Initial Value Problem Ontology

flow "InitialValueProblemOntology" v1.0

# ============================================================
# INITIAL VALUE PROBLEM ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/differential-equations/initial-value-problem-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Differential Equations"
ENTITY := "Initial Value Problem"

# ---------- CORE ENTITY ----------
INITIAL_VALUE_PROBLEM := {
    type: DifferentialEquationProblem,
    domain: DifferentialEquations,
    abbreviation: "IVP",
    purpose: "Determine a solution of a differential equation that satisfies specified data at an initial point or initial time."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    INITIAL_VALUE_PROBLEM,
    DIFFERENTIAL_EQUATION,
    INITIAL_CONDITION,
    INITIAL_POINT,
    INITIAL_TIME,
    UNKNOWN_FUNCTION,
    SOLUTION,
    EXISTENCE,
    UNIQUENESS,
    PICARD_LINDELOF_THEOREM,
    LOCAL_SOLUTION,
    MAXIMAL_SOLUTION,
    NUMERICAL_SOLUTION,
    STABILITY
}

# ---------- BASIC STRUCTURE ----------
INITIAL_VALUE_PROBLEM
    := DIFFERENTIAL_EQUATION
       + INITIAL_DATA

for first-order ODE:

y_prime
    = f(x,y)

y(x_0)
    = y_0

x_0
    -> INITIAL_POINT

y_0
    -> INITIAL_VALUE

# ---------- INITIAL CONDITION ----------
INITIAL_CONDITION
    -> constrains(SOLUTION_AT_INITIAL_POINT)

INITIAL_DATA
    -> selects(SOLUTION_FROM_GENERAL_FAMILY)
       when(SOLUTION_FAMILY_AND_UNIQUENESS_STRUCTURE_ALLOW)

# ---------- NTH-ORDER ODE ----------
for nth-order ODE:

y^(n)
    = F(
        x,
        y,
        y_prime,
        ...,
        y^(n-1)
      )

INITIAL_DATA
    := {
        y(x_0) = c_0,
        y_prime(x_0) = c_1,
        ...
        y^(n-1)(x_0) = c_(n-1)
    }

NTH_ORDER_IVP
    -> commonly_requires(n initial conditions)
       for(LOCAL_UNIQUE_SOLUTION_UNDER_STANDARD_REGULARITY)

# ---------- FIRST-ORDER SYSTEM ----------
SYSTEM_IVP
    := x_prime(t)
       = F(t,x)

with

x(t_0)
    = x_0

STATE_VECTOR
    -> initialised_at(t_0)

SYSTEM_FORM
    -> standard_framework_for(IVP_ANALYSIS)

# ---------- HIGHER-ORDER TO SYSTEM ----------
HIGHER_ORDER_ODE
    -> transformed_to(FIRST_ORDER_SYSTEM)

define:

x_1 = y
x_2 = y_prime
...
x_n = y^(n-1)

then

STATE_VECTOR
    := (x_1,...,x_n)

INITIAL_DERIVATIVE_DATA
    -> becomes(INITIAL_STATE_VECTOR)

# ---------- SOLUTION ----------
SOLUTION
    := function_satisfying(
        DIFFERENTIAL_EQUATION,
        INITIAL_CONDITIONS
    )

IVP_SOLUTION
    -> must_satisfy(EQUATION)
    -> must_satisfy(INITIAL_DATA)

# ---------- EXISTENCE ----------
EXISTENCE
    -> asks(Whether_at_least_one_solution_passes_through_initial_data)

for
    y_prime = f(x,y)

if
    f continuous_near(x_0,y_0)
then

PEANO_EXISTENCE_THEOREM
    -> may_guarantee(LOCAL_EXISTENCE)
       under(STANDARD_CONDITIONS)

EXISTENCE
    -/> automatically_implies(UNIQUENESS)

# ---------- UNIQUENESS ----------
UNIQUENESS
    -> asks(Whether_exactly_one_local_solution_satisfies_initial_data)

if
    f continuous
    AND locally_lipschitz_in_y
    near(x_0,y_0)
then

PICARD_LINDELOF_THEOREM
    -> LOCAL_EXISTENCE_AND_UNIQUENESS

LIPSCHITZ_CONDITION
    -> controls(SOLUTION_BRANCH_SEPARATION)

# ---------- NONUNIQUENESS ----------
if
    uniqueness_conditions_fail
then

IVP
    -> may_have(MULTIPLE_SOLUTIONS)

FAILURE_OF_THEOREM_HYPOTHESIS
    -/> proves(NONUNIQUENESS)

NONUNIQUENESS
    -> must_be_established_by(ANALYSIS_OR_COUNTEREXAMPLE)

# ---------- LOCAL SOLUTION ----------
LOCAL_SOLUTION
    -> defined_on(INTERVAL_AROUND_INITIAL_POINT)

LOCAL_EXISTENCE
    -/> necessarily_implies(GLOBAL_EXISTENCE)

SOLUTION
    -> may_terminate_due_to(BLOW_UP)
    -> may_terminate_due_to(DOMAIN_BOUNDARY)
    -> may_extend_to(MAXIMAL_INTERVAL)

# ---------- MAXIMAL SOLUTION ----------
MAXIMAL_SOLUTION
    -> solution_that_cannot_be_extended_further
       while_remaining_solution_of(IVP)

MAXIMAL_INTERVAL_OF_EXISTENCE
    -> largest_connected_interval_for(SOLUTION_BRANCH)

# ---------- CONTINUOUS DEPENDENCE ----------
WELL_POSED_IVP
    -> solution_exists
    -> solution_unique
    -> solution_depends_continuously_on(initial_data)

CONTINUOUS_DEPENDENCE
    -> supports(STABILITY_OF_SOLUTION_WITH_RESPECT_TO_DATA)

# ---------- LINEAR IVP ----------
LINEAR_FIRST_ORDER_IVP:

y_prime
    + P(x)y
    = Q(x)

y(x_0)
    = y_0

INTEGRATING_FACTOR
    -> solves(DIFFERENTIAL_EQUATION)

INITIAL_CONDITION
    -> determines(CONSTANT_OF_INTEGRATION)

# ---------- SEPARABLE IVP ----------
y_prime
    = g(x)h(y)

y(x_0)
    = y_0

SEPARATION_OF_VARIABLES
    -> may_generate(IMPLICIT_OR_EXPLICIT_SOLUTION)

EQUILIBRIUM_SOLUTIONS
    -> must_be_preserved_WHEN_DIVIDING_BY_h(y)

INITIAL_DATA
    -> selects(RELEVANT_BRANCH)

# ---------- AUTONOMOUS IVP ----------
x_prime
    = F(x)

x(0)
    = x_0

INITIAL_STATE
    -> determines(TRAJECTORY)
       when(UNIQUENESS_HOLDS)

FLOW_MAP
    -> maps(INITIAL_STATE_TO_LATER_STATE)

# ---------- PDE INITIAL VALUE PROBLEM ----------
for evolution PDE:

partial_u/partial_t
    = L[u]

u(x,0)
    = u_0(x)

u_0
    -> INITIAL_PROFILE

PDE_IVP
    -> evolves(INITIAL_DATA_IN_TIME)

INITIAL_DATA_REQUIREMENTS
    -> depend_on(PDE_ORDER_AND_TYPE)

# ---------- CAUCHY PROBLEM ----------
CAUCHY_PROBLEM
    -> general_initial_data_problem_for(DIFFERENTIAL_EQUATION)

INITIAL_DATA
    -> prescribed_on(INITIAL_HYPERSURFACE)
       in(PDE_CONTEXT)

CAUCHY_PROBLEM
    -> generalises(ODE_INITIAL_VALUE_PROBLEM)

# ---------- NUMERICAL SOLUTION ----------
NUMERICAL_IVP_SOLVER
    -> starts_from(INITIAL_DATA)
    -> advances(SOLUTION_APPROXIMATION)

EULER_METHOD:
    y_(n+1)
        = y_n
          + h*f(x_n,y_n)

RUNGE_KUTTA_METHODS
    -> use(MULTIPLE_STAGE_EVALUATIONS)

STEP_SIZE
    -> affects(ACCURACY)
    -> affects(NUMERICAL_STABILITY)

# ---------- LOCAL TRUNCATION ERROR ----------
LOCAL_TRUNCATION_ERROR
    -> error_introduced_by(ONE_NUMERICAL_STEP)

GLOBAL_ERROR
    -> accumulated_solution_error_over(MANY_STEPS)

CONVERGENCE
    -> requires(ERROR_DECREASE_WITH_REFINEMENT)

# ---------- STIFF IVP ----------
STIFF_IVP
    -> contains(STRONGLY_SEPARATED_TIME_SCALES)

EXPLICIT_METHODS
    -> may_require(SMALL_STEP_FOR_STABILITY)

IMPLICIT_METHODS
    -> often_preferred_for(STIFF_SYSTEMS)

# ---------- SENSITIVITY ----------
SOLUTION
    -> depends_on(INITIAL_DATA)

SENSITIVITY
    -> measures(RESPONSE_TO_INITIAL_PERTURBATION)

VARIATIONAL_EQUATION
    -> may_describe(LINEARISED_PERTURBATION_DYNAMICS)

# ---------- APPLICATIONS ----------
INITIAL_VALUE_PROBLEM
    -> applied_to(MECHANICS)
    -> applied_to(POPULATION_DYNAMICS)
    -> applied_to(CHEMICAL_KINETICS)
    -> applied_to(ELECTRICAL_CIRCUITS)
    -> applied_to(CONTROL_SYSTEMS)
    -> applied_to(EPIDEMIOLOGY)
    -> applied_to(TIME_DEPENDENT_PDEs)

# ---------- RELATION GRAPH ----------
DIFFERENTIAL_EQUATION
    + INITIAL_DATA
    -> INITIAL_VALUE_PROBLEM

INITIAL_CONDITION
    -> selects(SOLUTION_BRANCH)

CONTINUITY
    -> may_support(EXISTENCE)

LIPSCHITZ_CONDITION
    -> supports(UNIQUENESS)

LOCAL_SOLUTION
    -> may_extend_to(MAXIMAL_SOLUTION)

NUMERICAL_METHOD
    -> propagates(INITIAL_STATE)

INITIAL_DATA
    -> determines(TRAJECTORY)
       when(UNIQUENESS_HOLDS)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "initial value problem" ~= "differential equation together with initial data",
    "IVP" ~= "initial value problem",
    "initial condition" ~= "constraint imposed at an initial point or time",
    "initial state" ~= "state vector specified at the starting time",
    "Cauchy problem" ~= "general initial-data problem, especially in PDE contexts",
    "maximal solution" ~= "solution extended over its largest admissible connected interval"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    ivp_requires_differential_equation_and_initial_data: true,
    solution_must_satisfy_equation_and_initial_conditions: true,
    existence_does_not_imply_uniqueness: true,
    theorem_hypotheses_must_be_verified: true,
    local_existence_does_not_guarantee_global_existence: true,
    numerical_solution_is_approximate_unless_exactness_proven: true,
    initial_data_requirements_depend_on_equation_order_and_type: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    initial_condition_relation: 0.99,
    existence_relation: 0.98,
    uniqueness_relation: 0.99,
    maximal_solution_relation: 0.98,
    numerical_ivp_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(differential_equation)
    -> identify(initial_point_or_time)
    -> identify(initial_data)
    -> determine(equation_order_and_type)
    -> convert_to(first_order_system_if_useful)
    -> test(existence_conditions)
    -> test(uniqueness_conditions)
    -> select(analytical_or_numerical_method)
    -> construct(solution_from_initial_data)
    -> determine(local_or_maximal_interval_if_required)
    -> verify(equation_and_initial_conditions)
    -> analyse(stability_sensitivity_or_error_if_requested)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed initial-value-problem semantic graph",
    primary_entity: INITIAL_VALUE_PROBLEM,
    preserve: [
        differential_equation,
        initial_point,
        initial_time,
        initial_values,
        derivative_initial_data,
        domain,
        existence_conditions,
        uniqueness_conditions
    ],
    infer_through: [
        ode_systems,
        cauchy_problems,
        picard_lindelof,
        peano_existence,
        numerical_solvers,
        maximal_solutions,
        sensitivity
    ],
    avoid: [
        assuming_existence_implies uniqueness,
        claiming_uniqueness_without lipschitz_or_equivalent conditions,
        assuming_local_solution_is global,
        losing_equilibrium_solution during separation,
        ignoring_initial conditions after general solution,
        treating_numerical approximation as exact solution
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + INITIAL_VALUE_PROBLEM
    + INITIAL_CONDITION
    + PICARD_LINDELOF_THEOREM
    + LOCAL_SOLUTION
    + MAXIMAL_SOLUTION
    + CAUCHY_PROBLEM
    + NUMERICAL_IVP_SOLVER
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> INITIAL_VALUE_PROBLEM
    -> DIFFERENTIAL_EQUATION_IDENTIFICATION
    -> INITIAL_DATA_IDENTIFICATION
    -> ORDER_TYPE_ANALYSIS
    -> SYSTEM_FORM_ANALYSIS
    -> EXISTENCE_ANALYSIS
    -> UNIQUENESS_ANALYSIS
    -> METHOD_SELECTION
    -> SOLUTION_CONSTRUCTION
    -> INTERVAL_OF_EXISTENCE_ANALYSIS
    -> SOLUTION_VALIDATION
    -> STABILITY_SENSITIVITY_ERROR_ANALYSIS
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW