Briefly, we propose a time-dependent Poisson PDE expressed by:
dU/dt)partial=D Nabla^2 U +S
And,
Laplace's time-dependent PDE is proposed as follows:
dU/dt)partial=D Nabla^2 U
(In normal conventions.)
Subject to the boundary conditions of Dirichlet or any other appropriate BC.
As a general rule, any physical equation must contain time.
Classical time-independent PPDEs and LPDEs where electromagnetic energy transfer is assumed to be instantaneous are incomplete and violate special theory of relativity.
They only take place because the speed of transfer of their energy is close to that of light.
Moreover, it is quite surprising that the solution of time-dependent PPDEs and LPDEs is more accessible than that of time-independent solutions.
Below is an introductory answer to illustrate the question and its proposed answer.
Briefly, we propose a time-dependent Poisson PDE expressed by:
dU/dt)partial=D Nabla^2 U +S
And,
Laplace's time-dependent PDE is proposed as follows:
dU/dt)partial=D Nabla^2 U
(In normal conventions.)
Subject to the boundary conditions of Dirichlet or any other appropriate BC.
As a general rule, any physical equation should contain time as a fourth dimension.
Classical time-independent PPDEs and LPDEs where electromagnetic energy transfer is assumed to occur instantaneously are incomplete and violate special theory of relativity.
The current Poisson PDE is expressed by:
L U=S
and the current Laplace PDE is expressed as:
LU=0
Subject to the boundary conditions of Dirichlet or any other appropriate BC. There are only time-independent steady equilibrium solutions here where the part dU/dt) vanishes.
These time-independent equations only take place because the speed of their energy transfer is close to that of light and therefore equilibrium is reached almost instantaneously.
Incredibly, it is quite surprising that the solution of time-dependent PPDEs and LPDEs is more accessible than that of time-independent solutions.
The above statement can be validated via the following interesting applications:
i- Many mathematicians introduce "a time step dt" as the only way to get the time in the image where dt is the successive overrelaxation iteration step. Unfortunately they think dt is not real time and time is not a variable in the equation itself.
ii- The proposed time-dependent PPDEs and LPDEs have already been
successfully used to solve the energy density distribution of the electric potential field as well as the thermal energy in its most general time-dependent three-dimensional situation [1,2].
But the question arises;
How important is the time-dependent PPDE and
Laplace's time-dependent PDE when we live happily without it.
This will be the subject of the next answer.
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
II-Answer II continued.
From the moment we reformulate the LPDE and PPDE equation into time-dependent equations, the world is turned upside down.
Both equations become similar to the heat equation and all three are self-solving nature equations i.e. they do not need mathematics to solve them.
It is obvious that the solutions via the B-Matrix chain technique, called Cairo technique, do not need to implement the FDM finite difference methods or even the PDE itself as it was followed in the previous classical methods. [3,4] . .
It is quite striking that the B matrix chain technique is unbeatable and unstoppable even in the "seemingly" most complicated situations like double and triple integration, with or without singularities, as well as solving 3D geometry PPDE, LPDE versus time also as heat diffusion/conduction equations.
The numerical solution of this type of PDE belongs to the type of statistical numerical solution of chains of B-transition matrices where nature works in the 4D unit space according to the recurrence relation,
U(x,y,z,t+dt)= B.U(x,y,z,t) . . . . . . . (1)
The transition matrix B is well defined via four conditions,
For Cartesian coordinates in 1,2 and 3D space, the inputs B i , j respect or are subject to the following conditions:
i- B i , j = 1/2-RO/2,1/4-RO/4 and 1/6-RO/6 in 1D, 2D and 3D for i adjacent to j and B i, j = 0 otherwise. Condition (i) translates an equal a priori probability of all directions in space, ie no preferred direction.
ii- B i, i = RO, i.e. the main diagonal consists of equal or constant entries RO .
RO can take any value in the interval [0,1].
Condition (ii) corresponds to the assumption of an equal residue after each jump or time step dt for all the free elementary nodes.
For emw in free space RO=0 and for a perfect insulator in the heat diffusion/conduction equation RO=1.
iii- B i, j = B j, i, for all i, j.
Matrix B is symmetrical to conform to nature's symmetry and physical principles of reciprocity and detailed balance.
iv- The sum of B i, j = 1 for all rows (or columns) away from the borders and the sum B i, j < 1 for all the rows connected to the borders.
The condition iv means that the probability of the whole space = 1.
Obviously, the statistical matrix B is very different from the mathematical Laplacian matrix and the mathematical statistical matrix of the Markov transition probability.
The physical nature of B is clear and briefly explained above through its four conditions i-iv which support the hypothesis of being an accurate model of nature itself.
The question arises, why are statistical forms of numerical integration faster and more accurate than mathematical forms?
To be continued.
This question can best be answered by two elegant examples that show that the reformulation of LPDE and PPDE is no longer a matter of debate but rather a matter of decision.
Many physicists believe that time-independent equations are based on the assumption that the physical system is in equilibrium, and the Poisson and Laplace equations remain widely used in physics and mathematics. because these equations are relatively simple to understand and can be used to solve a wide variety of problems with sufficient accuracy in many equilibrium cases.
The above statement is not true because even in the time-independent equilibrium system, B-matrix strings produce much better solutions in terms of simplicity, speed and accuracy as shown below in the following two physical situations:
case A
Two dimensional energy density field subject to Dirichlet boundary conditions Fig1.
📷
Fig.1 A 2D rectangular domain with 9 equidistant free nodes.
Consider the simple case of a rectangular domain with 9 equidistant free nodes, u1, u2, u3, ... u9 subject to 12 Dirichlet boundary conditions BC1 to BC12 as shown in Figure 1
The 12 boundary conditions can be reduced to 9 BC for the 9 free nodes as follows,
BC1 = BC1X + BC1Y
BC2 = BC2X + BC2Y
. . . . . . . . . . . . . . . .
BC9 = BC9X + BC9Y .
The construction Now of the transition matrix B 9x9 complies with the i-iv conditions for RO = 0 and therefore the corresponding matrix E can be calculated.
Now just multiply the matrix D=E-I (E and D are not a function of BC) by any arbitrary vector BC (b) to get the required solution for the electrostatic voltage distribution V or equivalently the temperature distribution T c.
J. Mathews [3] classically solved the resulting system of 9 linear algebraic equations using the Gaussian elimination method in a more efficient scheme by extending the tridiagonal algorithm to the more sophisticated pentadiagonal algorithm for its arbitrarily chosen vector BC(b),
b = [100,20,20,80,0,0,260,180,180] T.. . . . . (1) and arrived at the solution vector: U=[55,7143,43,2143,27,1429,79,6429,70,0000,45,3571,112,357,111,786,84,2857]T.. . .(2)
Alternatively, the vector BC for the proposed statistical solution of matrix B corresponding to (1) reduces to
b=[100/4, 20/4, 20/4, 20/4, 80/4, 0, 0.260/4, 180 /4 , 180/ 4 ] T. . . (3)
Further, the calculated transfer matrix D=E-I can be multiplied by the vector BC(b) of and we get the statistical solution which is,
U=[55.7132187 43.2126846 27.1417885 79.6412506 69.9978638 45.3555412112.856079 111.784111 84.2846451]T. . . (4)
If we compare the proposed statistical solution (4) with that of Mathews (2), we find a striking simplicity, speed and precision.
To be continued
Answer III- Continued.
Below we present the 3D numerical statistical solution for the TRANSIENT HEATING CURVE (OR equivalently Poisson voltage distribution of the time-dependent reformulated PPDE) 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 as a solution of the time-dependent PPDE) when held in a contaner of 0c and heated by a 100 unit source placed at node 1 (or equivalently a 100 unit voltage source is placed at node 1).
The 3D solution as a function of time is represented by the following transient heating curve or, equivalently, the transient voltage distribution 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.
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?
Researchgatem, June 2023.Researchgatem, June 2023.
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