Partial Differential Equations (PDE) Solver Calculator
Solve and analyze partial differential equations including heat, wave, and Laplace equations
Category: Mathematics
Partial Differential Equations (PDE) Solver Calculator Inputs
Partial Differential Equations (PDE) Solver Calculator Formula
Equation
∂u/∂t = α∇²u (heat), ∂²u/∂t² = c²∇²u (wave), ∇²u = 0 (Laplace)
Excel Formula
=∂u/∂t=α∇^2u(heat),∂^2u/∂t^2=c^2∇^2u(wave),∇^2u=0(Laplace)
Variables
- PDE Type — Choose the PDE Type option used by the Partial Differential Equations (PDE) Solver.
- Spatial Dimension — Choose the Spatial Dimension option used by the Partial Differential Equations (PDE) Solver.
- Boundary Conditions — Enter the Boundary Conditions text used by the Partial Differential Equations (PDE) Solver.
- Initial Conditions — Enter the Initial Conditions text used by the Partial Differential Equations (PDE) Solver.
How the Partial Differential Equations (PDE) Solver Calculator Works
Partial Differential Equations (PDEs) involve unknown functions of multiple variables and their partial derivatives. Unlike ODEs (one independent variable), PDEs model spatially extended systems: diffusion, wave propagation, steady-state fields, quantum mechanics, fluid flow, and more. The three classical PDEs are: **Heat equation** $\frac{\partial u}{\partial t} = \alpha \nabla^2 u$ (parabolic, models diffusion), **Wave equation** $\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$ (hyperbolic, models vibrations/waves), and **Laplace equation** $\nabla^2 u = 0$ (elliptic, models steady states). Solutions require boundary conditions (BC) and initial conditions (IC). Solution methods include separation of variables (Fourier series), Green's functions, transform methods (Laplace/Fourier), characteristics (for hyperbolic), numerical methods (finite differences, finite elements). PDEs are arguably the most important equations in applied mathematics, governing virtually all continuous physical phenomena.
The core relationship is ∂u/∂t = α∇²u (heat), ∂²u/∂t² = c²∇²u (wave), ∇²u = 0 (Laplace). Typical inputs include PDE Type, Spatial Dimension, Boundary Conditions, Initial Conditions.
Enter your values in the partial differential equations (pde) solver calculator above, review the step-by-step solution, and compare against the worked examples below so you can see how each input changes the result. This free online mathematics tool is built for homework, design checks, and professional verification.
Partial Differential Equations (PDE) Solver Calculator Theory & Explanation
Heat Equation: Diffusion Processes
The heat equation models diffusion—temperature, concentration, probability density: (\partial u)/(\partial t) = α \nabla^2 u where u(\mathbfr, t) is temperature (or concentration), α is diffusivity (thermal diffusivity for heat, diffusion coefficient for mass). In 1D: (\partial u)/(\partial t) = α (\partial^2 u)/(\partial x^2). Laplacian \nabla^2 u measures curvature—if u is locally curved (high second derivative), it flattens out (diffuses). Parabolic PDE: information propagates at finite speed, but influence extends infinitely fast (instantaneous far-field effect, physically approximation). Solution via separation: u(x,t) = Σ_n A_n e^-α\lambda_n^2 t \sin(√(\lambda_n) x) (Fourier series). Initial "bump" spreads out, eventually uniform. Heat equation governs thermal engineering, chemical diffusion, financial options (Black-Scholes), image processing (blurring).
(\partial u)/(\partial t) = α \nabla^2 u \quad (\textheat/diffusion)
Wave Equation: Propagation Phenomena
The wave equation models vibrations and wave propagation: (\partial^2 u)/(\partial t^2) = c^2 \nabla^2 u where u(\mathbfr, t) is displacement, c is wave speed. In 1D: (\partial^2 u)/(\partial t^2) = c^2 (\partial^2 u)/(\partial x^2) (string/sound waves). Hyperbolic PDE: information propagates at finite speed c (causality). General solution (1D): u(x,t) = f(x - ct) + g(x + ct) (d'Alembert)—left and right traveling waves. For bounded domain with BCs: standing waves u(x,t) = Σ_n [A_n\cos(\omega_n t) + B_n\sin(\omega_n t)]\sin((nπ x)/(L)) where \omega_n = (nπ c)/(L) (natural frequencies). Wave equation governs acoustics, EM waves (light), seismology, vibrating structures (bridges, buildings), quantum fields.
(\partial^2 u)/(\partial t^2) = c^2 \nabla^2 u \quad (\textwave)
Laplace and Poisson Equations: Steady States
Laplace equation \nabla^2 u = 0 and Poisson equation \nabla^2 u = f model steady-state (time-independent) phenomena. In 2D Cartesian: (\partial^2 u)/(\partial x^2) + (\partial^2 u)/(\partial y^2) = 0 or = f(x,y). Elliptic PDEs: no time evolution, solution depends on boundary conditions all around (any BC change affects entire domain). Solutions are harmonic functions (Laplace) or satisfy maximum principle (extrema on boundary only). Physical applications: electrostatic potential (\nabla^2 \phi = -\rho/\epsilon_0, Poisson), gravitational potential, steady-state heat (\nabla^2 T = -Q/k with heat source), fluid potential flow (\nabla^2 \phi = 0), membrane equilibrium. Solution methods: separation of variables (rectangular domain), Fourier series, complex analysis (conformal mapping), Green's functions, numerical (finite elements).
\nabla^2 u = 0 \quad (\textLaplace), \quad \nabla^2 u = f \quad (\textPoisson)
Separation of Variables Method
For linear PDEs on rectangular domains, assume solution is product: u(x,y,t) = X(x)Y(y)T(t). Substitute into PDE, separate variables (each side depends on different variable, so both equal constant). Example: 2D heat equation u_t = α(u_xx + u_yy) on rectangle [0,L] × [0,W] with u=0 on boundary. Assume u = X(x)Y(y)T(t). Then: (T')/(α T) = (X'')/(X) + (Y'')/(Y). Left depends only on t, right only on x,y, so both equal -\lambda (constant). Further separate X,Y: (X'')/(X) = -\lambda_x, (Y'')/(Y) = -\lambda_y with \lambda = \lambda_x + \lambda_y. Solutions: X_n = \sin(nπ x/L), Y_m = \sin(mπ y/W), T_nm = e^-α(\lambda_n+\lambda_m)t. General solution: u = Σ_n,m A_nm \sin((nπ x)/(L))\sin((mπ y)/(W)) e^-α(...) t (double Fourier series). Separation reduces PDE to ODEs—powerful technique.
u(x,y,t) = Σ_n,m A_nm X_n(x) Y_m(y) T_nm(t)
Classification: Elliptic, Parabolic, Hyperbolic
Second-order linear PDE: Au_xx + 2Bu_xy + Cu_yy + \textlower order = 0. Classification by discriminant B^2 - AC: **Elliptic** (B^2 - AC < 0): Laplace, Poisson. Steady-state, boundary-value problems. Solutions smooth, max principle holds. **Parabolic** (B^2 - AC = 0): Heat equation. Time evolution, initial-boundary value. Smoothing, irreversible. **Hyperbolic** (B^2 - AC > 0): Wave equation. Propagation, characteristics. Reversible, energy-conserving. Each type has different mathematical properties and numerical methods. Elliptic: solve entire domain simultaneously (linear systems). Parabolic: march forward in time (stable explicit/implicit schemes). Hyperbolic: characteristics, upwind methods. Recognizing type guides solution approach.
B^2 - AC \begincases < 0 & \textElliptic \\ = 0 & \textParabolic \\ > 0 & \textHyperbolic \endcases
Boundary and Initial Conditions
PDEs require auxiliary conditions for unique solutions. **Boundary conditions** (BCs): Dirichlet: u = g on boundary (specified value). Neumann: (\partial u)/(\partial n) = g (specified derivative/flux). Robin/Mixed: α u + β (\partial u)/(\partial n) = g (combination). **Initial conditions** (ICs): For time-dependent PDEs: u(\mathbfr, 0) = f(\mathbfr) and (for second-order in time) (\partial u)/(\partial t)(\mathbfr, 0) = g(\mathbfr). Well-posed problem: solution exists, unique, depends continuously on data (Hadamard). Example: heat equation on rod [0,L] needs 2 BCs (one at each end) + 1 IC (temperature at t=0). Wave equation needs 2 BCs + 2 ICs (position and velocity at t=0). Proper BCs/ICs ensure physically meaningful, unique solutions.
\textDirichlet: u = g, \quad \textNeumann: (\partial u)/(\partial n) = g, \quad \textRobin: α u + β (\partial u)/(\partial n) = g
Applications Across Science and Engineering
PDEs model continuous systems. **Physics**: Quantum (Schrödinger i\hbar(\partial\psi)/(\partial t) = -(\hbar^2)/(2m)\nabla^2\psi + V\psi), electromagnetism (Maxwell), general relativity (Einstein field equations). **Engineering**: Heat transfer (thermal analysis), structural vibrations (elastic waves), fluid flow (Navier-Stokes), acoustics (sound propagation). **Finance**: Black-Scholes (option pricing): (\partial V)/(\partial t) + (1)/(2)\sigma^2 S^2 (\partial^2 V)/(\partial S^2) + rS(\partial V)/(\partial S) - rV = 0. **Biology**: Population diffusion, chemotaxis, pattern formation (Turing). **Climate**: Atmosphere/ocean dynamics. **Image processing**: Denoising, segmentation. Virtually every physical field (temperature, pressure, concentration, displacement, potential) satisfies a PDE. Understanding PDEs is essential for modeling, simulation, and prediction in modern science and engineering.
\textPDEs govern: heat, waves, fluids, quantum, finance, biology, climate, ...
Partial Differential Equations (PDE) Solver Calculator Worked Examples
Worked Example
Inputs
- pdeType: heat
- dimension: 1
- boundaryConditions: u(0,t)=0, u(1,t)=0
- initialConditions: u(x,0)=sin(πx)
Result: u(x,t) = sin(πx)e^(-απ²t)
Explanation
Separation of variables with BCs gives single mode solution
Wave on String
Inputs
- pdeType: wave
- dimension: 1
- boundaryConditions: u(0,t)=0, u(L,t)=0
- initialConditions: u(x,0)=sin(πx/L), u_t(x,0)=0
Result: u(x,t) = sin(πx/L)cos(πct/L)
Explanation
Standing wave, fundamental frequency ω=πc/L
Common Partial Differential Equations (PDE) Solver Calculator Use Cases
- Homework and exam practice
- Engineering and science coursework
- Quick verification of hand calculations
- Wave
- And Laplace equations
Partial Differential Equations (PDE) Solver Calculator FAQs
What is a partial differential equation?
A PDE involves unknown function of multiple variables (e.g., u(x,y,t)) and its partial derivatives (∂u/∂x, ∂²u/∂x², ∂u/∂t, etc.). Unlike ODEs (one variable), PDEs model spatial phenomena. Example: heat equation ∂u/∂t = α∂²u/∂x² describes temperature u(x,t) along rod. Requires boundary conditions (BCs) and initial conditions (ICs) for unique solution. PDEs are harder than ODEs—infinite-dimensional problem. Applications: physics, engineering, finance, biology. Most continuous physical laws are PDEs. Essential for modeling real-world systems.
What are the three classical PDEs?
Heat equation: ∂u/∂t = α∇²u (diffusion, parabolic). Models temperature, concentration, probability. Wave equation: ∂²u/∂t² = c²∇²u (propagation, hyperbolic). Models vibrations, sound, light. Laplace equation: ∇²u = 0 (steady-state, elliptic). Models potentials, equilibria. Each has distinct mathematical character: heat (smoothing, irreversible), wave (oscillating, reversible), Laplace (boundary-determined, no time). These three exemplify parabolic/hyperbolic/elliptic types. Understanding them provides foundation for all PDEs.
How do I solve a PDE?
Methods depend on PDE type and domain: (1) Separation of variables: assume u=X(x)Y(y)T(t), reduce to ODEs, solve each, combine via superposition (Fourier series). Works for linear PDEs, rectangular domains. (2) Transform methods: Laplace transform (time), Fourier transform (space), convert PDE to algebraic equation. (3) Green's functions: integral representation. (4) Characteristics: for hyperbolic PDEs. (5) Numerical: finite differences, finite elements, spectral methods. Example: heat equation on rod with u(0,t)=0, u(L,t)=0, u(x,0)=sin(πx/L). Solution: u(x,t)=sin(πx/L)e^(-α(π/L)²t) (separation + BC/IC). Choose method based on PDE linearity, domain shape, BC type.
What is separation of variables?
Assume solution is product: u(x,t)=X(x)T(t) (or u(x,y)=X(x)Y(y)). Substitute into PDE, rearrange so left side depends only on x, right only on t. Since independent, both equal constant (separation constant). Get two ODEs, solve each with BCs. Combine solutions via superposition (sum over all separation constants). Example: heat equation ∂u/∂t=α∂²u/∂x², assume u=XT. Then T'/T = α X''/X = -λ (constant). Solve: X''=-λX (with BCs), T'=-αλT. Get Fourier series. Separation works for linear PDEs on "nice" domains (rectangular, circular with symmetry). Standard technique taught in PDE courses.
What are boundary conditions and why do they matter?
BCs specify solution behavior at domain boundaries, ensuring uniqueness. Dirichlet: u=g on boundary (fixed temperature/potential). Neumann: ∂u/∂n=g (fixed flux/derivative). Robin: combination. Example: rod [0,L]—ends held at 0°C (Dirichlet u(0)=u(L)=0) vs. insulated ends (Neumann u'(0)=u'(L)=0). Different BCs give different solutions! Without BCs, infinitely many solutions. Physically, BCs represent how system interacts with environment. Choosing correct BCs is crucial—wrong BCs give wrong physics. Number needed: one per boundary per dimension (2 for 1D rod, surface integral worth for higher dimensions).
What's the difference between parabolic, hyperbolic, and elliptic PDEs?
Classification by mathematical character: Elliptic (Laplace): steady-state, all boundaries affect solution everywhere, smooth solutions, max principle. Solve entire domain simultaneously. Parabolic (heat): time evolution, smoothing/diffusion, irreversible, infinite propagation speed (artifact). March forward in time. Hyperbolic (wave): time evolution, wave propagation, reversible, finite speed, characteristics. March along characteristics. Physical examples: elliptic (electrostatics), parabolic (heat flow), hyperbolic (sound waves). Numerical methods differ: elliptic (linear systems, iterative), parabolic (explicit/implicit time stepping), hyperbolic (upwind, characteristics). Recognition guides solution approach.
Can PDEs be solved exactly?
Only for special cases! Linear PDEs with constant coefficients on simple domains (rectangles, disks, spheres) and simple BCs: yes, via separation/transforms (closed-form Fourier/Bessel series). General cases (nonlinear, complicated domains/BCs, variable coefficients): no exact solution, require numerical methods. Examples with exact solutions: heat on rod (Fourier series), wave on string (d'Alembert), Laplace on disk (Poisson integral formula). Examples needing numerics: Navier-Stokes (turbulence), nonlinear Schrödinger, Einstein equations. In practice, most real-world PDEs solved numerically. Exact solutions provide benchmarks, intuition, limiting cases.
How are PDEs used in machine learning?
Neural networks as function approximators solve PDEs numerically (Physics-Informed Neural Networks, PINNs). Loss function includes PDE residual + BC residual. Training finds u that satisfies PDE. For Burgers equation: train NN to minimize ||∂u/∂t + u∂u/∂x - ν∂²u/∂x²||². Also, diffusion models (image generation) use heat equation: add noise (reverse diffusion), denoise (solve backward heat equation). Optimal transport, HJB equations (reinforcement learning). PDEs describe continuous processes, NNs approximate solutions—active research area connecting deep learning and scientific computing.