# Partial differential equation ## MicroSim spec - **Recommended sim type:** vector field / flow - **Microsimmability score:** 72/100 - **Layout:** drawing region (canvas) on top; control region (sliders/buttons) below. ### Parameters (tunable controls) - `Diffusion rate` - `Wave speed` - `Grid size` ### What animates A field evolves on a grid according to a PDE, spreading or propagating as parameters change. ### Learning objective Show how a PDE governs the spread or propagation of a field in space and time. ## Links (Wikipedia order) <!-- injected from _registry/childlinks/Partial_differential_equation.json (2026-07-30T02:09:12Z) --> `Abel's_identity` · `Acoustic_wave_equation` · `Adomian_decomposition_method` · `Advances_in_Mathematics` · `Alain_Connes` · `Aleksandr_Lyapunov` · `Alexander_Hrennikoff` · `Alexandre_Mikhailovich_Vinogradov` · `Alexis_Clairaut` · `Algebraic_analysis` · `Andrei_Polyanin` · `Ansatz` · `Applied_mathematics` · `Astronomy` · `Asymptotic_expansion` · `Asymptotic_homogenization` · `Augustin-Louis_Cauchy` · `Autonomous_system_(mathematics)` · `Bernstein's_problem` · `Bertil_Gustafsson` · `Biology` · `Black–Scholes_equation` · `Boltzmann_equation` · `Boundary_value_problem` · `Burgers'_equation` · `Bäcklund_transform` · [[Calculus]] · [[Calculus_of_variations]] · `Carathéodory's_existence_theorem` · `Carl_Neumann` · `Cauchy_problem` · `Cauchy–Kovalevskaya_theorem` · `Change_of_variables_(PDE)` · [[Chaos_theory]] · [[Chemistry]] · `Classical_mechanics` · [[Complex_analysis]] · `Computational_fluid_dynamics` · `Computer` · `Conic_section` · `Continuity_equation` · `Continuous_function` · `Continuum_mechanics` · [[Convolution]] · `Crank–Nicolson_method` · `David_Hilbert` · `Deep_learning` · `Delay_differential_equation` · `Dependent_and_independent_variables` · `Derivative` · `Derivative_(multivariable_calculus)` · `Differential-algebraic_system_of_equations` · [[Differential_equation]] · `Differential_geometry` · `Differential_operator` · [[Diffusion]] · `Dirac_delta_function` · `Dirichlet_boundary_condition` · `Discontinuous_Galerkin_method` · `Discriminant` · `Dispersive_partial_differential_equation` · `Divergence_theorem` · `Domain_of_a_function` · `Economics` · `Elasticity_(physics)` · `Electromagnetism` · `Electrostatics` · `Elliptic_partial_differential_equation` · `Encyclopedia_of_Mathematics` · [[Engineering]] · `Ernst_Leonard_Lindelöf` · `Euclidean_vector` · `Euler_method` · `Euler–Tricomi_equation` · `Exact_differential_equation` · `Exponential_response_formula` · `Exponential_stability` · `Extended_finite_element_method` · `Felix_Browder` · `Finite_difference` · `Finite_difference_method` · `Finite_element_method` · `Finite_volume_method` · `First-order_partial_differential_equation` · `Floquet_theory` · `Florian_Cajori` · [[Fluid_dynamics]] · `Fluid_mechanics` · [[Fourier_analysis]] · `Fourier_series` · `Fourier_transform` · `Fractional_calculus` · `Fritz_John` · `Function_(mathematics)` · `Function_of_a_real_variable` · `Function_of_several_real_variables` · [[Functional_analysis]] · `Fundamental_solution` · `Fundamental_theorem_of_calculus` · `Galerkin_method` · `General_relativity` · `Geology` · `Geometric_topology` · `George_Adomian` · [[George_Boole]] · `George_Green_(mathematician)` · `Gottfried_Wilhelm_Leibniz` · `Green's_function` · `Group_theory` · `Hans_Stephani` · `Harmonic_analysis` · `Harmonic_function` · `Harold_Jeffreys` · `Haïm_Brezis` · `Heat` · `Heat_equation` · `Helmholtz_equation` · `Hessian_matrix` · `Holonomic_function` · `Homogeneous_differential_equation` · `Homogeneous_polynomial` · `Homotopy_analysis_method` · `Hp-FEM` · `Hyperbolic_partial_differential_equation` · `Hypercomplex_analysis` · `Hypersurface` · [[Impulse_response]] · `Infinite_element_method` · `Infinitesimal_transformation` · `Infinity` · `Initial_condition` · `Initial_value_problem` · `Integral` · [[Integral_transform]] · `Integrating_factor` · `Integration_by_substitution` · `Integro-differential_equation` · [[Isaac_Newton]] · `Ivan_Petrovsky` · `Jacob_Bernoulli` · `Jacques_Charles_François_Sturm` · `Jean_Bourgain` · `Jean_Le_Rond_d'Alembert` · `Jet_bundle` · `John_Crank` · `Joseph-Louis_Lagrange` · `Joseph_Fourier` · `Joseph_Liouville` · `Józef_Maria_Hoene-Wroński` · `Jürgen_Jost` · `Klein–Gordon_equation` · `Laplace's_equation` · `Laplace_operator` · `Laplace_transform_applied_to_differential_equations` · `Lawrence_C._Evans` · `Lax_pair` · [[Least-squares_spectral_analysis]] · `Leonhard_Euler` · `Liao_Shijun` · [[Lie_group]] · `Lie_theory` · `Limit_(mathematics)` · `Linear_differential_equation` · `List_of_dynamical_systems_and_differential_equations_topics` · `List_of_linear_ordinary_differential_equations` · `List_of_named_differential_equations` · `List_of_nonlinear_ordinary_differential_equations` · `List_of_nonlinear_partial_differential_equations` · `Lists_of_integrals` · `Louis_Nirenberg` · [[Lyapunov_stability]] · `Martin_Kutta` · `Mathematical_analysis` · `Mathematics` · `Matrix_calculus` · `Matrix_differential_equation` · `Maxwell's_equations` · `Measure_(mathematics)` · `Meshfree_methods` · `Method_of_characteristics` · `Method_of_undetermined_coefficients` · `Microlocal_analysis` · `Millennium_Prize_Problems` · `Monge–Ampère_equation` · `Multi-index_notation` · `Nail_H._Ibragimov` · `Natural_science` · `Navier–Stokes_equations` · `Navier–Stokes_existence_and_smoothness` · `Neil_Gershenfeld` · `Neumann_boundary_condition` · `Noga_Alon` · `Nonlinear_partial_differential_equation` · [[Nonlinear_system]] · `Notation_for_differentiation` · `Numerical_analysis` · [[Numerical_integration]] · `Numerical_methods_for_partial_differential_equations` · [[Ordinary_differential_equation]] · `Overdetermined_system` · `P-adic_analysis` · `P-adic_number` · `Parabolic_partial_differential_equation` · `Partial_derivative` · `Partial_differential_algebraic_equation` · `Pauli_equation` · `Peano_existence_theorem` · `Periodic_boundary_conditions` · `Perturbation_theory` · `Peter_Gustav_Lejeune_Dirichlet` · `Peter_J._Olver` · `Petrov–Galerkin_method` · `Phase_portrait` · [[Phase_space]] · `Phyllis_Nicolson` · [[Physics]] · `Picard–Lindelöf_theorem` · `Pierre-Simon_Laplace` · `Poincaré_conjecture` · `Poisson's_equation` · [[Population_dynamics]] · `Power_series_solution_of_differential_equations` · `Pure_mathematics` · `Quadratic_form` · [[Quantum_mechanics]] · `Quaternionic_analysis` · `Rate_of_convergence` · [[Reaction–diffusion_system]] · `Real_analysis` · [[Recurrence_relation]] · `Reduction_of_order` · `Regularity_theory` · `Representation_theory` · `Richard_Courant` · `Robin_boundary_condition` · `Rudolf_Lipschitz` · `Runge–Kutta_methods` · [[Schrödinger_equation]] · `Separable_partial_differential_equation` · `Separation_of_variables` · `Sergiu_Klainerman` · `Series_(mathematics)` · [[Signal_processing]] · `Smoothness` · `Sobolev_space` · `Social_science` · `Sofya_Kovalevskaya` · `Sophus_Lie` · `Sound` · `Special_functions` · `Spectral_element_method` · `Stochastic_differential_equation` · `Stochastic_partial_differential_equation` · `Stochastic_processes_and_boundary_value_problems` · `Sturm–Liouville_theory` · `Supercomputer` · `Superposition_principle` · [[Thermodynamics]] · `Tropical_analysis` · `Underdetermined_system` · `Variation_of_parameters` · `Vector_calculus` · `Victor_Gustave_Robin` · `Vitali_Milman` · `WKB_approximation` · [[Wave_equation]] · [[Wayback_Machine]] · `Weak_solution` · `Well-posed_problem` · `Wronskian` · `YouTube` · `Émile_Picard` ## From the Real GENERATIVE library  *Partial differential equation — placed from the Real G.E.N.E.R.A.T.I.V.E. course library (STEM and Music room). Source: Wikimedia Commons (via Wikipedia article media). [Details & license](https://commons.wikimedia.org/wiki/File:Commons-logo.svg).*  *Animated: Partial differential equation — placed from the Real G.E.N.E.R.A.T.I.V.E. course library (STEM and Music room). Source: Wikimedia Commons (via Wikipedia article media). [Details & license](https://commons.wikimedia.org/wiki/File:Heat.gif).* > In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function. ([Wikipedia](https://en.wikipedia.org/wiki/Partial_differential_equation)) <!-- REAL-GENERATIVE-MEDIA:END --> > p5.js MicroSim stub · part of Systems Science · [Wikipedia source](https://en.wikipedia.org/wiki/Partial_differential_equation) > Relation: subfield of systems science. ## Concept summary In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. ## Build checklist - [ ] Claim it: set `status: in-progress` + `lease` + `leased_at` - [ ] Write the child page explaining the concept (tie it back to systems science) - [ ] Finalize parameter ranges and defaults - [ ] Implement the p5.js sketch (drawing + control regions) - [ ] Add caption + the learning objective on the page - [ ] Set `status: done` ## p5.js sketch ```javascript // MicroSim: Partial differential equation let controls = {}; function setup() { // createCanvas(...); create sliders for the parameters above. } function draw() { // background(...); read controls; render vector field / flow; respond live. } ``` --- Back to Systems Science · Wikipedia: [Partial differential equation](https://en.wikipedia.org/wiki/Partial_differential_equation) --- <!-- SEMIOTIC-PROFILE:START --> ## Semiotic profile > *The semiotic universals this article invokes, machine-derived from the crossref — **unverified** (born so). Populated 2026-07-06 for the Systems room.* **Universals (5):** 🟡 science (5) · 🟡 system (5) · 🟢 field (3) · 🟢 flow (3) · 🟢 diffusion (1) **Enter by sign:** Systems Semiotic Gateway · Alphabetum · Icon Registry · ← Systems Portal <!-- SEMIOTIC-PROFILE:END --> <!-- REAL-GENERATIVE-MEDIA:START --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Partial_differential_equation) : [Wikitube](https://en.wikitube.io/wiki/Partial_differential_equation) ## Previous hub tags Tree parents: [[Cellular_automaton]] · [[Complex_system]] · [[Cybernetics]] · [[Decision_theory]] · [[Dynamical_system]] · [[Emergence]] · [[Feedback]] · [[Phase_space]] · [[Self-organization]] · [[Systems_science]] · [[Systems_theory]]. Legacy hubs: none. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*