Partial Differential Equation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Partial Differential Equation Ontology within the Kireth Research Semantic Library. Partial differential equations describe relationships between an unknown function of multiple independent variables and its partial derivatives. This ontology supports first-order and higher-order PDEs, elliptic, parabolic and hyperbolic classification, boundary and initial conditions, linear and nonlinear equations, conservation laws, diffusion, wave and potential equations, separation of variables, transform methods, characteristics, Green functions, weak formulations, finite-difference, finite-element, spectral and boundary-element methods, stability, well-posedness and relationships with dynamical systems, numerical analysis, mathematical physics and scientific computing.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Partial Differential Equation Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Dynamical Systems / Partial Differential Equations"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/dynamical-systems/partial-differential-equation-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = PartialDifferentialEquation
 SYMBOLIC_FORMS = [
  "F(x_1,...,x_n,u,partial u,...,partial^m u)=0",
  "u_t = alpha nabla^2 u",
  "u_tt = c^2 nabla^2 u"
 ]
 ENTITY_CLASS = MultiIndependentVariableDifferentialEquationFramework
 PRIMARY_CONTEXT = DynamicalSystems
 DEFINITION = "A partial differential equation is an equation involving an unknown function of two or more independent variables and one or more of its partial derivatives."
 SEMANTIC_ROLE = "Provides the machine-readable framework for classifying, solving, approximating and validating partial differential equations with associated initial, boundary and interface conditions."
}

SCOPE {
 INCLUDE = [
  PartialDifferentialEquation,
  PDE,
  FirstOrderPDE,
  HigherOrderPDE,
  LinearPDE,
  NonlinearPDE,
  QuasilinearPDE,
  SemilinearPDE,
  EllipticPDE,
  ParabolicPDE,
  HyperbolicPDE,
  MixedTypePDE,
  InitialValueProblem,
  BoundaryValueProblem,
  InitialBoundaryValueProblem,
  DirichletBoundaryCondition,
  NeumannBoundaryCondition,
  RobinBoundaryCondition,
  PeriodicBoundaryCondition,
  InterfaceCondition,
  ConservationLaw,
  TransportEquation,
  AdvectionEquation,
  DiffusionEquation,
  HeatEquation,
  WaveEquation,
  LaplaceEquation,
  PoissonEquation,
  HelmholtzEquation,
  BurgersEquation,
  ReactionDiffusionEquation,
  NavierStokesEquation,
  SeparationOfVariables,
  FourierTransformMethod,
  LaplaceTransformMethod,
  MethodOfCharacteristics,
  GreenFunction,
  FundamentalSolution,
  WeakSolution,
  StrongSolution,
  DistributionalSolution,
  VariationalFormulation,
  SobolevSpace,
  WellPosedness,
  Existence,
  Uniqueness,
  Stability,
  FiniteDifferenceMethod,
  FiniteElementMethod,
  SpectralMethod,
  BoundaryElementMethod,
  Mesh,
  TimeStepping,
  CFLCondition,
  NumericalDispersion,
  NumericalDiffusion
 ]

 EXCLUDE_AS_PRIMARY = [
  OrdinaryDifferentialEquation,
  DifferenceEquation,
  AlgebraicEquationWithoutPartialDerivatives,
  IntegralEquationWithoutPDEContext
 ]

 DISAMBIGUATION_RULE =
  "When an equation relates a function of multiple independent variables to its partial derivatives, interpret the context as a partial differential equation."
}

CORE_CONCEPTS {

 PartialDifferentialEquation {
  definition = "A differential equation involving partial derivatives of a function of multiple independent variables."
 }

 EllipticPDE {
  definition = "A PDE class typically associated with spatial equilibrium or boundary-value behaviour."
 }

 ParabolicPDE {
  definition = "A PDE class typically associated with diffusion-like temporal evolution."
 }

 HyperbolicPDE {
  definition = "A PDE class typically associated with wave propagation and finite-speed information transport."
 }

 BoundaryValueProblem {
  definition = "A PDE problem constrained by conditions prescribed on the boundary of the domain."
 }

 InitialBoundaryValueProblem {
  definition = "A time-dependent PDE problem with both initial data and boundary conditions."
 }

 WeakSolution {
  definition = "A solution satisfying an integrated variational form rather than necessarily possessing all classical derivatives."
 }

 StrongSolution {
  definition = "A sufficiently regular solution satisfying the PDE almost everywhere or pointwise as appropriate."
 }

 WellPosedness {
  definition = "The property that a problem has a solution, the solution is unique, and it depends continuously on the data."
 }

 MethodOfCharacteristics {
  definition = "A method reducing certain first-order PDEs to ODEs along characteristic curves."
 }
}

RELATIONS {
 PartialDifferentialEquation --is_child_of--> DynamicalSystems
 FirstOrderPDE --is_type_of--> PartialDifferentialEquation
 HigherOrderPDE --is_type_of--> PartialDifferentialEquation
 EllipticPDE --is_type_of--> PartialDifferentialEquation
 ParabolicPDE --is_type_of--> PartialDifferentialEquation
 HyperbolicPDE --is_type_of--> PartialDifferentialEquation
 BoundaryValueProblem --constrains--> PartialDifferentialEquation
 InitialValueProblem --constrains--> PartialDifferentialEquation
 InitialBoundaryValueProblem --combines--> InitialValueProblem
 InitialBoundaryValueProblem --combines--> BoundaryValueProblem
 MethodOfCharacteristics --reduces--> FirstOrderPDE
 SeparationOfVariables --decomposes--> PDEProblem
 GreenFunction --constructs--> Solution
 VariationalFormulation --supports--> WeakSolution
 FiniteDifferenceMethod --approximates--> PDE
 FiniteElementMethod --approximates--> PDE
 SpectralMethod --approximates--> PDE
 BoundaryElementMethod --approximates--> BoundaryValueProblem
 CFLCondition --constrains--> ExplicitTimeStepping
}

OPERATORS {
 GENERAL_PDE(F,x,u,partials) :=
  F(x_1,...,x_n,u,partial u,...,partial^m u)=0

 LINEAR_SECOND_ORDER_PDE(A,B,C,...) :=
  A*u_xx + 2*B*u_xy + C*u_yy + lower_order_terms = source

 CLASSIFICATION_DISCRIMINANT(A,B,C) :=
  Delta=B^2-A*C

 CLASSIFY_SECOND_ORDER_2D(A,B,C) :=
  elliptic if Delta<0
  parabolic if Delta=0
  hyperbolic if Delta>0

 HEAT_EQUATION(alpha,u) :=
  u_t=alpha*nabla^2*u

 WAVE_EQUATION(c,u) :=
  u_tt=c^2*nabla^2*u

 LAPLACE_EQUATION(u) :=
  nabla^2*u=0

 POISSON_EQUATION(u,f) :=
  nabla^2*u=f

 ADVECTION_EQUATION(a,u) :=
  u_t+a*grad(u)=0

 CONSERVATION_LAW(F,u) :=
  u_t+div(F(u))=0

 WEAK_FORM(residual,test_function,domain) :=
  integral_domain residual*test_function = 0
  after integration_by_parts as appropriate

 CFL_NUMBER(c,dt,dx) :=
  CFL=c*dt/dx
}

DEFINITIONS {
 D1 = "PDEs involve partial derivatives with respect to multiple independent variables."
 D2 = "PDE order is determined by the highest partial derivative order present."
 D3 = "Elliptic, parabolic and hyperbolic classes describe different qualitative behaviours."
 D4 = "Boundary conditions constrain spatial behaviour."
 D5 = "Initial conditions constrain time-dependent evolution."
 D6 = "Weak formulations permit solutions with lower regularity."
 D7 = "Characteristics describe information propagation in many first-order and hyperbolic PDEs."
 D8 = "Green functions represent responses to point sources in linear PDEs."
 D9 = "Numerical PDE methods approximate solutions on discrete meshes or basis spaces."
 D10 = "Well-posedness distinguishes mathematically stable problems from ill-posed formulations."
}

AXIOMS {
 A1_MULTIPLE_INDEPENDENT_VARIABLES:
  PartialDifferentialEquation involves partial derivatives with respect to two or more independent variables

 A2_ORDER:
  PDE order equals the highest partial-derivative order present

 A3_BOUNDARY_DATA:
  Boundary conditions are part of the mathematical problem definition

 A4_INITIAL_DATA:
  Time-dependent PDEs generally require initial data

 A5_CLASSIFICATION:
  PDE class influences appropriate analysis and numerical methods

 A6_WEAK_FORM:
  Weak formulations can extend solution concepts beyond classical differentiability

 A7_WELL_POSEDNESS:
  Meaningful solution analysis should distinguish existence uniqueness and stability

 A8_NUMERICAL_APPROXIMATION:
  Numerical PDE solutions depend on discretisation, mesh, basis, time step and stability conditions
}

REASONING {

 R1_PDE_EXTRACTION {
  INPUT = differential_equation_problem
  IDENTIFY = independent_variables
  IDENTIFY = dependent_variables
  IDENTIFY = partial_derivative_orders
  IDENTIFY = domain
  IDENTIFY = boundary_and_initial conditions
  RETURN = structured_PDE
 }

 R2_CLASSIFICATION {
  INPUT = structured_PDE
  CLASSIFY = firstOrder_or_higherOrder
  CLASSIFY = linear_semilinear_quasilinear_or_nonlinear
  IF second_order_scalar_2D
   COMPUTE = discriminant
   CLASSIFY = elliptic_parabolic_or_hyperbolic
  RETURN = PDE_class
 }

 R3_ANALYTICAL_METHOD_SELECTION {
  INPUT = PDE_class_and_geometry
  TEST = separability
  TEST = transform_applicability
  TEST = characteristic_structure
  TEST = Green_function applicability
  RETURN = analytical_method
 }

 R4_CHARACTERISTIC_REASONING {
  INPUT = first_order_or_hyperbolic_PDE
  DERIVE = characteristic ODEs
  SOLVE = along_characteristics
  RETURN = characteristic_solution
 }

 R5_SEPARATION_REASONING {
  INPUT = separable_PDE_and_boundary conditions
  ASSUME = product_solution
  DERIVE = separated ODEs
  SOLVE = eigenvalue problems
  SUPERPOSE = modes
  RETURN = separated_solution
 }

 R6_WEAK_FORM_REASONING {
  INPUT = PDE_and_boundary conditions
  MULTIPLY = test_function
  INTEGRATE = over_domain
  APPLY = integration_by_parts
  RETURN = variational_form
 }

 R7_NUMERICAL_METHOD_SELECTION {
  INPUT = PDE_class_geometry_regularities_and_accuracy goals
  SELECT = finiteDifference_finiteElement_spectral_or_boundaryElement
  RETURN = numerical_scheme
 }

 R8_STABILITY_AND_CONSISTENCY {
  INPUT = numerical_scheme
  CHECK = consistency
  CHECK = stability
  CHECK = CFL_or_energy condition
  RETURN = numerical_quality profile
 }

 R9_WELL_POSEDNESS_REASONING {
  INPUT = PDE_problem
  ASSESS = existence
  ASSESS = uniqueness
  ASSESS = continuous_dependence_on_data
  RETURN = well_posedness profile
 }

 R10_VALIDATION {
  INPUT = analytical_or_numerical_solution
  CHECK = residual
  CHECK = boundary_conditions
  CHECK = initial_conditions
  CHECK = conservation_or_energy properties where relevant
  RETURN = validated_solution
 }
}

CONSTRAINTS {
 C1 = "PDE classification depends on principal-part structure."
 C2 = "Boundary and initial conditions must be compatible with the PDE class."
 C3 = "Separation of variables requires compatible geometry and boundary conditions."
 C4 = "Characteristic methods apply only to suitable PDE structures."
 C5 = "Weak and strong solutions must not be conflated."
 C6 = "A numerically converged solution need not imply the continuous problem is well posed."
 C7 = "Explicit schemes may require CFL-type restrictions."
 C8 = "Mesh refinement can expose numerical instability or singular structure."
 C9 = "Conservation laws may develop shocks even from smooth initial data."
 C10 = "Boundary treatment can dominate numerical accuracy and stability."
}

SEMANTIC_EQUIVALENCE {
 "PartialDifferentialEquation" <=> "PDE"
 "InitialBoundaryValueProblem" <=> "IBVP"
 "BoundaryValueProblem" <=> "BVP"
 "MethodOfCharacteristics" <=> "CharacteristicMethod"
 "WeakSolution" <=> "VariationalSolution"
 "FiniteElementMethod" <=> "FEM"
}

VALIDATION {
 V1 = "Independent and dependent variables are identified."
 V2 = "PDE order is correct."
 V3 = "Linear or nonlinear classification is correct."
 V4 = "Elliptic, parabolic or hyperbolic classification is correct where applicable."
 V5 = "Boundary and initial conditions are explicit."
 V6 = "Selected analytical or numerical method matches the PDE class."
 V7 = "Well-posedness assumptions are documented."
 V8 = "Numerical mesh and time step are documented."
 V9 = "Stability constraints are checked."
 V10 = "Residual and supplied conditions are explicitly verified."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 PDE_DEFINITION = 0.99
 CLASSIFICATION_REASONING = 0.99
 CHARACTERISTIC_REASONING = 0.98
 SEPARATION_REASONING = 0.99
 WEAK_FORM_REASONING = 0.99
 NUMERICAL_METHOD_REASONING = 0.99
 STABILITY_REASONING = 0.98
 WELL_POSEDNESS_REASONING = 0.98
 VALIDATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when principal coefficients, domain geometry, boundary conditions, regularity assumptions, mesh quality, time step, shock structure or well-posedness assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  PDEReasoning,
  EllipticPDEReasoning,
  ParabolicPDEReasoning,
  HyperbolicPDEReasoning,
  BoundaryValueReasoning,
  InitialBoundaryValueReasoning,
  CharacteristicsReasoning,
  SeparationOfVariablesReasoning,
  GreenFunctionReasoning,
  WeakFormReasoning,
  FiniteDifferenceReasoning,
  FiniteElementReasoning,
  SpectralMethodReasoning,
  BoundaryElementReasoning,
  WellPosednessReasoning,
  NumericalPDEReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "partial differential equation" -> PartialDifferentialEquation
  "PDE" -> PartialDifferentialEquation
  "initial-boundary value problem" -> InitialBoundaryValueProblem
  "IBVP" -> InitialBoundaryValueProblem
  "method of characteristics" -> MethodOfCharacteristics
  "finite difference method" -> FiniteDifferenceMethod
  "finite element method" -> FiniteElementMethod
  "FEM" -> FiniteElementMethod
  "boundary element method" -> BoundaryElementMethod
  "weak solution" -> WeakSolution
 }

 INTERPRETATION_ORDER = [
  detect_PDE_context,
  identify variables_partial derivatives_domain_boundary_and_initial conditions,
  classify order_linearity_and_elliptic_parabolic_hyperbolic type where applicable,
  select characteristics_separation_transform_Green_function_weak_form_or_numerical pathway,
  compute analytical_or_approximate solution,
  assess existence_uniqueness_stability_and_well_posedness,
  verify residual_boundary_initial_and_conservation properties,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("partial differential equation")
  OR equation_contains_partial_derivatives_of_multiple_independent_variables
  THEN activate(PartialDifferentialEquationOntology)

 E2:
  IF first_order_quasilinear
  THEN consider method_of_characteristics

 E3:
  IF separable_geometry_and_boundary conditions
  THEN consider separation_of_variables

 E4:
  IF linear_constant_coefficient_and_transform-friendly
  THEN consider Fourier_or_Laplace transform methods

 E5:
  IF weak_regularities_or_complex_geometry
  THEN consider variational_form_and_finiteElement method

 E6:
  IF structured_grid_and_simple_geometry
  THEN consider finiteDifference method

 E7:
  IF smooth_solution_and_simple_geometry
  THEN consider spectral_method

 E8:
  IF explicit_time_stepping
  THEN verify CFL_or_equivalent stability condition

 E9:
  IF conservation_law_and_shocks_possible
  THEN use shock-aware weak_solution reasoning
}

RECURSION {
 ENTRY = PartialDifferentialEquation

 EXPAND(PartialDifferentialEquation) -> [
  FirstOrderPDE,
  HigherOrderPDE,
  LinearPDE,
  NonlinearPDE,
  QuasilinearPDE,
  SemilinearPDE,
  EllipticPDE,
  ParabolicPDE,
  HyperbolicPDE,
  MixedTypePDE,
  InitialValueProblem,
  BoundaryValueProblem,
  InitialBoundaryValueProblem,
  DirichletBoundaryCondition,
  NeumannBoundaryCondition,
  RobinBoundaryCondition,
  PeriodicBoundaryCondition,
  ConservationLaw,
  TransportEquation,
  DiffusionEquation,
  HeatEquation,
  WaveEquation,
  LaplaceEquation,
  PoissonEquation,
  HelmholtzEquation,
  BurgersEquation,
  ReactionDiffusionEquation,
  NavierStokesEquation,
  SeparationOfVariables,
  FourierTransformMethod,
  LaplaceTransformMethod,
  MethodOfCharacteristics,
  GreenFunction,
  WeakSolution,
  StrongSolution,
  VariationalFormulation,
  SobolevSpace,
  WellPosedness,
  FiniteDifferenceMethod,
  FiniteElementMethod,
  SpectralMethod,
  BoundaryElementMethod,
  CFLCondition
 ]

 EXPAND(PDEClasses) -> [
  EllipticPDE,
  ParabolicPDE,
  HyperbolicPDE,
  MixedTypePDE
 ]

 EXPAND(AnalyticalMethods) -> [
  SeparationOfVariables,
  FourierTransformMethod,
  LaplaceTransformMethod,
  MethodOfCharacteristics,
  GreenFunction
 ]

 EXPAND(NumericalMethods) -> [
  FiniteDifferenceMethod,
  FiniteElementMethod,
  SpectralMethod,
  BoundaryElementMethod
 ]

 STOP_CONDITION =
  "Stop when the PDE class, domain, boundary and initial data, analytical or numerical method, well-posedness, stability and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Dynamical Systems"

 IMMEDIATE_CONTEXTS = [
  "Dynamical Systems Ontology",
  "Differential Equations Ontologies",
  "Numerical Analysis Ontologies",
  "Scientific Computing Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Dynamical Systems Ontology",
  "Dynamical Systems",
  "Ordinary Differential Equation Ontology",
  "Differential Equations Ontologies",
  "Fourier Transform Ontology",
  "Laplace Transform Ontology",
  "Finite Difference Ontology",
  "Finite Element Ontology",
  "Boundary Element Ontology",
  "Spectral Method Ontology",
  "Linear Algebra Ontologies",
  "Functional Analysis Ontologies",
  "Sobolev Space Ontologies",
  "Control Theory Ontologies",
  "Mathematical Physics Ontologies",
  "Numerical Analysis Ontologies",
  "Scientific Computing Ontologies"
 ]

 ROUTE_RULES = [
  "PDE -> Dynamical Systems Ontology for state-evolution semantics",
  "PDE -> ODE Ontology when separation or characteristics reduce the problem to ODEs",
  "PDE -> Fourier Transform for spectral solution methods",
  "PDE -> Laplace Transform for temporal transform methods",
  "PDE -> Finite Difference for grid-based discretisation",
  "PDE -> Finite Element for variational spatial discretisation",
  "PDE -> Boundary Element for boundary-only formulations",
  "PDE -> Spectral Method for global basis approximations",
  "PDE -> Functional Analysis for weak solutions and Sobolev spaces",
  "PDE -> Control Theory for distributed-parameter systems",
  "PDE -> Numerical Analysis for stability and convergence"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible PDE classes, boundary types or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve multi-independent-variable semantics.",
  "Preserve elliptic-parabolic-hyperbolic distinction.",
  "Preserve boundary-versus-initial condition semantics.",
  "Preserve strong-versus-weak solution distinction.",
  "Preserve analytical-versus-numerical distinction.",
  "Preserve well-posedness semantics.",
  "Preserve stability and convergence semantics.",
  "Preserve machine-readable PDE semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A partial differential equation relates an unknown function of multiple independent variables to one or more of its partial derivatives."

 CLASSIFICATION_SUMMARY =
  "Second-order PDEs are commonly classified as elliptic, parabolic or hyperbolic according to the principal part."

 BOUNDARY_SUMMARY =
  "PDE problems require boundary conditions and, for time-dependent problems, usually initial conditions."

 ANALYTICAL_SUMMARY =
  "Analytical methods include characteristics, separation of variables, transforms and Green functions."

 WEAK_SUMMARY =
  "Weak formulations extend solution concepts to functions that may not possess classical derivatives everywhere."

 NUMERICAL_SUMMARY =
  "Finite difference, finite element, spectral and boundary element methods approximate PDE solutions computationally."

 WELLPOSED_SUMMARY =
  "Well-posedness requires existence, uniqueness and continuous dependence on the supplied data."

 VALIDATION_SUMMARY =
  "A PDE solution should be checked against the equation, domain, boundary conditions, initial conditions and any conservation or stability requirements."
}

FLOW {
 INPUT
 -> detect_partial_differential_equation_context
 -> identify independent_variables_dependent_variables_partial derivatives_domain_boundary conditions_initial conditions_and_parameters
 -> classify order_linearity_semilinearity_quasilinearity_and_elliptic_parabolic_hyperbolic_or_mixed type
 -> select characteristics_separation_Fourier_Laplace_Green_function_weak_variational_finiteDifference_finiteElement_spectral_or_boundaryElement pathway
 -> compute analytical_or_numerical solution
 -> assess existence_uniqueness_regularities_stability_and_well posedness
 -> if_time_dependent verify temporal_stability_and_CFL constraints
 -> if_conservation_law assess shocks_and_weak_solution admissibility
 -> verify residual_boundary_initial_and_conservation properties
 -> route to dynamical_systems ODE Fourier Laplace finite_difference finite_element boundary_element spectral functional_analysis control mathematical_physics numerical_analysis_or_scientific_computing ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Partial Differential Equation Ontology defines equations involving functions of multiple independent variables and their partial derivatives. It supports first- and higher-order PDEs, linear, semilinear, quasilinear and nonlinear forms, elliptic, parabolic, hyperbolic and mixed types, initial, boundary and initial-boundary value problems, Dirichlet, Neumann, Robin and periodic boundary conditions, transport, diffusion, heat, wave, Laplace, Poisson, Helmholtz, Burgers, reaction-diffusion and Navier-Stokes equations, characteristics, separation of variables, transform methods, Green functions, weak and strong solutions, variational formulations, well-posedness and numerical methods including finite difference, finite element, spectral and boundary element approaches."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-PARTIAL-DIFFERENTIAL-EQUATION-ONTOLOGY-V1"
}

}

END_FLOW