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  2. Partial differential equation - Wikipedia

    en.wikipedia.org/wiki/Partial_differential_equation

    In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.. The function is often thought of as an "unknown" that solves the equation, similar to how x is thought of as an unknown number solving, e.g., an algebraic equation like x 2 − 3x + 2 = 0.

  3. Duhamel's principle - Wikipedia

    en.wikipedia.org/wiki/Duhamel's_principle

    Duhamel's principle is the result that the solution to an inhomogeneous, linear, partial differential equation can be solved by first finding the solution for a step input, and then superposing using Duhamel's integral. Suppose we have a constant coefficient, m-th order inhomogeneous ordinary differential equation.

  4. Collocation method - Wikipedia

    en.wikipedia.org/wiki/Collocation_method

    In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...

  5. Method of characteristics - Wikipedia

    en.wikipedia.org/wiki/Method_of_characteristics

    Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation. The method is to reduce a partial differential equation (PDE) to a family of ordinary differential equations (ODE) along which the solution can be integrated from some initial data ...

  6. Method of lines - Wikipedia

    en.wikipedia.org/wiki/Method_of_lines

    Thus it cannot be used directly on purely elliptic partial differential equations, such as Laplace's equation. However, MOL has been used to solve Laplace's equation by using the method of false transients. [1] [8] In this method, a time derivative of the dependent variable is added to Laplace’s equation. Finite differences are then used to ...

  7. List of unsolved problems in mathematics - Wikipedia

    en.wikipedia.org/wiki/List_of_unsolved_problems...

    Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.

  8. Calculus of variations - Wikipedia

    en.wikipedia.org/wiki/Calculus_of_Variations

    These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification = ˙ ˙. We conclude that the function ψ {\displaystyle \psi } is the value of the minimizing integral A {\displaystyle A} as a function of the upper end point.

  9. Pierre-Louis Lions - Wikipedia

    en.wikipedia.org/wiki/Pierre-Louis_Lions

    In 1988, François Golse, Lions, Benoît Perthame, and Rémi Sentis studied the transport equation, which is a first-order linear partial differential equation. [GLPS88] They showed that if the first-order coefficients are randomly chosen according to some probability distribution , then the corresponding function values are distributed with ...