General topics

• Partial differential equation
  Partial differential equation
  In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
  
  
• Boundary condition
• Boundary value problem
  Boundary value problem
  In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions...
  
  
  • Dirichlet problem
    Dirichlet problem
    In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of the region....
    
    , Dirichlet boundary condition
    Dirichlet boundary condition
    In mathematics, the Dirichlet boundary condition is a type of boundary condition, named after Johann Peter Gustav Lejeune Dirichlet who studied under Cauchy and succeeded Gauss at University of Göttingen. When imposed on an ordinary or a partial differential equation, it specifies the values a...
    
    
  • Neumann boundary condition
    Neumann boundary condition
    In mathematics, the Neumann boundary condition is a type of boundary condition, named after Carl Neumann.When imposed on an ordinary or a partial differential equation, it specifies the values that the derivative of a solution is to take on the boundary of the domain.* For an ordinary...
    
    
  • Stefan problem
    Stefan problem
    In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem is a particular kind of boundary value problem for a partial differential equation , adapted to the case in which a phase boundary can move with time...
    
    
  • Wiener–Hopf problem
• Separation of variables
  Separation of variables
  In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
  
  
• Green's function
  Green's function
  In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
  
  
• Elliptic partial differential equation
• Singular perturbation
  Singular perturbation
  In mathematics, more precisely in perturbation theory, a singular perturbation problem is a problem containing a small parameter that cannot be approximated by setting the parameter value to zero...
  
  
• Cauchy–Kovalevskaya theorem
• H-principle
  H-principle
  In mathematics, the homotopy principle is a very general way to solve partial differential equations , and more generally partial differential relations...
  
  
• Atiyah–Singer index theorem
  Atiyah–Singer index theorem
  In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...
  
  
• Bäcklund transform
  Bäcklund transform
  In mathematics, Bäcklund transforms or Bäcklund transformations relate partial differential equations and their solutions. They are an important tool in soliton theory and integrable systems...
  
  
• Viscosity solution
  Viscosity solution
  In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation...
  
  
• Weak solution
  Weak solution
  In mathematics, a weak solution to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for...
  
  

Specific partial differential equations

• Euler equations
  Euler equations
  In fluid dynamics, the Euler equations are a set of equations governing inviscid flow. They are named after Leonhard Euler. The equations represent conservation of mass , momentum, and energy, corresponding to the Navier–Stokes equations with zero viscosity and heat conduction terms. Historically,...
  
  
• Hamilton–Jacobi equation
  Hamilton–Jacobi equation
  In mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations. In physics, the Hamilton–Jacobi equation is a reformulation of classical mechanics and, thus, equivalent to other formulations such as...
  
  , Hamilton–Jacobi–Bellman equation
• Heat equation
  Heat equation
  The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...
  
  
• Laplace's equation
  Laplace's equation
  In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...
  
  
  • Laplace operator
    Laplace operator
    In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...
    
    
  • Harmonic function
    Harmonic function
    In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
    
    
  • Spherical harmonic
    Spherical Harmonic
    Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. It tells the story of Dyhianna Selei , the Ruby Pharaoh of the Skolian Imperialate, as she strives to reform her government and reunite her family in the aftermath of a devastating interstellar...
    
    
  • Poisson integral formula
• Klein–Gordon equation
• Korteweg–de Vries equation
  Korteweg–de Vries equation
  In mathematics, the Korteweg–de Vries equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified...
  
  
• Maxwell's equations
  Maxwell's equations
  Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
  
  
• Navier–Stokes equations
• Poisson's equation
  Poisson's equation
  In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering and theoretical physics...
  
  
• Primitive equations
  Primitive equations
  The primitive equations are a set of nonlinear differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models...
  
   (hydrodynamics)
• Schrödinger equation
  Schrödinger equation
  The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
  
  
• Wave equation
  Wave equation
  The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
  
  

Numerical methods for PDEs

• Finite difference
  Finite difference
  A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
  
  
• Finite element method
  Finite element method
  The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
  
  
• Finite volume method
  Finite volume method
  The finite volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]....
  
  
• Boundary element method
  Boundary element method
  The boundary element method is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations . It can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics, and fracture...
  
  
• Multigrid
• Spectral method
  Spectral method
  Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain Dynamical Systems, often involving the use of the Fast Fourier Transform. Where applicable, spectral methods have excellent error properties, with the so called "exponential...
  
  
• Computational fluid dynamics
  Computational fluid dynamics
  Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...
  
  
• Alternating direction implicit

Related areas of mathematics

• Calculus of variations
  Calculus of variations
  Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
  
  
• Harmonic analysis
  Harmonic analysis
  Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
  
  
• Ordinary differential equation
  Ordinary differential equation
  In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
  
  
• Sobolev space
  Sobolev space
  In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...