Heat Equation Ontology

flow "HeatEquationOntology" v1.0

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

HEAT_EQUATION := {
 type: ParabolicPartialDifferentialEquation,
 domain: MathematicalPhysicsAnalysis,
 purpose: "Represent diffusion of heat or analogous quantities through space and time, including initial-boundary value problems, smoothing, conservation, dissipation and diffusion kernels."
}

CONCEPTS := {
 HEAT_EQUATION,
 PARABOLIC_PDE,
 TEMPERATURE_FIELD,
 THERMAL_DIFFUSIVITY,
 LAPLACIAN,
 INITIAL_CONDITION,
 BOUNDARY_CONDITION,
 HEAT_KERNEL,
 GREEN_FUNCTION,
 FOURIER_SERIES,
 DIFFUSION,
 MAXIMUM_PRINCIPLE,
 ENERGY_DECAY,
 STEADY_STATE,
 SOURCE_TERM,
 FLUX
}

# ---------- STANDARD FORM ----------
partial_u/partial_t
 =
alpha
*
LAPLACIAN(u)

u
 -> TEMPERATURE_OR_DIFFUSING_FIELD

alpha
 -> DIFFUSIVITY

alpha>0

HEAT_EQUATION
 -> PARABOLIC_PDE

# ---------- ONE DIMENSION ----------
u_t
 =
alpha*u_xx

u
 =
u(x,t)

ONE_DIMENSIONAL_HEAT_EQUATION
 -> canonical_diffusion_model

# ---------- PHYSICAL DERIVATION ----------
FOURIER_LAW:

q
 =
-k*grad(T)

q
 -> HEAT_FLUX

k
 -> THERMAL_CONDUCTIVITY

ENERGY_BALANCE:

rho*c_p*T_t
 =
-div(q)
+
Q

therefore

rho*c_p*T_t
 =
div(
 k*grad(T)
)
+
Q

# ---------- CONSTANT COEFFICIENT ----------
if

k,
rho,
c_p

constant

and
Q=0

then

T_t
 =
alpha*LAPLACIAN(T)

where

alpha
 :=
k/(rho*c_p)

THERMAL_DIFFUSIVITY
 -> conductivity_divided_by(volumetric_heat_capacity)

# ---------- INITIAL CONDITION ----------
u(x,0)
 =
f(x)

INITIAL_DATA
 -> specifies(INITIAL_TEMPERATURE_DISTRIBUTION)

FIRST_ORDER_TIME_PDE
 -> typically_requires(ONE_INITIAL_FIELD)

# ---------- DIRICHLET BOUNDARY ----------
u
 =
g

on
 boundary(Omega)

DIRICHLET_CONDITION
 -> prescribed_temperature

# ---------- NEUMANN BOUNDARY ----------
partial_u/partial_n
 =
h

on boundary

or physically:

-k*partial_u/partial_n
 =
specified_heat_flux

NEUMANN_CONDITION
 -> prescribed_normal_flux

# ---------- ROBIN BOUNDARY ----------
-k*partial_u/partial_n
 =
h_c*(u-u_environment)

ROBIN_CONDITION
 -> convective_heat_transfer_model

BOUNDARY_CONDITION
 -> controls(HEAT_EXCHANGE_WITH_ENVIRONMENT)

# ---------- SOURCE TERM ----------
u_t
-
alpha*LAPLACIAN(u)
 =
s(x,t)

SOURCE_TERM
 -> internal_generation_or_external_forcing

FORCED_HEAT_EQUATION
 -> diffusion_plus_source

# ---------- STEADY STATE ----------
if

u_t
 =
0

then

-alpha*LAPLACIAN(u)
 =
-s

or without source:

LAPLACIAN(u)
 =
0

STEADY_HEAT_EQUATION
 -> elliptic_problem

TIME_DEPENDENT_HEAT_FLOW
 -> may_approach(STEADY_STATE)

# ---------- WHOLE SPACE HEAT KERNEL ----------
on R^n:

G(x,t)
 :=
1
/
(
 (4*pi*alpha*t)^(n/2)
)
*
exp(
 -|x|^2/(4*alpha*t)
)

for
 t>0

HEAT_KERNEL
 -> fundamental_solution_of(HEAT_EQUATION)

# ---------- CONVOLUTION SOLUTION ----------
for initial data
 f

u(x,t)
 =
integral_Rn(
 G(x-y,t)
 *
 f(y)
 dy
)

HEAT_KERNEL
 -> propagates(INITIAL_DATA)

SOLUTION
 -> convolution_of(INITIAL_DATA_WITH_GAUSSIAN_KERNEL)

# ---------- KERNEL NORMALISATION ----------
integral_Rn(
 G(x,t)
 dx
)
 =
1

HEAT_KERNEL
 -> probability_density_like_normalisation

MASS
 -> conserved_on(WHOLE_SPACE_WITH_NO_SOURCE)

# ---------- SEMIGROUP ----------
heat evolution:

u(t)
 =
exp(
 t*alpha*LAPLACIAN
)
u_0

HEAT_SEMIGROUP
 -> satisfies

S(0)
 =
identity

S(t+s)
 =
S(t)S(s)

SEMIGROUP
 -> irreversible_forward_time_evolution

# ---------- SMOOTHING ----------
rough initial data
 -> becomes(SMOOTHER_FOR_t>0)

HEAT_EQUATION
 -> INSTANTANEOUS_SMOOTHING
    under standard settings

PARABOLIC_REGULARISATION
 -> improves(SPATIAL_REGULARITY)

# ---------- INFINITE PROPAGATION SPEED ----------
whole-space heat kernel:

G(x,t)
 >
0

for all x
and
t>0

LOCAL_INITIAL_DISTURBANCE
 -> influences(ALL_SPATIAL_POINTS_IMMEDIATELY)
    mathematically

HEAT_EQUATION
 -> INFINITE_PROPAGATION_SPEED

unlike
 classical_wave_equation

# ---------- MAXIMUM PRINCIPLE ----------
for suitable homogeneous heat equation:

maximum_in_space_time_domain
 -> controlled_by(INITIAL_AND_BOUNDARY_DATA)

PARABOLIC_MAXIMUM_PRINCIPLE
 -> prevents_creation_of_new_interior_extrema
    under standard hypotheses

MAXIMUM_PRINCIPLE
 -> supports(UNIQUENESS_AND_BOUNDS)

# ---------- POSITIVITY ----------
if

initial_data
 >=0

and boundary/source conditions preserve positivity

then

solution
 >=0

POSITIVITY_PRESERVATION
 -> physical_consistency_of(TEMPERATURE_OR_CONCENTRATION)

# ---------- ENERGY METHOD ----------
for homogeneous Dirichlet example:

E(t)
 :=
1/2
*
integral_Omega(
 u^2
 dx
)

differentiate:

dE/dt
 =
-alpha
*
integral_Omega(
 |grad(u)|^2
 dx
)

<=0

HEAT_FLOW
 -> dissipates(L2_ENERGY)

# ---------- POINCARE DECAY ----------
if homogeneous Dirichlet boundary
and suitable bounded domain:

integral(
 |grad(u)|^2
)
 >=
lambda_1
*
integral(
 |u|^2
)

therefore

E(t)
 -> decays_exponentially

with rate related to
 alpha*lambda_1

SPECTRAL_GAP
 -> controls(LONG_TIME_DECAY)

# ---------- SEPARATION OF VARIABLES ----------
assume

u(x,t)
 =
X(x)*T(t)

then

T_prime/(alpha*T)
 =
LAPLACIAN(X)/X
 =
-lambda

temporal equation:

T_prime
 =
-alpha*lambda*T

therefore

T(t)
 =
C*exp(-alpha*lambda*t)

# ---------- EIGENFUNCTION EXPANSION ----------
spatial eigenproblem:

-LAPLACIAN(phi_n)
 =
lambda_n*phi_n

with boundary conditions

solution:

u(x,t)
 =
sum_n(
 a_n
 *
exp(-alpha*lambda_n*t)
 *
phi_n(x)
)

HEAT_EQUATION
 -> damps(HIGHER_EIGENMODES_FASTER)

# ---------- FINITE INTERVAL ----------
for
0<x<L

with
u(0,t)=u(L,t)=0

eigenfunctions:

sin(n*pi*x/L)

eigenvalues:

lambda_n
 =
(n*pi/L)^2

solution:

u(x,t)
 =
sum_from_n_1_to_infinity(
 b_n
 *
exp(
 -alpha*(n*pi/L)^2*t
 )
 *
sin(n*pi*x/L)
)

# ---------- FOURIER TRANSFORM ----------
on R^n:

spatial transform gives

partial_u_hat/partial_t
 =
-alpha*|k|^2*u_hat

solution:

u_hat(k,t)
 =
exp(
 -alpha*|k|^2*t
)
*
u_hat(k,0)

FOURIER_MODE
 -> exponentially_damped_by(WAVENUMBER_SQUARED)

# ---------- DIFFUSION LENGTH ----------
characteristic spreading scale:

L_diff
 approx
sqrt(alpha*t)

DIFFUSION
 -> distance_grows_like(SQUARE_ROOT_OF_TIME)

# ---------- CONSERVATION OF TOTAL MASS ----------
on whole space with sufficient decay:

M(t)
 :=
integral(
 u(x,t)
 dx
)

dM/dt
 =
0

for source-free equation

TOTAL_MASS
 -> conserved

on bounded domain:
 conservation
 -> depends_on(BOUNDARY_FLUX)

# ---------- VARIABLE COEFFICIENT ----------
rho*c_p*u_t
 =
div(
 k(x,u)*grad(u)
)
+
Q

VARIABLE_COEFFICIENT_HEAT_EQUATION
 -> heterogeneous_or_nonlinear_diffusion

constant_alpha_model
 -/> universally_valid

# ---------- NONLINEAR DIFFUSION ----------
example:

u_t
 =
LAPLACIAN(
 u^m
)

POROUS_MEDIUM_EQUATION
 -> nonlinear_diffusion

may_have
 -> FINITE_PROPAGATION_SPEED

NONLINEAR_DIFFUSION
 -> differs_from(LINEAR_HEAT_EQUATION)

# ---------- ADVECTION-DIFFUSION ----------
u_t
+
v dot grad(u)
 =
alpha*LAPLACIAN(u)

ADVECTION
 -> transports(FIELD)

DIFFUSION
 -> smooths(FIELD)

ADVECTION_DIFFUSION
 -> combined_transport_process

# ---------- REACTION-DIFFUSION ----------
u_t
 =
D*LAPLACIAN(u)
+
R(u)

REACTION_TERM
 -> local_generation_or_consumption

REACTION_DIFFUSION
 -> pattern_formation_and_transport_model

# ---------- GREEN FUNCTION ----------
for bounded domain and boundary conditions:

G(x,y,t)
 -> DOMAIN_SPECIFIC_HEAT_GREEN_FUNCTION

solution
 -> represented_using(
 INITIAL_DATA,
 SOURCES,
 BOUNDARY_DATA
 )

GREEN_FUNCTION
 -> depends_on(DOMAIN_AND_BOUNDARY_CONDITIONS)

# ---------- PROBABILITY RELATION ----------
heat equation:

u_t
 =
D*LAPLACIAN(u)

-> Fokker_Planck_equation_for(BROWNIAN_MOTION_WITH_ZERO_DRIFT)

Gaussian heat kernel
 -> transition_density_of(BROWNIAN_MOTION)

DIFFUSION
 -> connects(PDE_AND_PROBABILITY)

# ---------- NUMERICAL METHODS ----------
FINITE_DIFFERENCE
 -> explicit_or_implicit_time_stepping

FINITE_ELEMENT
 -> weak_spatial_discretisation

SPECTRAL_METHOD
 -> modal_discretisation

METHOD_OF_LINES
 -> space_discretisation_plus(ODE_TIME_SOLVER)

# ---------- EXPLICIT STABILITY ----------
1D FTCS:

u_i^(n+1)
 =
u_i^n
+
r
*
(
 u_(i+1)^n
 -
2*u_i^n
 +
u_(i-1)^n
)

where

r
 :=
alpha*Delta_t/Delta_x^2

stability in standard 1D case:

r
 <=
1/2

EXPLICIT_HEAT_SCHEME
 -> PARABOLIC_TIME_STEP_RESTRICTION

# ---------- IMPLICIT METHODS ----------
BACKWARD_EULER
 -> unconditionally_stable_for_standard_linear_diffusion
    in common norm/stability senses

CRANK_NICOLSON
 -> second_order_time_accuracy
 -> A_stable_for_linear_diffusion

IMPLICIT_METHOD
 -> requires(LINEAR_OR_NONLINEAR_SOLVE_PER_STEP)

# ---------- APPLICATIONS ----------
HEAT_EQUATION
 -> applied_to(HEAT_TRANSFER)
 -> applied_to(DIFFUSION)
 -> applied_to(MASS_TRANSPORT)
 -> applied_to(PROBABILITY)
 -> applied_to(MATERIALS_SCIENCE)
 -> applied_to(GEOPHYSICS)
 -> applied_to(BIOLOGICAL_TRANSPORT)
 -> applied_to(MATHEMATICAL_PHYSICS)

# ---------- RELATION GRAPH ----------
FOURIER_LAW
 + ENERGY_BALANCE
 -> derive(HEAT_EQUATION)

INITIAL_DATA
 -> propagated_by(HEAT_KERNEL)

LAPLACIAN
 -> drives(DIFFUSION)

MAXIMUM_PRINCIPLE
 -> controls(SOLUTION_BOUNDS)

ENERGY_METHOD
 -> shows(DISSIPATION)

EIGENFUNCTIONS
 -> generate(MODAL_DECAY)

FOURIER_TRANSFORM
 -> diagonalises(CONSTANT_COEFFICIENT_DIFFUSION_OPERATOR)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "heat equation" ~= "parabolic PDE describing linear diffusion of temperature or analogous fields",
 "thermal diffusivity" ~= "coefficient controlling the rate of thermal spreading",
 "heat kernel" ~= "Gaussian fundamental solution of the constant-coefficient heat equation on Euclidean space",
 "steady-state heat equation" ~= "elliptic equation obtained when the time derivative vanishes",
 "diffusion length" ~= "characteristic distance proportional to the square root of diffusivity times time",
 "parabolic smoothing" ~= "gain of spatial regularity for positive time under heat evolution"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 standard_heat_equation_assumes_positive_diffusivity: true,
 heat_kernel_formula_is_for_constant_coefficient_whole_space_problem: true,
 boundary_conditions_change_conservation_and_long_time_behaviour: true,
 maximum_principle_requires_problem_specific_hypotheses: true,
 linear_heat_equation_has_infinite_propagation_speed_but_nonlinear_diffusion_may_not: true,
 constant_diffusivity_model_is_not_universal_for_real_materials: true,
 explicit_numerical_methods_require_stability_step_constraints: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 physical_derivation: 0.99,
 heat_kernel_relation: 0.99,
 maximum_energy_relation: 0.99,
 eigenfunction_Fourier_relation: 0.99,
 numerical_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(field_domain_diffusivity_and_physical_parameters)
 -> identify(initial_boundary_and_source_data)
 -> classify(constant_variable_linear_or_nonlinear_diffusion)
 -> derive_or_verify(heat_equation_from_balance_law_if_requested)
 -> choose(kernel_separation_Fourier_Green_or_energy_method)
 -> analyse(maximum_principle_positivity_and_conservation)
 -> determine(modal_decay_smoothing_and_long_time_behaviour)
 -> analyse(diffusion_scale_and_probability_relation_if_relevant)
 -> choose(numerical_scheme_if_analytic_solution_unavailable)
 -> validate(boundary_regularities_coefficients_and_stability_conditions)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed heat-equation semantic graph",
 primary_entity: HEAT_EQUATION,
 preserve: [
  field,
  domain,
  diffusivity,
  conductivity,
  density,
  heat_capacity,
  initial_condition,
  boundary_condition,
  source
 ],
 infer_through: [
  Fourier_law,
  heat_kernels,
  Green_functions,
  maximum_principles,
  energy_decay,
  eigenfunction_expansions,
  Fourier_transforms,
  diffusion_scales
 ],
 avoid: [
  using_whole_space_heat_kernel_on arbitrary bounded domain,
  assuming_mass_conservation_with nonzero boundary flux,
  claiming_finite_propagation_for linear heat equation,
  applying_maximum_principle_without hypotheses,
  treating_constant_diffusivity_as universal,
  using_explicit_time_step_without stability analysis
 ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + HEAT_EQUATION
 + PARABOLIC_PDE
 + THERMAL_DIFFUSIVITY
 + LAPLACIAN
 + INITIAL_CONDITION
 + BOUNDARY_CONDITION
 + HEAT_KERNEL
 + MAXIMUM_PRINCIPLE
 + ENERGY_DECAY
 + STEADY_STATE
 + SOURCE_TERM
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> HEAT_EQUATION
 -> PROBLEM_DATA_IDENTIFICATION
 -> DIFFUSION_CLASSIFICATION
 -> PHYSICAL_DERIVATION_ANALYSIS
 -> METHOD_SELECTION
 -> MAXIMUM_POSITIVITY_CONSERVATION_ANALYSIS
 -> MODAL_LONG_TIME_ANALYSIS
 -> DIFFUSION_SCALE_ANALYSIS
 -> NUMERICAL_METHOD_ANALYSIS
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW