MATHEMATICS & NATURE
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⃝ Mathematics & Nature–Free Media of Eternal Truth, China, 2021 https://orcid.org/0000-0002-3644-5170
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Article
.
Mathematics
Dongfang Special Entangled Spherical Solution of Laplace
Equation
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
Abstract: The latest discovery of sp ecial entangled spherical harmonics indicates that
traditional spherical harmonics are only a very narrow partial solution of spherical har-
monic partial differential equations, and the theory of exact solutions for similar linear
partial differential equations is therefore subject to disruptive effects. Here is a detailed
introduction to the special entangled solution of the Laplace equation for the Dirichlet
problem in a spherical region. Returning to the analytical form of the solution of spher-
ical harmonic partial differential equations, it is first explained that in the special case
where the magnetic quantum is zero, the spherical harmonic function partially degener-
ates into a Legendre function with respect to the angle θ, resulting in the existence of
missing solutions in the Laplace equation. Then provide two special entangled solution
sets for the Laplace equation with zero magnetic quantum number, including sine and
cosine functions for angles θ and ϕ. The integral constant of the zero magnetic quantum
number general solution in the special case formed by linear superposition of three types
of special solution sets is not unique, indicating that the specific form of the exact solu-
tion of the Laplace equation cannot be determined. It can be seen that the conclusions
of scientific theories described by partial differential equations are not absolutely reliable,
and the theory of partial differential equations has serious flaws that urgently need to be
improved.
Keywords: Spherical harmonic partial differential equation; Special entangled spherical
harmonic function; Laplace equation; Exact solution uncertainty.
MSC(2020) Subject Classification: 35J05, 43A90, 33E10
Contents
1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 2
2 Traditional solution of Laplace equation · · · · · · · · · · · · · · · · · · · · · · · · · · · 2
3 Sp ecial solution of Laplace equation with zero magnetic quantum number· · · · · 4
4 Complete sp ecial solution set of zero magnetic quantum number Laplace equa-
tion· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 9
5 Conclusion · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 12
Citation: Dongfang, X. D. Dongfang Special Entangled Spherical Solution of Laplace Equation. Mathematics & Nature 3,
202305 (2023).
2 X. D. Dongfang Dongfang Special Entangled Spherical Solution of Laplace Equation
1 Introduction
The Laplace equation
[1-5]
is widely used to describe various physical phenomena and engi-
neering techniques
[6-10]
. Electromagnetics uses Laplace’s equation to describe the distribution of
electromagnetic fields; Thermodynamics uses Laplace equation to describe steady-state temper-
ature distribution; The Laplace equation is used in fluid mechanics to describe the steady-state
pressure and velocity distribution; The theory of gravity uses the Lass equation to describe the
distribution of the gravitational field; The Laplace equation is used in engineering technology to
calculate the dynamic changes of interacting systems; Biology even uses Laplace’s equation to
describe the laws of motion and development. In fact, the causal relationships and confidence
levels of these theories are not reliable in traditional cognition. According to the latest discovery
of special entangled spherical harmonics, there are a large number of missing solutions in the
spherical harmonic partial differential equations decomposed from the Laplace equation in the
spherical coordinate system
[11]
. All applications related to the Laplace equation also obviously
have missing solutions. The conclusion of the widespread application of the Laplace equation
needs to be systematically reprocessed
[12, 13]
. Here we introduce a special set of solutions to the
Laplace equation represented by a special entangled spherical harmonic function that goes be-
yond the scope described by traditional standard solutions for the special case of zero magnetic
quantum numbers.
2 Traditional solution of Laplace equation
In a three-dimensional Cartesian coordinate system, the Laplace equation is the simplest sym-
metric second-order linear partial differential equation satisfied by a time independent function
u = u (x, y, z),
∂
2
u
∂x
2
+
∂
2
u
∂y
2
+
∂
2
u
∂z
2
= 0 (1)
Usually, when dealing with the Laplace equation in a spherical coordinate system, a transfor-
mation relationship is introduced between the Cartesian coordinate parameters (x, y, z) and the
spherical coordinate parameters (r, θ, ϕ),
x = r sin θ cos ϕ
y = r sin θ sin ϕ
z = r cos θ
(2)
Thus, the Cartesian coordinate form of Laplace equation (1) is transformed into spherical coor-
dinate form,
∂
2
u
∂r
2
+
2
r
∂u
∂r
+
1
r
2
∂
2
u
∂θ
2
+
cos θ
sin θ
∂u
∂z
+
1
r
2
sin
2
θ
∂
2
u
∂ϕ
2
= 0 (3)
In history, the Dirichlet condition was proposed
u (θ , ϕ )|
r=R
= f (θ, ϕ) (4)
The purpose is to obtain the spherical solution of the Laplace equation, and the additional
condition implied by the Dirichlet condition is that the exact solution has no singularities inside
the sphere. This implicit condition is worth discussing. If the Laplace equation describes the
electromagnetic field of the source inside the sphere, the absence of singularities inside the sphere
is actually not true. Therefore, it is not logical to abandon the existence of singularities within
the Laplace sphere in advance for the singularity problem. Let’s first list the general solution
form and then determine the integral constant based on the actual problem.
The solution to the Dirichlet problem of the Laplace equation (3) given in standard textbooks is
Mathematics & Nature Vol. 3 (2023) 3
composed of the traditional associated Legendre spherical harmonic function Y
n
(θ , ϕ)
[14]
. For any
integer n, the expression of the traditional solution’s implementable programmatic verification is
as follows,
u (r, θ, φ) =
n
l=0
r
l
Y
l
(θ, φ)
Y
l
(θ, φ) =
l
m=0
(a
m
l
cos mφ + b
m
l
sin mφ) P
m
l
(cos θ)
(5)
This solution abandons the solution with the coordinate origin as the singularity. If the field
source is at the coordinate origin, this traditional solution is unreasonable.
The above form is convenient for memory, but it hides many details, which is not conducive
to problem discovery, and therefore limits the innovation and development of theory. (5) The
associative Legendre function P
m
l
(cos θ ) in is a polynomial function,
P
m
l
(cos θ ) =
l
m=0
sin
m
θ
l−m+2
2
n=0
n
k=1
(2k − l + m − 2) (2k + l + m − 1)
2k (2k − 1)
cos
2n
θ
l−m=0,2,4,···
+
l
m=0
sin
m
θ
l−m+1
2
n=0
n
k=1
(2k − l + m − 1) (2k + l + m)
(2k + 1) 2k
cos
2n+1
θ
l−m=1,3,5,···
The analytical form of the traditional solution (5) for the Dirichlet problem of Laplace equation
containing singular terms is as follows.
Lemma 1 (Traditional Solution of Ball Laplace): If the integer m, natural number l, and
positive integer n satisfy the relationship |m| 6 l and l 6 n, then the spherical coordinate function
u (r, θ, φ) =
n
1
l=0
l
m=0
α
l
r
l
+
α
−l
r
l+1
a
⟨m⟩
n,l
cos mφ + a
⟨−m⟩
n,l
sin mφ
sin
m
θ
×
l−m+2
2
j=0
j
k=1
(2k − l + m − 2) (2k + l + m − 1)
2k (2k − 1)
cos
2j
θ
l−m=0,2,4,···
+
n
2
l=0
l
m=0
β
l
r
l
+
β
−l
r
l+1
b
⟨m⟩
n,l
cos mφ + b
⟨−m⟩
n,l
sin mφ
sin
m
θ
×
l−m+1
2
j=0
j
k=1
(2k − l + m − 1) (2k + l + m)
(2k + 1) 2k
cos
2j+1
θ
l−m=1,3,5,···
(6)
with undetermined coefficients a
⟨m⟩
n,l
and a
⟨−m⟩
n,l
or b
⟨m⟩
n,l
and b
⟨−m⟩
n,l
satisfying the natural periodic
condition u (r, θ + 2π, ϕ + 2π) = u (r, θ, ϕ) is a subset of the exact solutions of Laplace equation
∂
2
u
∂r
2
+
2
r
∂u
∂r
+
1
r
2
∂
2
u
∂θ
2
+
cos θ
sin θ
∂u
∂z
+
1
r
2
sin
2
θ
∂
2
u
∂ϕ
2
= 0
The finite term summation in expression (6) can also be written as an infinite term summation.
During the calculation process, the infinite summation will be interrupted, and the result will
4 X. D. Dongfang Dongfang Special Entangled Spherical Solution of Laplace Equation
still be the finite term summation,
u (r, θ, φ) =
n
1
l=0
l
m=0
α
l
r
l
+
α
−l
r
l+1
a
⟨m⟩
n,l
cos mφ + a
⟨−m⟩
n,l
sin mφ
sin
m
θ
×
∞
j=0
j
k=1
(2k − l + m − 2) (2k + l + m − 1)
2k (2k − 1)
cos
2j
θ
l−m=0,2,4,···
+
n
2
l=0
l
m=0
β
l
r
l
+
β
−l
r
l+1
b
⟨m⟩
n,l
cos mφ + b
⟨−m⟩
n,l
sin mφ
sin
m
θ
×
∞
j=0
j
k=1
(2k − l + m − 1) (2k + l + m)
(2k + 1) 2k
cos
2j+1
θ
l−m=1,3,5,···
(7)
3 Special solution of Laplace equation with zero magnetic quantum number
The concepts of direct product function and entanglement function were defined when intro-
ducing special entangled spherical harmonics earlier
[11]
. The special entanglement function of zero
magnetic quantum number is independent of traditional spherical harmonic functions, indicating
that the traditional methods for solving spherical harmonic partial differential equations are not
systematic, and their conclusions cannot be complete. The solution to the Dirichlet problem of
the Laplace equation is inevitably only a local solution. Here, we first provide the traditional spe-
cial solution for the Dirichlet problem of the Laplace equation corresponding to the special case of
zero magnetic quantum numbers, and then present two different special solutions represented by
two types of entangled spherical harmonics. The conclusions are expressed in the form of lemmas
and theorems, respectively. Refer to the detailed proof in spherical harmonics, these lemmas and
theorems do not need to be proven separately. But for the convenience of verification, traditional
solutions and special entangled solutions will list some specific functions separately.
3.1 Complete Legendre Special Traditional Solution of Zero Magnetic Quantum
Number Laplace Equation
When the magnetic quantum number m = 0, the traditional exact solution (7) of the Laplace
equation’s Dirichlet problem degenerates into a binary function containing only radial parameters
r and angle θ, known as the Legendre special traditional solution of the Laplace equation’s
Dirichlet problem
[15, 16]
.
Lemma 2 (special traditional solution of Laplace equation Dirichlet problem): The special
direct product function
u (r, θ, φ) =
n
1
l=0
l
m=0
α
n,l
r
l
+
α
n,−l
r
l+1
l+2
2
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
cos
2j
θ
l=0,2,4,···
+
n
2
l=0
l
m=0
β
n,l
r
l
+
β
n,−l
r
l+1
l+1
2
j=0
j
k=1
(2k − l − 1) (2k + l )
(2k + 1) 2k
cos
2j+1
θ
l=1,3,5,···
(8)
with zero magnetic quantum numbers is a subset of the solutions of the Dirichlet problem
∂
2
u
∂r
2
+
2
r
∂u
∂r
+
1
r
2
∂
2
u
∂θ
2
+
cos θ
sin θ
∂u
∂θ
+
1
r
2
sin
2
θ
∂
2
u
∂φ
2
= 0
u (θ , φ )|
r=R
= f (θ, φ)
Mathematics & Nature Vol. 3 (2023) 5
for the ball Laplace equation.
The special traditional solution of the Dirichlet problem for the Laplace equation in the special
case of zero magnetic quantum number mentioned above does not include angle ϕ. Table 3.1
lists the specific functions calculated from the special traditional solution (8) withn
1
= n
2
=
0, 1, 2, · · · , 6 respectively. It can be verified that these functions satisfy the Laplace equation of
football.
Table 3.1 Partial Legendre Special Traditional Solutions for the Dirichlet Problem of the
Laplace Equation with Zero Magnetic Quantum Number
u
0
(r, θ, φ) = β
0
,
0
+
β
0,0
r
u
1
(r, θ, φ) = cos θ
α
1,−1
r
2
+ α
1,1
r
+ β
1,0
+
β
1,0
r
u
2
(r, θ, φ) = cos θ
α
2,−1
r
2
+ α
2,1
r
+ β
2,0
+
β
2,0
r
+
1 − 3cos
2
θ
β
2,−2
r
3
+ β
2,2
r
2
u
3
(r, θ, φ) =
cos θ
α
3,−1
r
2
+ α
3,1
r
+
cos θ −
5cos
3
θ
3
α
3,−3
r
4
+ α
3,3
r
3
+β
3,0
+
β
3,0
r
+
1 − 3cos
2
θ
β
3,−2
r
3
+ β
3,2
r
2
u
4
(r, θ, φ) =
cos θ
α
4,−1
r
2
+ rα
4,1
+
cos θ −
5cos
3
θ
3
α
4,−3
r
4
+ α
4,3
r
3
+β
4,0
+
β
4,0
r
+
1 − 3cos
2
θ
β
4,−2
r
3
+ r
2
β
4,2
+
1 − 10cos
2
θ +
35cos
4
θ
3
β
4,−4
r
5
+ r
4
β
4,4
u
5
(r, θ, φ) =
cos θ
α
5,−1
r
2
+ α
5,1
r
+
cos θ −
5cos
3
θ
3
α
5,−3
r
4
+ α
5,3
r
3
+
cos θ −
14
3
cos
3
θ +
21cos
5
θ
5
α
5,−5
r
6
+ α
5,5
r
5
+β
5,0
+
β
5,0
r
+
1 − 3cos
2
θ
β
5,−2
r
3
+ β
5,2
r
2
+
1 − 10cos
2
θ +
35cos
4
θ
3
β
5,−4
r
5
+ β
5,4
r
4
u
6
(r, θ, φ) =
cos θ
α
6,−1
r
2
+ α
6,1
r
+
cos θ −
5cos
3
θ
3
α
6,−3
r
4
+ α
6,3
r
3
+
cos θ −
14
3
cos
3
θ +
4
15
63cos
5
θ
4
α
6,−5
r
6
+ α
6,5
r
5
+β
6,0
+
β
6,0
r
+
1 − 3cos
2
θ
β
6,−2
r
3
+ β
6,2
r
2
+
1 − 10cos
2
θ +
35cos
4
θ
3
β
6,−4
r
5
+ β
6,4
r
4
+
1 −
315
15
cos
2
θ +
945cos
4
θ
15
−
693cos
6
θ
15
β
6,−6
r
7
+ β
6,6
r
6
It can be seen that, except for u
0
(r, θ , ϕ ), even in the special case where the magnetic quantum
number is zero, the traditional special integral constant of the Dirichlet problem of the Laplace
6 X. D. Dongfang Dongfang Special Entangled Spherical Solution of Laplace Equation
equation increases with the increase of n, with 2n + 1 > 1. Normalization conditions are far from
sufficient to determine the specific function of such solutions. The traditional theory of using
normalization conditions to determine the solution of the Laplace equation is not necessarily an
inference.
3.2 The first type of special entangled solution for the zero magnetic quantum
number Laplace equation
If cos θ in the special traditional solution (8) of the Laplace equation Dirichlet problem is
replaced with sin θ cos ϕ, a ternary entanglement function containing three spherical coordinate
parameters r, θ, ϕ satisfying the Laplace equation with zero magnetic quantum number will be
obtained, which is called the first type of special entanglement solution of the Laplace equation
Dirichlet problem.
Theorem 1 (First Class Special Entangled Solution of Laplace Equation Dirichlet Problem)
The first type of special entanglement function
v (r, θ, φ) =
p
1
l=0
χ
n,l
r
l
+
χ
n,−l
r
l+1
l+2
2
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(sin θ cos φ)
2j
l=0,2,4,···
+
p
2
l=0
γ
n,l
r
l
+
γ
n,−l
r
l+1
l+1
2
j=0
j
k=1
(2k − l − 1) (2k + l )
(2k + 1) 2k
(sin θ cos φ)
2j+1
l=1,3,5,···
(9)
with zero magnetic quantum numbers is a subset of the solutions to the Dirichlet problem
∂
2
v
∂r
2
+
2
r
∂v
∂r
+
1
r
2
∂
2
v
∂θ
2
+
cos θ
sin θ
∂v
∂θ
+
1
r
2
sin
2
θ
∂
2
v
∂φ
2
= 0
v (θ, φ)|
r=R
= f (θ, φ)
of the ball Laplace equation.
The first type of special entangled solution for the Dirichlet problem of the Laplace equation
with zero magnetic quantum numbers mentioned above is not included in the Legendre special
traditional solution (8). The shortcomings of the traditional theory of partial differential equa-
tions are obvious. Table 3.2 lists the specific functions calculated from the first type of special
entanglement solution (9) with p
1
= p
2
= 0, 1, 2, · · · , 6. It can be verified that these functions all
satisfy the Laplace equation.
Table 3.2 Partial first type special entangled solutions of the zero magnetic quantum number
Laplace equation Dirichlet problem
v
0
(r, θ, φ) = γ
0,0
+
γ
0,0
r
v
1
(r, θ, φ) = cos φ sin θ
χ
1,−1
r
2
+ χ
1,1
r
+ γ
1,0
+
γ
1,0
r
v
2
(r, θ, φ) = cos φ sin θ
χ
2,−1
r
2
+ χ
2,1
r
+ γ
2,0
+
γ
2,0
r
+
1 − 3cos
2
φsin
2
θ
γ
2,−2
r
3
+ γ
2,2
r
2
v
3
(r, θ, φ) =
cos φ sin θ
χ
3,−1
r
2
+ χ
3,1
r
+
cos φ sin θ −
5
3
cos
3
φsin
3
θ
χ
3,−3
r
4
+ χ
3,3
r
3
+γ
3,0
+
γ
3,0
r
+
1 − 3cos
2
φsin
2
θ
γ
3,−2
r
3
+ γ
3,2
r
2
Mathematics & Nature Vol. 3 (2023) 7
v
4
(r, θ, φ) =
cos φ sin θ
χ
4,−1
r
2
+ χ
4,1
r
+
cos φ sin θ −
5
3
cos
3
φsin
3
θ
χ
4,−3
r
4
+ χ
4,3
r
3
+γ
4,0
+
γ
4,0
r
+
1 − 3cos
2
φsin
2
θ
γ
4,−2
r
3
+ γ
4,2
r
2
+
1 − 10 cos φ
2
sin
2
θ +
35
3
cos
4
φsin
4
θ
γ
4,−4
r
5
+ γ
4,4
r
4
v
5
(r, θ, φ) =
cos φ sin θ
χ
5,−1
r
2
+ χ
5,1
r
+
cos φ sin θ −
5
3
cos
3
φsin
3
θ
χ
5,−3
r
4
+ χ
5,3
r
3
+
cos φ sin θ −
14
3
cos
3
φsin
3
θ +
21
5
cos
5
φsin
5
θ
χ
5,−5
r
6
+ χ
5,5
r
5
+γ
5,0
+
γ
5,0
r
+
1 − 3cos
2
φsin
2
θ
γ
5,−2
r
3
+ γ
5,2
r
2
+
1 − 10cos
2
φsin
2
θ +
35
3
cos
4
φsin
4
θ
γ
5,−4
r
5
+ γ
5,4
r
4
v
6
(r, θ, φ) =
cos φ sin θ
χ
6,−1
r
2
+ χ
6,1
r
+
cos φ sin θ −
5
3
cos
3
φsin
3
θ
χ
6,−3
r
4
+ χ
6,3
r
3
+
cos φ sin θ −
14
3
cos
3
φsin
3
θ +
21
5
cos
5
φsin
5
θ
χ
6,−5
r
6
+ χ
6,5
r
5
+γ
6,0
+
γ
6,0
r
+
1 − 3cos
2
φsin
2
θ
γ
6,−2
r
3
+ γ
6,2
r
2
+
1 − 10cos
2
φsin
2
θ +
35
3
cos
4
φsin
4
θ
γ
6,−4
r
5
+ γ
6,4
r
4
+
1 − 21cos
2
φsin
2
θ + 63cos
4
φsin
4
θ −
231
5
cos
6
φsin
6
θ
γ
6,−6
r
7
+ γ
6,6
r
6
Similar to Legendre’s special traditional solution, even in the special case where the magnetic
quantum number is zero, except for v
0
(r, θ, ϕ), the integral constant of the first type of special
entangled solution of the Laplace equation Dirichlet problem increases with the increase of p,
with a total of 2p + 1 > 1. Normalization conditions are far from sufficient to determine the
specific function of such solutions. Moreover, traditional theories overlook the first type of special
entangled solutions, and the conclusions described do not conform to the principle of existence
and uniqueness of solutions.
3.3 The second type entangled special solution of the zero magnetic quantum num-
ber Laplace equation
Further replacing cos θ with sin ϕ sin θ in the special traditional solution (8) of the Laplace
equation Dirichlet problem, another type of ternary function with three spherical coordinate
parameters r, θ, ϕ that also satisfies the Laplace equation with zero magnetic quantum numbers
is obtained, which is called the second type of special entangled solution of the Laplace equation
Dirichlet problem.
Theorem 2 (Second type special entanglement solution of Laplace equation Dirichlet problem)
8 X. D. Dongfang Dongfang Special Entangled Spherical Solution of Laplace Equation
The second type special entanglement function
w (r, θ, φ) =
q
1
l=0
η
n,l
r
l
+
η
n,−l
r
l+1
l+2
2
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(sin φ sin θ)
2j
l=0,2,4,···
+
q
2
l=0
µ
n,l
r
l
+
µ
n,−l
r
l+1
l+1
2
j=0
j
k=1
(2k − l − 1) (2k + l)
(2k + 1) 2k
(sin φ sin θ)
2j+1
l=1,3,5,···
(10)
for zero magnetic quantum numbers is a subset of the solutions to the Dirichlet problem
∂
2
w
∂r
2
+
2
r
∂w
∂r
+
1
r
2
∂
2
w
∂θ
2
+
cos θ
sin θ
∂w
∂θ
+
1
r
2
sin
2
θ
∂
2
w
∂φ
2
= 0
w (θ, φ)|
r=R
= f (θ, φ)
of the Laplace equation.
The second type of special entangled solution for the Dirichlet problem of the Laplace equation
with zero magnetic quantum numbers mentioned above does not include the Legendre special
traditional solution (8) and the first type of special entangled solution (9), which expands the
Laplace equation solution. The new conclusion further illustrates the serious flaws in the tra-
ditional theory of partial differential equations. Table 3.3 lists the specific functions calculated
from the second type of special entanglement solution (9) with q
1
= q
2
= 0, 1, 2, · · · , 6. It can be
verified that these functions all satisfy the Laplace equation.
Table 3.3 Part of the second type of special entangled solution for the zero magnetic quantum
number Laplace equation Dirichlet problem
w
0
(r, θ, φ) = µ
0,0
+
µ
0,0
r
w
1
(r, θ, φ) = sin θ sin φ
η
1,−1
r
2
+ η
1,1
r
+ µ
1,0
+
µ
1,0
r
w
2
(r, θ, φ) = sin θ sin φ
η
2,−1
r
2
+ η
2,1
r
+ µ
2,0
+
µ
2,0
r
+
1 − 3sin
2
θ sin
2
φ
µ
2,−2
r
3
+ µ
2,2
r
2
w
3
(r, θ, φ) =
sin θ sin φ
η
3,−1
r
2
+ η
3,1
r
+
sin θ sin φ −
5
3
sin
3
θ sin
3
φ
η
3,−3
r
4
+ η
3,3
r
3
+µ
3,0
+
µ
3,0
r
+
1 − 3sin
2
θ sin
2
φ
µ
3,−2
r
3
+ µ
3,2
r
2
w
4
(r, θ, φ) =
sin θ sin φ
η
4,−1
r
2
+ η
4,1
r
+
sin θ sin φ −
5
3
sin
3
θ sin
3
φ
η
4,−3
r
4
+ η
4,3
r
3
+µ
4,0
+
µ
4,0
r
+
1 − 3sin
2
θ sin
2
φ
µ
4,−2
r
3
+ µ
4,2
r
2
+
1 − 10sin
2
θ sin
2
φ +
35
3
sin
4
θ sin
4
φ
µ
4,−4
r
5
+ µ
4,4
r
4
w
5
(r, θ, φ) =
sin θ sin φ
η
5,−1
r
2
+ η
5,1
r
+
sin θ sin φ −
5
3
sin
3
θ sin
3
φ
η
5,−3
r
4
+ η
5,3
r
3
+
sin θ sin φ −
14
3
sin
3
θ sin
3
φ +
21
5
s sin
5
θ sin
5
φ
η
5,−5
r
6
+ η
5,5
r
5
+µ
5,0
+
µ
5,0
r
+
1 − 3sin
2
θ sin
2
φ
µ
5,−2
r
3
+ µ
5,2
r
2
+
1 − 10sin
2
θ sin
2
φ +
35
3
sin
4
θ sin
4
φ
µ
5,−4
r
5
+ µ
5,4
r
4
Mathematics & Nature Vol. 3 (2023) 9
w
6
(r, θ, φ) =
sin θ sin φ
η
6,−1
r
2
+ η
6,1
r
+
sin θ sin φ −
5
3
sin
3
θ sin
3
φ
η
6,−3
r
4
+ η
6,3
r
3
+
sin θ sin φ −
14
3
sin
3
θ sin
3
φ +
21
5
sin
5
θ sin
5
φ
η
6,−5
r
6
+ η
6,5
r
5
+µ
6,0
+
µ
6,0
r
+
1 − 3sin
2
θ sin
2
φ
µ
6,−2
r
3
+ µ
6,2
r
2
+
1 − 10sin
2
θ sin
2
φ +
35
3
sin
4
θ sin
4
φ
µ
6,−4
r
5
+ µ
6,4
r
4
+
1 − 21sin
2
θ sin
2
φ + 63sin
4
θ sin
4
φ −
231
5
sin
6
θ sin
6
φ
µ
6,−6
r
7
+ r
6
µ
6,6
The table is similar to the Legendre special traditional solution and the first type of special
entangled solution. Except for w
0
(r, θ, ϕ), the integral constant of the second type of special
entangled solution of the Laplace equation Dirichlet problem increases with the increase of q,
with a total of 2q + 1 > 1. Normalization conditions are far from sufficient to determine the
specific function of this type of solution. Traditional theory neglects the first and second types
of special entangled solutions, and the conclusions described do not conform to the principle of
existence and uniqueness of solutions.
4 Complete special solution set of zero magnetic quantum number Laplace
equation
Based on the two types of special entangled solutions of the Dirichlet problem of the Laplace
equation with zero magnetic quantum numbers mentioned above, a preliminary conclusion can
be drawn that the general solution of the Dirichlet problem of the Laplace equation described
by traditional theory is far less than one-third of the total solution set. In mathematical terms,
all functions that satisfy linear partial differential equations and meet the conditions for definite
solutions are solutions to partial differential equations. The special traditional solution and two
types of entangled solutions each form a solution basis for the Dirichlet problem of the Laplace
equation. Any linear combination of solution bases is the general solution of the Laplace equation
under the defined conditions. By superimp osing three special functions (8), (9), and (10), the
complete set of special solutions for the Dirichlet problem of the Laplace equation with zero
magnetic quantum numbers is obtained.
Theorem 3 (Special General Solution of Laplace Equation Dirichlet Problem): The zero
10 X. D. Dongfang Dongfang Special Entangled Spherical Solution of Laplace Equation
magnetic quantum number mixed special entanglement function
U (r, θ, φ) =
n
1
l=0
l
m=0
α
n,l
r
l
+
α
n,−l
r
l+1
l+2
2
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
cos
2j
θ
+
p
1
l=0
χ
n,l
r
l
+
χ
n,−l
r
l+1
l+2
2
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(sin θ cos φ)
2j
+
q
1
l=0
η
n,l
r
l
+
η
n,−l
r
l+1
l+2
2
j=0
j
k=1
(2k − l − 2) (2k + l − 1)
2k (2k − 1)
(sin φ sin θ)
2j
l=0,2,4,···
+
n
2
l=0
l
m=0
β
n,l
r
l
+
β
n,−l
r
l+1
l+1
2
j=0
j
k=1
(2k − l − 1) (2k + l )
(2k + 1) 2k
cos
2j+1
θ
+
p
2
l=0
γ
n,l
r
l
+
γ
n,−l
r
l+1
l+1
2
j=0
j
k=1
(2k − l − 1) (2k + l )
(2k + 1) 2k
(sin θ cos φ)
2j+1
+
q
2
l=0
µ
n,l
r
l
+
µ
n,−l
r
l+1
l+1
2
j=0
j
k=1
(2k − l − 1) (2k + l )
(2k + 1) 2k
(sin φ sin θ)
2j+1
l=1,3,5,···
(11)
is the complete set of special solutions for the Dirichlet problem
∂
2
U
∂r
2
+
2
r
∂U
∂r
+
1
r
2
∂
2
U
∂θ
2
+
cos θ
sin θ
∂U
∂θ
+
1
r
2
sin
2
θ
∂
2
U
∂φ
2
= 0
U (θ, φ)|
r=R
= f (θ, φ)
of the zero magnetic quantum number ball Laplace equation.
The complete special solution set of the Laplace equation Dirichlet problem with zero magnetic
quantum numbers consists of n
1
+ n
2
+ p
1
+ p
2
+ q
1
+ q
2
+ 4 undetermined integral constants, which
cannot be determined by normalization conditions. Therefore, the specific form of the solution to
the Laplace equation Dirichlet problem cannot be determined. In (11), take n
1
= 1, n
2
= 2, p
1
=
3, p
2
= 4, q
1
= 5, q
2
= 6, the results obtained are as follows
U
66
(r, θ, φ) =
cos θ
α
n,−1
r
2
+ α
n,1
r
+ β
n,0
+
β
n,0
r
+
1 − 3cos
2
θ
β
n,−2
r
3
+ β
n,2
r
2
+γ
n,0
+
γ
n,0
r
+
1 − 3cos
2
φsin
2
θ
γ
n,−2
r
3
+ γ
n,2
r
2
+
1 − 10cos
2
φsin
2
θ +
35
3
cos
4
φsin
4
θ
γ
n,−4
r
5
+ γ
n,4
r
4
+ sin θ sin φ
η
n,−1
r
2
+ η
n,1
r
+
sin θ sin φ −
5
3
sin
3
θsin
3
φ
η
n,−3
r
4
+ η
n,3
r
3
+
sin θ sin φ −
14
3
sin
3
θsin
3
φ +
21
5
sin
5
θsin
5
φ
η
n,−5
r
6
+ η
n,5
r
5
+µ
n,0
+
µ
n,0
r
+
1 − 3sin
2
θsin
2
φ
µ
n,−2
r
3
+ µ
n,2
r
2
+
1 − 10sin
2
θsin
2
φ +
35
3
sin
4
θsin
4
φ
µ
n,−4
r
5
+ µ
n,4
r
4
+
1 − 21sin
2
θsin
2
φ + 63sin
4
θsin
4
φ −
231
5
sin
6
θsin
6
φ
µ
n,−6
r
7
+ µ
n,6
r
6
+ cos φ sin θ
χ
n,−1
r
2
+ χ
n,1
r
+
cos φ sin θ −
5
3
cos
3
φsin
3
θ
χ
n,−3
r
4
+ χ
n,3
r
3
(12)
Mathematics & Nature Vol. 3 (2023) 11
In (11), take n1=3, n2=1, p1=2, p2=6, q2=5, The results obtained are as follows
U
66
(r, θ, φ) =
cos θ
α
n,−1
r
2
+ rα
n,1
+
cos θ −
5
3
cos
3
θ
α
n,−3
r
4
+ r
3
α
n,3
+β
n,0
+
β
n,0
r
+ γ
n,0
+
γ
n,0
r
+
1 − 3cos
2
φsin
2
θ
γ
n,−2
r
3
+ r
2
γ
n,2
+
1 − 10cos
2
φsin
2
θ +
35
3
cos
4
φsin
4
θ
γ
n,−4
r
5
+ r
4
γ
n,4
+
1 − 21cos
2
φsin
2
θ + 63cos
4
φsin
4
θ −
231
5
cos
6
φsin
6
θ
γ
n,−6
r
7
+ r
6
γ
n,6
+ sin θ sin φ
η
n,−1
r
2
+ rη
n,1
+
sin θ sin φ −
5
3
sin
3
θsin
3
φ
η
n,−3
r
4
+ r
3
η
n,3
+µ
n,0
+
µ
n,0
r
+
1 − 3sin
2
θsin
2
φ
µ
n,−2
r
3
+ r
2
µ
n,2
+ cos φ sin θ
χ
n,−1
r
2
+ rχ
n,1
+
1 − 10sin
2
θsin
2
φ +
35
3
sin
4
θsin
4
φ
µ
n,−4
r
5
+ r
4
µ
n,4
(13)
According to Theorem 3, it is possible to write functions similar to (12) and (13) that satisfy the
Laplace equation and contain arbitrary terms. After combining similar terms, there are sufficient
uncertain integral constants. Therefore, the specific functional form of the special entangled
solution of the Laplace equation with zero magnetic quantum number is uncertain.
The college system enables successive generations of students to proficiently memorize the
solutions of the Laplace equation represented by the function defined by the retraction symbol in
textbooks, but the hidden problems behind those solutions cannot be discovered, which reflects
the shortcomings of academic education. Society needs cloned talents, but scientific problems
also require independent research and creation. Even a specific example of the solution to the
Dirichlet problem of the Laplace equation with zero magnetic quantum numbers is much more
complex than the seemingly perfect universal form in memory. If different solutions such as
Laplace’s equation have received special attention in history and more people are accustomed
to manual verification, then the theory of partial differential equations will not hide numerous
significant flaws in principles that are difficult to discover like the established traditional forms.
The college system enables successive generations of students to proficiently memorize the
solutions of the Laplace equation represented by the function defined by the retraction symbol in
textbooks, but the hidden problems behind those solutions cannot be discovered, which reflects
the shortcomings of academic education. Society needs cloned talents, but scientific problems
also require independent research and creation. Even a specific example of the solution to the
Dirichlet problem of the Laplace equation with zero magnetic quantum numbers is much more
complex than the seemingly perfect universal form in memory. If different solutions such as
Laplace’s equation have received special attention in history and more people are accustomed
to manual verification, then the theory of partial differential equations will not hide numerous
significant flaws in principles that are difficult to discover like the established traditional forms.
Many details of the Laplace equation solution cannot be reflected by formal standard solutions,
but can be inferred from various concrete solutions constructed. Using Mathematica to verify
(12), the format and results that satisfy the Laplace equation should be
D
r
2
D [U, r]
, r
1
Sin[θ]
D [Sin [θ ] D [U, θ] , θ] +
1
(Sin[θ])
2
D [D [U, φ ] , φ]
= −1 (14)
However, the specific form of the Laplace equation solution with zero magnetic quantum numbers
(12) cannot be determined. All the conclusions of theories described by Laplace equations are only
non inevitable inferences hidden under special descriptive techniques, with very low confidence.
12 X. D. Dongfang Dongfang Special Entangled Spherical Solution of Laplace Equation
5 Conclusion
The Laplace equation is the simplest second-order homogeneous linear partial differential equa-
tion with symmetry, but its solution is the most complex. This only discusses the very special case
of zero magnetic quantum numbers, but the description of the problem and conclusions needed
are already cumbersome enough. As for the general case where the magnetic quantum number
is not zero, the generalized entangled spherical harmonic function can provide a richer and more
accurate solution to the Laplace equation. There seems to be no exhaustive method to find all
solutions to the Laplace equation that are not described by existing theories and differ from the
latest discoveries. Whether in the field of mathematics or applied science
[17]
, the conclusions
of Laplace equation solutions advocated by traditional theories and the descriptions of natural
phenomena are obviously unrealistic.
References
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Ltd, 2004).
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Business Media, 2006).
[3] Strauss, W. A. Partial differential equations: An introduction. (John Wiley & Sons, 2007).
[4] Evans, L. C. Partial differential equations. Vol. 19 (American Mathematical Society, 2022).
[5] Garabedian, P. R. Partial differential equations. Vol. 325 (American Mathematical Society, 2023).
[6] Silverman, R. A. Special functions and their applications. (Courier Corporation, 1972).
[7] Shortley, G. H. & Weller, R. The numerical solution of Laplace’s equation. Journal of Applied Physics 9,
334-348 (1938).
[8] Bland, D. R. Solutions of Laplace’s equation. (Springer Science & Business Media, 2012).
[9] Medkov? D. The Laplace equation. Boundary value problems on bounded and unbounded Lipschitz domains.
Springer, Cham (2018).
[10] Pinsky, M. A. Partial differential equations and boundary-value problems with applications. Vol. 15 (Amer-
ican Mathematical Soc., 2011).
[11] Dongfang, X. D. Dongfang Special Entangled Spherical Harmonic Functions. Mathematics & Nature 3,
202302 (2023).
[12] Dongfang, X. D. Dongfang Special Entangled Solution of Schr¨odinger Hydrogen Equation. Mathematics &
Nature 3, 202303 (2023).
[13] Dongfang, X. D. Dongfang Special Entangled Schr¨odinger Wave Function of Hydrogen. Mathematics &
Nature 3, 202304 (2023).
[14] Zachmanoglou, E. C. & Thoe, D. W. Introduction to partial differential equations with applications. (Courier
Corporation, 1986).
[15] Wang, Z. X. & Guo, D. R. Special functions. (world scientific, 1989).
[16] Simmons, G. F. Differential equations with applications and historical notes. (CRC Press, 2016).
[17] Zachmanoglou, E. C. & Thoe, D. W. Introduction to partial differential equations with applications. (Courier
Corporation, 1986).
Latest findings
Dongfang General Entangled Spherical Harmonic Functions
The unitary principle, which is widely applicable to the test of mathematical logic, natural science logic and social science logic, is elaborated. The traditional associated Legendre spherical harmonic function with spherical symmetry satisfying the condition of function boundedness has been extended to a general associated Legendre spherical harmonic function that includes non spherically symmetric unbounded functions. Furthermore, two types of special entangled spherical harmonic functions have been extended to include both bounded functions with spherical symmetry and non spherically symmetric unbounded functions. The linear combination of three kinds of generalized spherical harmonics constitutes the integrated generalized entangled spherical harmonics as the general solution of spherical harmonic partial differential equations. The results clearly show that the traditional theory of solving partial differential equations leads to the serious lack of exact solutions, and the given solutions of spherical harmonic partial differential equations and related wave equations are seriously distorted. The undetermined integral coefficients contained in the integrated general entangled spherical harmonic function increase with the increase of the redefined periodic quantum number. Based on the basic principles of general and specific solutions of differential equations, even if the function is required to satisfy bounded conditions, there is no specific form for the complete set of spherical harmonic functions that represent the general solution of spherical harmonic partial differential equations. This conclusion overturns the traditional theory of determining the specific solution of spherical harmonic partial differential equations using normalization conditions. From this, the uncertain principle of special solution of partial differential equations is summarized, and the uncertain principle of observation for experimental tests of quantitative inference in natural sciences is proposed, so as to avoid wasting time and effort on complex refutation of all claims of obtaining so-called experimental tests based on conclusions that are essentially miscalculations. So far, even the most conservative mathematicians and scientists have had to face up to the fact that both physical theories based on the characteristic solutions of spherical harmonic partial differential equations and certain other partial differential equations, and experimental reports testing the inferences that partial differential equations describe physical phenomena, are being challenged, and that mathematics and natural sciences will usher in a new period of great prosperity.