{c}\bigg)$$ Subsequently applying orthogonality, the follwowing is the result:
$$A_{nm} = \frac{-4 T_w \tanh(\sqrt{\gamma}a)}{nm\pi^2}(1-\cos{(n\pi)})(1-\cos{(m\pi)})$$