Nature is linear, binary and symmetrical.
We assume that any statistical transition matrix M that satisfies the above conditions would generate the solution of PPDE with Dirichlet boundary conditions simply as,
W(x,y,z,t)= M(N).b
where b is the boundary vector working as the solution-generating agent and N is the number of time steps.
The boundary conditions b are no longer problematic but rather a source of simplified solutions.
No, because the statement of the problem just doesn’t make sense. It would be a good idea to actually study the subjects, instead of asking questions at random.
This is just a preliminary introductory answer to shed some light on the question and its proposed answer.
And recommend readers to disregard misleading comments such as the problem statement doesn't make sense.
These kinds of comments show an extremely low level of understanding and a lack of basic knowledge.
The significance is that an elegant answer exists and has been published many times with successful applications in double and triple integration as well as solving the time-dependent Poisson and Laplace PDE without the need for the PDE it -even or even the use of FDM Techniques [1,2].
The required statistical transition matrix is called the matrix B which satisfies the conditions of binary, symmetry and linearity.
And the requested solution is:
U(x,y,z,t)= D(N).(b+S)
where b is the Dirichlet boundary condition vector and S is the source term vector.
The transfer matrix D(N) is defined as the matrix power series,
D(N)= B + B^2 +B^3+.....+B^N
Which gives the transitory solution dependent on time at time t=N dt.
For sufficiently large values of N, it can be shown that,
D(N)=E(N)-I
which gives the time-independent stationary solution.
Notice that,
E(N)=(I-B)^-1
In a following answer we provide illustrative numerical examples in 1D, 2D and 3D solutions.
1-I.M.Abbas, journal IJISRT, A Numerical Statistical Solution to theLaplace and Poisson Partial Differential Equations, Volume 5, Issue11, November – 2020.
2-I.M.Abbas, IJISRT, Time Dependent Numerical Statistical Solution of the Partial Differential Equation for Heat Diffusion, Volume 6, Number 1, January - 2021
The Poisson equation is L.f=s, where L is the Laplacian on the domain and s is the source. Of course, boundary conditions must be imposed, in order for the solution to be uniquely defined-and Dirichlet boundary conditions are but one example. It isn’t exactly front page news that it can be solved by the method of successive overrelaxation, when using the method of finite differences, so the only way time enters the picture is in the iteration step of successive overrelaxation; time isn’t a variable in the equation itself.
It would be a good idea not to use ChatGPT, or any similar program-``Nature is linear, binary and symmetrical" is grammatically and syntactically correct, but meaningless. And that’s but one example.
Answer II- Continued.
Below we present the 3D numerical statistical solution for the TRANSIENT HEATING CURVE (OR equivalently Poisson voltage distribution) inside a cube Fig 1 initially at 0c (or equivalently at zero voltage) placed in a 76 c container (equivalent to a 76 voltage unit "Dirichlet BC" equipotential container)
📷
Fig 1. A cube subdivided into 27 equidistant nodes.
The cube in Figure 1 is subdivided into 27 equidistant nodes.
*RO is set equal to zero which corresponds to the diffusion of heat in the perfect conductor and the distribution of voltage in free space.
The numerical calculation results are as follows:
time=dt
30.400 20.270 30.400 20.270 10.133 20.270 30.400 20.270 30.400
20.270 10.133 20.270 10.133 0.000 10.133 20.270 10.133 20.270
30.400 20.270 30.400 20.270 10.133 20.270 30.400 20.270 30.400 .
time= 2 dt
44.587 35.129 44.587 35.129 22.969 35.129 44.587 35.129 44.587
35.129 22.969 35.129 22.969 8.107 22.969 35.129 22.969 35.129
44.587 35.129 44.587 35.129 22.969 35.129 44.587 35.129 44.587
time= 3 dt
53.369 45.307 53.369 45.307 34.543 45.307 53.369 45.307 53.369
45.307 34.543 45.307 34.543 19.996 34.543 45.307 34.543 45.307
53.369 45.307 53.369 45.307 34.543 45.307 53.369 45.307 53.369 .
time= 4 dt
59.197 52.771 59.197 52.771 43.872 52.771 59.197 52.771 59.197
52.771 43.872 52.771 43.872 31.634 43.872 52.771 43.872 52.771
59.197 52.771 59.197 52.771 43.872 52.771 59.197 52.771 59.197 .
time= 5 dt
63.348 58.306 63.348 58.306 51.270 58.306 63.348 58.306 63.348
58.306 51.270 58.306 51.270 41.424 51.270 58.306 51.270 58.306
63.348 58.306 63.348 58.306 51.270 58.306 63.348 58.306 63.348 .
time= 6 dt
66.392 62.493 66.392 62.493 57.007 62.493 66.392 62.493 66.392
62.493 57.007 62.493 57.007 49.301 57.007 62.493 57.007 62.493
66.392 62.493 66.392 62.493 57.007 62.493 66.392 62.493 66.392
. . . . . . .
..etc..
Another interesting physical situation is the case where the same cube in Figure 1 heats up (OR equivalently the Poisson voltage distribution builds up) when held in a reservoir of 0c and heated by a source of 100 units placed at node 1.
The time-dependent 3D solution is shown as the following transient heating curve,
,
time = dt
120.0 13.3 0.00 13.3 0.00 0.00 0.00 0.00 0.00
13.3 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.0 0.00 0.00.
Time = 4 dt
133.3 22.16 2.84 22.16 5.69 0.711 2.84 0.711 0.0
22.16 5.69 0.71 5.69 1.42 0.00 0.71 0.00 0.00000000
2.84 0.71 0.00 0.71 0.00 0.00 0.00 0.00 0.00.
Time = 7 dt
137.5 26.2 4.73 26.17 9.45 2.23 4.73 2.23 0.577
26.17 9.45 2.23 9.45 4.47 1.15 2.23 1.15 0.278
4.73 2.23 0.577 2.23 1.15 0.278 0.577 0.278 5.057E-02
Time = 15 dt
138.9 27.9 5.79 27.9 11.6 3.50 5.79 3.50 1.32
27.9 11.6 3.50 11.6 7.01 2.65 3.50 2.65 1.15
5.79 3.50 1.32 3.50 2.65 1.15 1.32 1.15 0.54
Time = 16 dt
138.95 27.96 5.814 27.96 11.63 3.54 5.81 3.54 1.35
27.96 11.63 3.54 11.63 7.09 2.705 3.54 2.70 1.18
5.81 3.54 1.35 3.54 2.70 1.18 1.35 1.18 0.567
Time = 19 dt
139.0 28.0 5.87 28.0 11.73 3.62 5.87 3.62 1.40
28.0 11.7 3.6 11.7 7.23 2.80 3.617 2.80 1.25
5.87 3.62 1.40 3.62 2.80 1.25 1.40 1.25 0.615
Note that the time-independent stationary equilibrium is almost reached on the 27 free nodes after 19 time steps.
In a following answer we provide examples of cooling curves for different metals where the RO value is between zero and 1 depending on the size and thermal diffusivity of the object under consideration.
To be continued.
I hope you find a more complete answer in the following article,
Is it time to reformulate the partial differential equations of Poisson and Laplace?
Researchgate, June 2023.
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