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Laplace equation problem with numerous non-homogeneous BC(s) [Linear Superposition]

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I need to solve the three-dimensional Laplace equation ($nabla^2T = 0$) where $nabla^2=fracpartial^2partial x^2 +fracpartial^2partial y^2+fracpartial^2partial z^2$ in the domain where $xin[0,L];yin[0,l]$ and $zin[0,w]$. The boundary conditions are
$$Tvert_0,y,z = T_hi, fracpartial Tpartial xvert_L,y,z = 0$$
$$Tvert_x,0,z = T_ci, fracpartial Tpartial yvert_x,l,z = 0$$
$$fracpartial Tpartial zvert_x,y,0 = k_c(Tvert_x,y,0-T_c,av )$$
$$fracpartial Tpartial zvert_x,y,w = k_h(T_h,av - Tvert_x,y,w)$$
Here, $T_c,av = frac12Bigg(T_ci+e^-b_cBigg[T_ci+fracb_clint_0^l e^b_c s/lT(x,s,z)mathrmdsBigg]Bigg)$

and, $T_h,av = frac12Bigg(T_hi+e^-b_hBigg[T_hi+fracb_hLint_0^L e^b_h s/LT(s,y,z)mathrmdsBigg]Bigg)$
As can be seen there are two non-homogeneous Dirichlet conditions and two non-homogeneous Robin conditions. Using, the linear nature of the Laplace operator I am supposed to divide the problem into sub-problems and finally add up all the solutions.

I need help on segregating this problem into smaller ones. I have encountered examples where each problem must have a single non-homogeneous BC.

Secondly, what could be a good guess initial solution form for such problems ?.

pde problem-solving boundary-value-problem heat-equation laplacian

share|cite|improve this question

edited Mar 15 at 2:34

Ethan Bolker

45.1k553120

asked Mar 13 at 5:25

Indrasis MitraIndrasis Mitra

80111

$endgroup$

add a comment | 

0

$begingroup$

I need to solve the three-dimensional Laplace equation ($nabla^2T = 0$) where $nabla^2=fracpartial^2partial x^2 +fracpartial^2partial y^2+fracpartial^2partial z^2$ in the domain where $xin[0,L];yin[0,l]$ and $zin[0,w]$. The boundary conditions are
$$Tvert_0,y,z = T_hi, fracpartial Tpartial xvert_L,y,z = 0$$
$$Tvert_x,0,z = T_ci, fracpartial Tpartial yvert_x,l,z = 0$$
$$fracpartial Tpartial zvert_x,y,0 = k_c(Tvert_x,y,0-T_c,av )$$
$$fracpartial Tpartial zvert_x,y,w = k_h(T_h,av - Tvert_x,y,w)$$
Here, $T_c,av = frac12Bigg(T_ci+e^-b_cBigg[T_ci+fracb_clint_0^l e^b_c s/lT(x,s,z)mathrmdsBigg]Bigg)$

and, $T_h,av = frac12Bigg(T_hi+e^-b_hBigg[T_hi+fracb_hLint_0^L e^b_h s/LT(s,y,z)mathrmdsBigg]Bigg)$
As can be seen there are two non-homogeneous Dirichlet conditions and two non-homogeneous Robin conditions. Using, the linear nature of the Laplace operator I am supposed to divide the problem into sub-problems and finally add up all the solutions.

I need help on segregating this problem into smaller ones. I have encountered examples where each problem must have a single non-homogeneous BC.

Secondly, what could be a good guess initial solution form for such problems ?.

pde problem-solving boundary-value-problem heat-equation laplacian

share|cite|improve this question

edited Mar 15 at 2:34

Ethan Bolker

45.1k553120

asked Mar 13 at 5:25

Indrasis MitraIndrasis Mitra

80111

$endgroup$

add a comment | 

$begingroup$

I need to solve the three-dimensional Laplace equation ($nabla^2T = 0$) where $nabla^2=fracpartial^2partial x^2 +fracpartial^2partial y^2+fracpartial^2partial z^2$ in the domain where $xin[0,L];yin[0,l]$ and $zin[0,w]$. The boundary conditions are
$$Tvert_0,y,z = T_hi, fracpartial Tpartial xvert_L,y,z = 0$$
$$Tvert_x,0,z = T_ci, fracpartial Tpartial yvert_x,l,z = 0$$
$$fracpartial Tpartial zvert_x,y,0 = k_c(Tvert_x,y,0-T_c,av )$$
$$fracpartial Tpartial zvert_x,y,w = k_h(T_h,av - Tvert_x,y,w)$$
Here, $T_c,av = frac12Bigg(T_ci+e^-b_cBigg[T_ci+fracb_clint_0^l e^b_c s/lT(x,s,z)mathrmdsBigg]Bigg)$

and, $T_h,av = frac12Bigg(T_hi+e^-b_hBigg[T_hi+fracb_hLint_0^L e^b_h s/LT(s,y,z)mathrmdsBigg]Bigg)$
As can be seen there are two non-homogeneous Dirichlet conditions and two non-homogeneous Robin conditions. Using, the linear nature of the Laplace operator I am supposed to divide the problem into sub-problems and finally add up all the solutions.

I need help on segregating this problem into smaller ones. I have encountered examples where each problem must have a single non-homogeneous BC.

Secondly, what could be a good guess initial solution form for such problems ?.

pde problem-solving boundary-value-problem heat-equation laplacian

share|cite|improve this question

edited Mar 15 at 2:34

Ethan Bolker

45.1k553120

asked Mar 13 at 5:25

Indrasis MitraIndrasis Mitra

80111

$endgroup$

share|cite|improve this question

edited Mar 15 at 2:34

Ethan Bolker

45.1k553120

asked Mar 13 at 5:25

Indrasis MitraIndrasis Mitra

80111

share|cite|improve this question

edited Mar 15 at 2:34

Ethan Bolker

45.1k553120

asked Mar 13 at 5:25

Indrasis MitraIndrasis Mitra

80111

edited Mar 15 at 2:34

Ethan Bolker

45.1k553120

edited Mar 15 at 2:34

Ethan Bolker

45.1k553120

asked Mar 13 at 5:25

Indrasis MitraIndrasis Mitra

80111

asked Mar 13 at 5:25

Indrasis MitraIndrasis Mitra

80111