Solving heat, wave, and Laplace equations with rigorous mathematical methods
← Back to MathematicsLearn the basic concepts, classifications, and examples of partial differential equations.
Study methods for solving first-order partial differential equations.
Master the fundamental technique of separation of variables for linear PDEs.
Apply Fourier series and transforms to solve PDEs on various domains.
Study the parabolic heat equation and its properties in detail.
Analyze the hyperbolic wave equation and wave phenomena.
Study the elliptic Laplace equation and harmonic function theory.
Learn Green's functions as a unified approach to solving linear PDEs.
Study computational approaches to solving PDEs numerically.
Explore advanced PDE theory and modern applications.
Build the foundation with basic concepts, classifications, and physical origins of partial differential equations.
Learn PDEs as equations involving partial derivatives of unknown functions of multiple variables. Study classic examples: heat, wave, and Laplace equations.
Classify PDEs by order (highest derivative) and linearity. Distinguish linear, quasilinear, and fully nonlinear equations and their different behaviors.
Learn the fundamental classification: elliptic (Laplace), parabolic (heat), hyperbolic (wave). Understand how classification affects solution methods.
Study Hadamard's criteria: existence, uniqueness, and continuous dependence on data. Learn why well-posedness is crucial for physical modeling.
Understand Dirichlet, Neumann, and Robin boundary conditions. Learn how initial/boundary conditions determine unique solutions to PDEs.
Discover how PDEs arise from physical principles: conservation laws, Newton's laws, thermodynamics. Connect mathematics to real-world phenomena.
Derive ut = kuxx from Fourier's law of heat conduction and conservation of energy. Understand the physical meaning of each term.
Derive utt = c²uxx from Newton's law applied to vibrating strings. Learn how wave speed c relates to physical parameters.
Master methods for solving first-order PDEs and understand the geometric method of characteristics.
Solve equations of the form aux + buy + cu = f using method of characteristics. Learn how characteristics are curves along which PDEs become ODEs.
Master the geometric approach: characteristics are curves where information propagates. Learn to find characteristic curves and solve along them.
Study nonlinear first-order PDEs where coefficients depend on u: F(x,y,u,ux,uy) = 0. Learn how nonlinearity can cause characteristics to intersect.
Study PDEs of the form ut + f(u)x = 0 arising from conservation principles. Learn how to find weak solutions when classical solutions break down.
Apply conservation laws to model traffic flow: ρt + (ρv)x = 0. Understand how traffic density and velocity interact to create traffic patterns.
Learn how smooth initial data can develop discontinuities (shocks) in finite time. Study shock conditions and entropy solutions.
Define weak solutions using integral formulation when classical derivatives don't exist. Learn how to handle discontinuous solutions rigorously.
Study nonlinear first-order PDEs ut + H(∇u) = 0 arising in optimal control. Learn viscosity solutions and dynamic programming connections.
Master the fundamental technique for solving linear PDEs by reducing them to ODEs.
Learn to assume solutions of the form u(x,t) = X(x)T(t) for linear PDEs. Understand when and why this ansatz works for linear equations.
Solve ut = kuxx on bounded intervals with homogeneous boundary conditions. Learn how initial conditions determine coefficients in series solutions.
Solve utt = c²uxx using separation of variables. Understand standing wave solutions and their relationship to normal modes of vibration.
Solve ∇²u = 0 in rectangular domains. Learn how to handle different boundary conditions on different sides of rectangles.
Study Sturm-Liouville problems X''+ λX = 0 arising from separation. Learn orthogonality of eigenfunctions and completeness properties.
Express solutions as infinite series of eigenfunctions. Learn how Fourier coefficients are determined by initial or boundary data.
Handle various boundary conditions: Dirichlet u = 0, Neumann ux = 0, Robin αu + βux = 0. Learn how boundary conditions affect eigenvalues.
Extend separation of variables to higher dimensions. Study problems in rectangular boxes, circular domains, and other separable coordinate systems.
Apply Fourier series and transforms as powerful tools for solving PDEs on various domains.
Review Fourier series expansion of periodic functions. Understand convergence properties and how to handle different types of boundary conditions.
Solve heat equation using Fourier series on bounded domains. Learn how different boundary conditions lead to sine, cosine, or general Fourier series.
Learn Fourier transforms for functions on infinite domains. Study properties: linearity, convolution theorem, Parseval's identity, and derivatives.
Solve heat and wave equations on infinite intervals using Fourier transforms. Learn how transforms convert PDEs to ODEs in frequency space.
Use convolution theorem to construct solutions via Green's functions. Learn how fundamental solutions relate to Fourier transforms.
Apply energy conservation in Fourier space. Learn how Parseval's theorem helps analyze energy dissipation in heat equations and energy conservation in wave equations.
Study DFT for numerical computation. Learn relationship between continuous and discrete Fourier transforms, and aliasing effects.
Understand FFT algorithm for efficient computation of DFT. Learn applications to numerical solution of PDEs and spectral methods.
Study the parabolic heat equation in depth, including advanced properties and solution techniques.
Study ut = k∇²u as the fundamental parabolic PDE. Learn smoothing properties, irreversibility, and infinite speed of information propagation.
Learn the heat kernel G(x,t) = (4πkt)^(-n/2) exp(-|x|²/4kt). Understand how it describes heat spreading from a point source.
Prove that solutions achieve maximum on the parabolic boundary. Learn weak maximum principle and its applications to uniqueness and comparison.
Use maximum principle to prove uniqueness of solutions. Learn comparison theorems and how they provide bounds on solutions.
Express general solutions as convolution with heat kernel: u(x,t) = ∫G(x-y,t)u₀(y)dy. Understand probabilistic interpretation.
Study energy E(t) = ∫u² dx and its monotonic decrease for heat equation. Learn how energy methods provide stability and convergence results.
Study long-time behavior: solutions approach mean value as t → ∞. Learn decay rates and self-similar solutions.
Solve ut = k∇²u + f(x,t) using Duhamel's principle. Learn how to handle time-dependent source terms and boundary conditions.
Analyze the hyperbolic wave equation and understand wave phenomena and propagation.
Derive utt = c²∇²u from physical principles. Understand wave speed c and its relationship to medium properties like tension and density.
Learn the explicit solution u(x,t) = ½[f(x+ct) + f(x-ct)] + 1/(2c)∫[x-ct to x+ct] g(s)ds for infinite string problems.
Understand characteristics x ± ct as paths of information propagation. Learn domain of dependence and range of influence concepts.
Study energy E = ½∫(u²t + c²u²x)dx and its conservation for wave equation. Learn how energy provides stability and uniqueness.
Study wave equation in 2D and 3D. Learn about cylindrical and spherical waves, and how dimensionality affects wave behavior.
Understand wave propagation in odd dimensions where signals have sharp wavefronts. Learn why 2D waves have "trailing" effects.
Solve utt = c²∇²u + F(x,t) using Green's functions and retarded potentials. Understand causality and finite propagation speed.
Learn that wave solutions have compact support if initial data has compact support. Understand the fundamental difference from heat equation.
Study the elliptic Laplace equation and the beautiful theory of harmonic functions.
Study functions satisfying ∇²u = 0. Learn that harmonic functions model steady-state heat distribution and gravitational potentials.
Learn that harmonic functions equal their average over any sphere: u(x) = (1/|∂B|)∫∂B u dS. This is equivalent to being harmonic.
Prove harmonic functions achieve maximum and minimum on the boundary. Learn strong maximum principle and its uniqueness consequences.
Study Dirichlet problem (specify u on boundary) and Neumann problem (specify ∂u/∂n on boundary). Learn existence and uniqueness conditions.
Construct Green's functions G(x,y) for Laplace equation in various domains. Learn how Green's functions solve Poisson equation ∇²u = f.
Learn explicit solution for Dirichlet problem in disks and balls using Poisson kernel. Understand harmonic measure interpretation.
Use maximum principle and energy methods to prove uniqueness. Learn how different boundary conditions affect uniqueness statements.
Study harmonic functions as gravitational or electrostatic potentials. Learn connections to complex analysis and conformal mapping.
Master Green's functions as a unified approach to solving inhomogeneous linear PDEs.
Review Green's functions for boundary value problems in ODEs. Understand how they invert differential operators and represent solutions.
Learn fundamental solutions Φ as Green's functions for the free-space operator. Study how they satisfy LΦ = δ in distributional sense.
Construct Green's functions using method of images for simple geometries. Learn how to satisfy boundary conditions using symmetric/antisymmetric images.
Study Green's first and second identities from integration by parts. Learn how they lead to integral representations of solutions.
Learn symmetry property G(x,y) = G(y,x) and its physical interpretation as reciprocity between source and observation points.
Express Green's functions as series in eigenfunctions: G(x,y) = Σ φₙ(x)φₙ(y)/λₙ. Learn bilinear formula and its applications.
Study specific Green's functions for heat and wave equations. Learn retarded Green's functions and causality for hyperbolic problems.
Use Green's functions to solve general boundary value problems. Learn how different boundary conditions modify the Green's function construction.
Learn computational approaches to solving PDEs when analytical methods are insufficient.
Approximate derivatives using finite differences. Learn forward, backward, and central difference schemes and their accuracy orders.
Study von Neumann stability analysis and Lax equivalence theorem. Learn that consistency + stability = convergence for linear schemes.
Compare explicit methods (easy to implement, stability restrictions) vs implicit methods (stable, require linear system solves).
Learn Courant-Friedrichs-Lewy condition for hyperbolic PDEs: numerical domain of dependence must contain analytical domain of dependence.
Study variational formulation and finite element approximation. Learn about basis functions, assembly of stiffness matrices, and mesh refinement.
Learn weak formulation of PDEs using integration by parts. Understand how weak solutions allow for less regular solutions and natural boundary conditions.
Use global polynomial approximations and FFT for high accuracy. Learn about spectral accuracy and applications to smooth periodic problems.
Discretize spatial derivatives to get ODE system in time. Learn how to apply ODE solvers to time-dependent PDEs.
Explore modern developments in PDE theory and applications to current research areas.
Study nonlinear phenomena: blow-up, pattern formation, traveling waves. Learn how nonlinearity fundamentally changes PDE behavior.
Study special nonlinear PDEs with soliton solutions. Learn KdV equation, inverse scattering transform, and exactly solvable systems.
Learn how PDEs arise as Euler-Lagrange equations from variational principles. Study critical points of functionals and minimal surface problems.
Study generalized solutions using distribution theory. Learn how to handle discontinuous solutions and shock waves rigorously.
Study function spaces with weak derivatives. Learn Sobolev embedding theorems and their applications to existence theory for PDEs.
Study PDEs with rapidly oscillating coefficients. Learn how microscopic structure affects macroscopic behavior through effective equations.
Study problems where domain boundary is unknown (Stefan problems, obstacle problems). Learn variational inequalities and regularity theory.
Apply PDEs to model biological phenomena: reaction-diffusion systems, population dynamics, pattern formation in development.