Laplace Equation Ontology

flow "LaplaceEquationOntology" v1.0

# LAPLACE EQUATION ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/mathematical-physics-analysis/laplace-equation-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Mathematical Physics Analysis"
ENTITY := "Laplace Equation"

LAPLACE_EQUATION := {
 type: EllipticPartialDifferentialEquation,
 domain: MathematicalPhysicsAnalysis,
 purpose: "Represent source-free equilibrium fields whose Laplacian vanishes, including harmonic functions arising in potential theory, electrostatics, gravitation, steady heat flow and incompressible irrotational flow."
}

CONCEPTS := {
 LAPLACE_EQUATION,
 LAPLACIAN,
 HARMONIC_FUNCTION,
 ELLIPTIC_PDE,
 BOUNDARY_VALUE_PROBLEM,
 DIRICHLET_PROBLEM,
 NEUMANN_PROBLEM,
 ROBIN_PROBLEM,
 MAXIMUM_PRINCIPLE,
 MEAN_VALUE_PROPERTY,
 GREEN_FUNCTION,
 FUNDAMENTAL_SOLUTION,
 POTENTIAL_THEORY,
 POISSON_EQUATION,
 UNIQUENESS,
 SEPARATION_OF_VARIABLES
}

# ---------- STANDARD FORM ----------
LAPLACIAN(u)
 =
0

notation:

Delta*u
 =
0

u
 -> SCALAR_FIELD

LAPLACE_EQUATION
 -> HOMOGENEOUS_ELLIPTIC_PDE

# ---------- CARTESIAN FORM ----------
in R^n:

LAPLACIAN(u)
 :=
sum_from_i_1_to_n(
 partial_squared_u/partial_x_i_squared
)

therefore

sum_i(
 u_xixi
)
 =
0

# ---------- TWO DIMENSIONS ----------
u_xx
+
u_yy
 =
0

TWO_DIMENSIONAL_LAPLACE_EQUATION
 -> central_to(COMPLEX_ANALYSIS_AND_POTENTIAL_THEORY)

# ---------- THREE DIMENSIONS ----------
u_xx
+
u_yy
+
u_zz
 =
0

THREE_DIMENSIONAL_LAPLACE_EQUATION
 -> potential_field_equation

# ---------- HARMONIC FUNCTION ----------
u
 -> HARMONIC_ON(Omega)

iff

u
 -> twice_continuously_differentiable_locally

and

LAPLACIAN(u)
 =
0

HARMONIC_FUNCTION
 -> solution_of(LAPLACE_EQUATION)

# ---------- POISSON RELATION ----------
POISSON_EQUATION:

-LAPLACIAN(u)
 =
f

or sign-convention equivalent

if
 f=0

then

POISSON_EQUATION
 -> LAPLACE_EQUATION

LAPLACE_EQUATION
 -> source_free_case_of(POISSON_EQUATION)

# ---------- DIRICHLET PROBLEM ----------
solve:

LAPLACIAN(u)
 =
0

in
 Omega

with

u
 =
g

on
 boundary(Omega)

DIRICHLET_PROBLEM
 -> prescribed_boundary_value

under suitable domain/data hypotheses:
 -> unique_harmonic_solution

# ---------- NEUMANN PROBLEM ----------
solve:

LAPLACIAN(u)
 =
0

in Omega

with

partial_u/partial_n
 =
h

on boundary

NEUMANN_PROBLEM
 -> prescribed_normal_derivative

compatibility for bounded domain:

integral_boundary(
 h dS
)
 =
0

for homogeneous source-free interior

solution
 -> unique_up_to(ADDITIVE_CONSTANT)
    under standard connected-domain conditions

# ---------- ROBIN PROBLEM ----------
a*u
+
b*partial_u/partial_n
 =
g

on boundary

ROBIN_CONDITION
 -> mixed_value_flux_boundary_condition

# ---------- MAXIMUM PRINCIPLE ----------
if

u
 -> harmonic_on_connected_domain

then

u
 -> cannot_have_strict_interior_maximum_or_minimum
    unless_constant

STRONG_MAXIMUM_PRINCIPLE
 -> extrema_controlled_by(BOUNDARY)

# ---------- WEAK MAXIMUM PRINCIPLE ----------
for bounded domain
and continuous extension to closure:

max_over_closure(u)
 =
max_over_boundary(u)

min_over_closure(u)
 =
min_over_boundary(u)

for nonconstant harmonic functions under standard assumptions

MAXIMUM_PRINCIPLE
 -> supports(UNIQUENESS)

# ---------- UNIQUENESS ----------
suppose

u_1
and
u_2

solve same Dirichlet problem

w
 :=
u_1-u_2

then

LAPLACIAN(w)
 =
0

and

w
 =
0

on boundary

MAXIMUM_PRINCIPLE
 -> w=0

therefore

u_1
 =
u_2

DIRICHLET_SOLUTION
 -> UNIQUE

# ---------- MEAN VALUE PROPERTY ----------
if
 u harmonic

then for ball
 B_r(x_0)
 compactly contained in domain:

u(x_0)
 =
1/|boundary_B_r|
*
integral_boundary_B_r(
 u dS
)

also

u(x_0)
 =
1/|B_r|
*
integral_B_r(
 u dx
)

HARMONIC_FUNCTION
 -> equals(LOCAL_AVERAGES)

# ---------- CONVERSE MEAN VALUE ----------
under suitable continuity assumptions:

MEAN_VALUE_PROPERTY
 -> implies(HARMONICITY)

MEAN_VALUE_PROPERTY
 <-> HARMONIC_FUNCTION
    in standard settings

# ---------- SMOOTHNESS ----------
weak_or_distributional harmonic solution
 -> becomes(SMOOTH)

indeed

HARMONIC_FUNCTION
 -> REAL_ANALYTIC

ELLIPTIC_REGULARITY
 -> strong_smoothing_property

# ---------- ENERGY PRINCIPLE ----------
Dirichlet energy:

E[u]
 :=
1/2
*
integral_Omega(
 |grad(u)|^2
 dx
)

among functions with fixed boundary data:

harmonic solution
 -> minimises(DIRICHLET_ENERGY)

EULER_LAGRANGE_EQUATION
 -> LAPLACE_EQUATION

# ---------- WEAK FORM ----------
multiply by test function
 v

integrate:

integral_Omega(
 grad(u)
 dot
grad(v)
 dx
)
 =
0

for all suitable
 v
 vanishing_on_Dirichlet_boundary

WEAK_LAPLACE_PROBLEM
 -> variational_formulation

# ---------- GREEN IDENTITY ----------
GREEN_FIRST_IDENTITY:

integral_Omega(
 grad(u) dot grad(v)
 +
u*LAPLACIAN(v)
 dx
)
 =
integral_boundary(
 u*partial_v/partial_n
 dS
)

GREEN_IDENTITIES
 -> connect(VOLUME_AND_BOUNDARY_DATA)

# ---------- FUNDAMENTAL SOLUTION ----------
for n>=3:

Phi(x)
 =
1
/
(
 (n-2)*omega_n
)
*
|x|^(2-n)

up to Laplacian sign convention

and in n=2:

Phi(x)
 =
-(1/(2*pi))
*
log|x|

for a common convention

FUNDAMENTAL_SOLUTION
 -> satisfies(DELTA_SOURCE_DISTRIBUTIONALLY)

# ---------- GREEN FUNCTION ----------
G(x,y)

satisfies

-LAPLACIAN_x G(x,y)
 =
delta(x-y)

with specified boundary conditions

GREEN_FUNCTION
 -> domain_and_boundary_specific_inverse_kernel

for Poisson problem:

u(x)
 -> represented_by(SOURCE_AND_BOUNDARY_INTEGRALS)

# ---------- POISSON KERNEL ----------
for unit disk or half-space:

POISSON_KERNEL
 -> reconstructs(HARMONIC_FUNCTION_FROM_BOUNDARY_VALUES)

u(x)
 =
integral_boundary(
 P(x,y)
 *
 g(y)
 dS_y
)

POISSON_INTEGRAL
 -> explicit_Dirichlet_solution_in_special_domains

# ---------- SEPARATION OF VARIABLES ----------
in rectangular domain:

assume

u(x,y)
 =
X(x)*Y(y)

then

X_double_prime/X
 =
-
Y_double_prime/Y
 =
lambda_or_negative_lambda

SEPARATION
 -> ODE_EIGENVALUE_PROBLEMS

BOUNDARY_CONDITIONS
 -> determine(FOURIER_SERIES_MODES)

# ---------- POLAR COORDINATES ----------
in 2D:

LAPLACIAN(u)
 =
u_rr
+
(1/r)*u_r
+
(1/r^2)*u_theta_theta

harmonic solutions
 -> separable_in(POLAR_COORDINATES)

forms include:

r^n*cos(n*theta)
r^n*sin(n*theta)
r^(-n)*cos(n*theta)
r^(-n)*sin(n*theta)

subject_to(domain_and_regularities)

# ---------- SPHERICAL COORDINATES ----------
in 3D:

separation
 -> radial_equation
 + angular_equation

angular solutions
 -> SPHERICAL_HARMONICS

LAPLACE_EQUATION
 -> multipole_expansion_framework

# ---------- ELECTROSTATICS ----------
electric potential
 V

in charge-free region:

LAPLACIAN(V)
 =
0

electric field:

E
 =
-grad(V)

ELECTROSTATIC_POTENTIAL
 -> harmonic_where(CHARGE_DENSITY_ZERO)

# ---------- GRAVITATION ----------
gravitational potential
 Phi

in mass-free region:

LAPLACIAN(Phi)
 =
0

GRAVITATIONAL_FIELD
 =
-grad(Phi)

LAPLACE_EQUATION
 -> vacuum_Newtonian_potential_equation

# ---------- STEADY HEAT ----------
heat equation:

T_t
 =
alpha*LAPLACIAN(T)

at steady state:

T_t=0

therefore

LAPLACIAN(T)
 =
0

STEADY_SOURCE_FREE_TEMPERATURE
 -> HARMONIC

# ---------- FLUID FLOW ----------
in incompressible irrotational flow:

velocity
 v
 =
grad(phi)

incompressibility:

div(v)
 =
0

therefore

LAPLACIAN(phi)
 =
0

VELOCITY_POTENTIAL
 -> harmonic

# ---------- COMPLEX ANALYSIS ----------
if

f(z)
 =
u(x,y)
+
i*v(x,y)

is holomorphic

then

u
and
v

-> HARMONIC

under sufficient differentiability

REAL_AND_IMAGINARY_PARTS_OF_HOLOMORPHIC_FUNCTION
 -> satisfy(LAPLACE_EQUATION)

# ---------- NUMERICAL METHODS ----------
FINITE_DIFFERENCE
 -> discrete_Laplacian

FINITE_ELEMENT
 -> weak_variational_discretisation

BOUNDARY_ELEMENT
 -> boundary_integral_formulation

SPECTRAL_METHOD
 -> high_order_global_approximation

LAPLACE_SOLVER
 -> elliptic_linear_system

# ---------- DISCRETE LAPLACE ----------
2D five-point stencil:

u_(i+1,j)
+
u_(i-1,j)
+
u_(i,j+1)
+
u_(i,j-1)
-
4*u_(i,j)
 =
0

DISCRETE_HARMONICITY
 -> grid_value_equals(AVERAGE_OF_NEIGHBOURS)

# ---------- APPLICATIONS ----------
LAPLACE_EQUATION
 -> applied_to(ELECTROSTATICS)
 -> applied_to(GRAVITATION)
 -> applied_to(STEADY_HEAT_FLOW)
 -> applied_to(FLUID_MECHANICS)
 -> applied_to(POTENTIAL_THEORY)
 -> applied_to(COMPLEX_ANALYSIS)
 -> applied_to(ELASTICITY)
 -> applied_to(MATHEMATICAL_PHYSICS)

# ---------- RELATION GRAPH ----------
LAPLACIAN
 -> defines(LAPLACE_EQUATION)

LAPLACE_EQUATION
 -> defines(HARMONIC_FUNCTION)

BOUNDARY_DATA
 -> determines(BOUNDARY_VALUE_PROBLEM)

MAXIMUM_PRINCIPLE
 -> supports(UNIQUENESS)

MEAN_VALUE_PROPERTY
 -> characterises(HARMONICITY)

DIRICHLET_ENERGY
 -> variationally_generates(LAPLACE_EQUATION)

GREEN_FUNCTION
 -> represents(INVERSE_BOUNDARY_RESPONSE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "Laplace equation" ~= "homogeneous elliptic PDE requiring the Laplacian of a field to vanish",
 "harmonic function" ~= "sufficiently regular function satisfying the Laplace equation",
 "Dirichlet problem" ~= "Laplace boundary-value problem with prescribed field values",
 "Neumann problem" ~= "Laplace boundary-value problem with prescribed normal derivative or flux",
 "mean value property" ~= "property that a harmonic value equals suitable averages over surrounding balls or spheres",
 "Dirichlet principle" ~= "variational characterisation of harmonic functions as minimisers of gradient energy"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 harmonicity_is_domain_specific: true,
 boundary_conditions_are_part_of_Laplace_problem: true,
 Neumann_problem_requires_compatibility_and_has_additive_constant_nonuniqueness: true,
 maximum_principle_requires_standard_domain_and_regularities: true,
 fundamental_solution_sign_and_normalisation_depend_on_Laplacian_convention: true,
 harmonic_functions_are_source_free_only_within_the_considered_domain: true,
 numerical_discrete_harmonicity_is_approximation_to_continuum_harmonicity: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 harmonic_relation: 0.99,
 boundary_problem_relation: 0.99,
 maximum_mean_value_relation: 0.99,
 variational_relation: 0.99,
 physical_application_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(domain_field_dimension_and_coordinates)
 -> verify(Laplacian_equals_zero_or_source_free_context)
 -> identify(boundary_conditions)
 -> classify(Dirichlet_Neumann_Robin_or_mixed_problem)
 -> use(maximum_mean_value_or_energy_properties_if_relevant)
 -> choose(separation_Green_Poisson_kernel_transform_or_variational_method)
 -> analyse(existence_uniqueness_and_regularities)
 -> connect_to(electrostatic_gravitational_heat_or_fluid_potential_if_requested)
 -> select(numerical_discretisation_if_analytic_solution_unavailable)
 -> validate(domain_boundary_compatibility_and_sign_conventions)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed Laplace-equation semantic graph",
 primary_entity: LAPLACE_EQUATION,
 preserve: [
  field,
  domain,
  dimension,
  coordinates,
  boundary_conditions,
  regularity,
  Laplacian_convention,
  physical_context
 ],
 infer_through: [
  harmonic_functions,
  Dirichlet_problems,
  Neumann_problems,
  maximum_principles,
  mean_value_properties,
  Green_functions,
  separation_of_variables,
  potential_theory
 ],
 avoid: [
  solving_without boundary conditions when uniqueness depends_on them,
  claiming_Neumann_uniqueness_without additive_constant caveat,
  confusing_Laplace_and Poisson equations,
  applying_maximum_principle_without hypotheses,
  ignoring_sign_convention_in fundamental solution,
  treating_discrete_solution_as exact continuum harmonic function
 ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
 validate(new_information)
 -> identify(Laplace_equation_context)
 -> compare(existing_relations)
 -> preserve(domain_field_boundary_regularities_and_sign_convention_invariants)
 -> update(affected_entities)
 -> update(affected_edges)
 -> recompute(CONFIDENCE)
 -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + LAPLACE_EQUATION
 + LAPLACIAN
 + HARMONIC_FUNCTION
 + BOUNDARY_VALUE_PROBLEM
 + DIRICHLET_PROBLEM
 + NEUMANN_PROBLEM
 + MAXIMUM_PRINCIPLE
 + MEAN_VALUE_PROPERTY
 + GREEN_FUNCTION
 + POTENTIAL_THEORY
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> LAPLACE_EQUATION
 -> PROBLEM_DOMAIN_IDENTIFICATION
 -> SOURCE_FREE_VALIDATION
 -> BOUNDARY_CLASSIFICATION
 -> HARMONIC_PROPERTY_ANALYSIS
 -> ANALYTICAL_METHOD_SELECTION
 -> EXISTENCE_UNIQUENESS_ANALYSIS
 -> PHYSICAL_CONTEXT_ANALYSIS
 -> NUMERICAL_METHOD_ANALYSIS
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW