Integral Equation Ontology

FLOW ALGEBRA
INTEGRAL EQUATION ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Equation and Statement Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/equation-and-statement-entities/integral-equation-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13
Status: Formal mathematical synthesis; source page not independently verified.

01. PURPOSE
flow "IntegralEquationOntology" v1.0
  domain: mathematical_equations
  entity: IntegralEquation
  purpose: represent, classify, transform, validate and reason over equations in which an unknown function appears under an integral operator
  semantics: typed, domain-aware, condition-preserving

02. CORE ENTITY SCHEMA
entity IntegralEquation:
  id: URI
  equation_form: symbolic_expression
  unknown_function: Function
  independent_variables: list[Variable]
  integration_variables: list[BoundVariable]
  domain: Set
  integration_region: Set | variable_bounds
  kernel: Function?
  forcing_term: Function?
  coupling_parameter: Scalar?
  equation_kind: enum[fredholm,volterra,other]
  linearity: enum[linear,nonlinear]
  kind: enum[first_kind,second_kind,third_kind,other]
  constraints: list[Predicate]
  function_space: FunctionSpace?
  solution_status: enum[unknown,exists,unique,multiple,none]
  provenance: Provenance
  confidence: ConfidenceRecord

03. FUNDAMENTAL DEFINITIONS
  Fredholm_first_kind: f(x) = integral_a^b K(x,t)*u(t) dt
  Fredholm_second_kind: u(x) = f(x) + lambda*integral_a^b K(x,t)*u(t) dt
  Volterra_first_kind: f(x) = integral_a^x K(x,t)*u(t) dt
  Volterra_second_kind: u(x) = f(x) + lambda*integral_a^x K(x,t)*u(t) dt
  NOTE: sign conventions may be absorbed into lambda.
  NOTE: these are canonical linear forms, not exhaustive definitions.

04. ENTITY TYPES
  UnknownFunction: function sought as solution
  Kernel: coupling function K(x,t)
  ForcingTerm: known input f(x)
  IntegralOperator: maps functions to functions
  IntegrationDomain: region over which integration occurs
  CouplingParameter: scalar controlling integral term
  BoundaryCondition: optional condition on solution
  RegularityCondition: continuity, integrability or differentiability requirement

05. TYPED RELATIONSHIPS
  IntegralEquation HAS_UNKNOWN_FUNCTION Function
  IntegralEquation HAS_KERNEL Kernel
  IntegralEquation HAS_INTEGRATION_DOMAIN IntegrationDomain
  IntegralEquation HAS_FORCING_TERM ForcingTerm
  IntegralEquation USES_OPERATOR IntegralOperator
  IntegralEquation HAS_CONSTRAINT Predicate
  IntegralEquation HAS_SOLUTION Function
  IntegralEquation CLASSIFIED_AS EquationKind
  IntegralOperator ACTS_ON FunctionSpace
  Solution SATISFIES IntegralEquation

06. OPERATOR REPRESENTATION
  T[u](x) := integral_D K(x,t)*u(t) dt
  second_kind: (I - lambda*T)u = f
  first_kind: T[u] = f
  valid_if: T is well-defined on the declared function space
  WARNING: inverse operators require proven invertibility and appropriate domains.

07. CLASSIFICATION RULES
  if upper_bound == independent_variable and lower_bound fixed:
    infer type = Volterra (for canonical one-dimensional form)
  if integration_bounds independent_of independent_variable:
    infer type = Fredholm (for canonical one-dimensional form)
  if unknown occurs only inside integral and equation is T[u]=f:
    infer kind = first_kind
  if unknown occurs outside integral as u and equation is u=f+T[u]:
    infer kind = second_kind
  if unknown appears nonlinearly under integral:
    infer linearity = nonlinear
  preserve original equation and assumptions during classification

08. LINEARITY VALIDATION
  linear_operator(T) iff forall u,v,a,b:
    T[a*u+b*v] = a*T[u]+b*T[v]
  kernel_independent_of_unknown: required for standard linear kernel form
  check: integration domain and operator definition fixed under linear combinations
  nonlinearity_example: u(x) = f(x) + integral_0^1 K(x,t)*u(t)^2 dt

09. FREDHOLM PROPERTIES
  fixed integration interval [a,b]
  compactness may follow for continuous kernels on compact domains in suitable function spaces
  Fredholm alternative applies under specified compact-operator hypotheses
  no blanket uniqueness claim without spectral and operator assumptions

10. VOLTERRA PROPERTIES
  variable upper integration limit commonly x
  causality-like ordering for time-oriented problems
  successive approximations may converge under suitable continuity and boundedness conditions on finite intervals
  differentiate only if Leibniz rule hypotheses hold

11. SOLUTION METHODS
  method: successive_approximation
    initialise u_0 = f for second-kind form
    iterate u_(n+1) = f + lambda*T[u_n]
    accept only with convergence evidence
  method: resolvent_kernel
    applicable under valid resolvent construction and parameter conditions
  method: degenerate_kernel
    if K(x,t)=sum_i a_i(x)*b_i(t), reduce to finite-dimensional system when justified
  method: quadrature_discretisation
    replace integrals by weighted sums; record mesh and approximation error
  method: projection_galerkin
    choose finite-dimensional trial and test spaces; record stability conditions
  method: laplace_transform
    useful for convolution-type Volterra equations with transformability assumptions

12. EXISTENCE AND UNIQUENESS
  result must identify function_space, operator_bounds and assumptions
  second_kind_unique_if: I-lambda*T is bijective with bounded inverse on declared Banach space
  sufficient_condition: |lambda|*norm(T) < 1 implies unique solution by Neumann series
  first_kind_warning: may be ill-posed; existence and stability not guaranteed
  avoid inferring existence from symbolic form alone

13. SYMBOLIC TRANSFORMATIONS
  rewrite second_kind u=f+lambda*T[u] to (I-lambda*T)u=f
  allowed_if: identical domain, same operator and no hidden boundary changes
  alpha_rename integration dummy variable without capture
  distribute integral over finite sums when integrability conditions hold
  exchange integral order only under Fubini/Tonelli hypotheses
  differentiate under integral only under appropriate regularity and domination conditions

14. SEMANTIC EQUIVALENCE
  structural_equivalence: same normalised operator tree and typed operands
  alpha_equivalence: renaming bound integration variable preserves meaning
  mathematical_equivalence: equal solution sets under recorded assumptions
  approximation_relation: numerical solution approximates exact solution within stated error model
  not_equivalent: replacing fixed bounds with variable bounds without justification

15. INFERENCE RULES
  R1: integral_kernel_form + fixed_bounds -> candidate_Fredholm
  R2: integral_kernel_form + variable_upper_bound -> candidate_Volterra
  R3: (I-lambda*T)u=f + invertible(I-lambda*T) -> u=(I-lambda*T)^(-1)f
  R4: |lambda|*norm(T)<1 -> Neumann_series_converges_in_operator_norm
  R5: candidate_solution + valid_integrals + equality_verified -> solution_valid
  R6: bound_variable_rename + capture_free -> semantic_equivalence
  R7: discretised_operator -> approximate_model, never exact_model by default

16. CONSTRAINTS AND VALIDATION
  require declared integration bounds or region
  require function signatures compatible with integral expression
  require measurable/integrable integrands in specified sense
  reject undefined singularities without improper or principal-value interpretation
  reject division by zero and incompatible domains
  validate unknown_function type and output codomain
  track whether kernel is singular, weakly singular or regular
  record numerical tolerance separately from proof of exact equality

17. KNOWLEDGE GRAPH MAPPING
  node: IntegralEquation
  node: IntegralOperator
  node: Kernel
  node: FunctionSpace
  node: Solution
  edge: HAS_KERNEL
  edge: OPERATES_ON
  edge: REQUIRES_ASSUMPTION
  edge: HAS_SOLUTION
  edge: EQUIVALENT_UNDER_CONDITIONS
  edge: APPROXIMATED_BY
  edge: DERIVED_FROM
  crosslinks: EquationOntology, DifferentialEquationOntology, OperatorOntology, FunctionOntology, NumericalAnalysisOntology

18. EXAMPLES AND TEST CASES
  E1: u(x)=1+integral_0^x u(t)dt on [0,1]
    expected: Volterra second kind, linear
    verified_solution: u(x)=exp(x)
  E2: integral_0^1 u(t)dt=1 for all x in [0,1]
    expected: Fredholm first kind with K(x,t)=1
    result: infinitely many integrable solutions satisfying integral_0^1 u(t)dt=1
  E3: u(x)=f(x)+lambda*integral_0^1 x*t*u(t)dt
    expected: Fredholm second kind, degenerate kernel
  E4: u(x)=f(x)+integral_0^x u(t)^2 dt
    expected: nonlinear Volterra second kind

19. CONFIDENCE MODEL
  confidence.source: verified | unverified
  confidence.mathematical: proven | conditional | heuristic | unknown
  confidence.numeric: error_bound | empirical_tolerance | unavailable
  confidence.rule: explicit_assumptions_required
  do_not_equate heuristic confidence with formal proof

20. FLOW HASH
  canonicalise: entity identifiers, operator trees, alpha-equivalent bound variables, assumptions, version
  flow_hash: SHA256(canonical_serialisation)
  recompute_if: -expression, kernel, bounds, domain, assumptions or inference rules change
  never claim a hash value without actually computing it

21. UPDATE MODEL
  update_model: versioned
  preserve_previous_versions: true
  change_detection: semantic_diff
  revalidate: dependent_inferences, proof obligations, solution claims
  provenance: source_url, authors, timestamp, review_status

22. AI CONSUMPTION
  input: natural_language | LaTeX | MathML | typed_AST
  pipeline: parse -> typecheck -> classify -> attach_assumptions -> normalise -> infer -> verify -> serialise
  output: ontology_entity, operator_form, solution_status, proof_obligations, provenance, confidence
  require: explicit uncertainty for unverified claims
  forbid: fabricated solutions, unjustified transformations, unproven convergence claims

23. QUALITY ASSURANCE
  test: correct first_kind/second_kind classification
  test: fixed/variable integration bounds distinguished
  test: dummy variable alpha-renaming invariant
  test: nonlinear kernels not marked linear
  test: ill-posed cases not assigned guaranteed unique solutions
  test: numerical results labelled approximate
  test: source verification status retained

24. AUTHORSHIP AND DISCLOSURE
  Authors: Mark Higgins — Sky (Recursive AI)
  Research site: https://kirethresearch.org
  Disclosure: AI-assisted ontology formalisation, subject to human mathematical review.
  This specification is a formal synthesis, not a claim of executing a solver.

END FLOW ALGEBRA