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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. Method of characteristics - Wikipedia

    en.wikipedia.org/wiki/Method_of_characteristics

    In mathematics, the method of characteristics is a technique for solving partial differential equations.Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation.

  4. First-order partial differential equation - Wikipedia

    en.wikipedia.org/wiki/First-order_partial...

    The general solution to the first order partial differential equation is a solution which contains an arbitrary function. But, the solution to the first order partial differential equations with as many arbitrary constants as the number of independent variables is called the complete integral. The following n-parameter family of solutions

  5. Parabolic partial differential equation - Wikipedia

    en.wikipedia.org/wiki/Parabolic_partial...

    The solution (,), as a function of for a fixed time >, is generally smoother than the initial data (,) = (). For a nonlinear parabolic PDE, a solution of an initial/boundary-value problem might explode in a singularity within a finite amount of time. It can be difficult to determine whether a solution exists for all time, or to understand the ...

  6. Weak solution - Wikipedia

    en.wikipedia.org/wiki/Weak_solution

    In mathematics, a weak solution (also called a generalized 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 different ...

  7. List of nonlinear partial differential equations - Wikipedia

    en.wikipedia.org/wiki/List_of_nonlinear_partial...

    Name Dim Equation Applications Landau–Lifshitz model: 1+n = + Magnetic field in solids Lin–Tsien equation: 1+2 + = Liouville equation: any + = Liouville–Bratu–Gelfand equation

  8. Separation of variables - Wikipedia

    en.wikipedia.org/wiki/Separation_of_variables

    The analytical method of separation of variables for solving partial differential equations has also been generalized into a computational method of decomposition in invariant structures that can be used to solve systems of partial differential equations.

  9. Calculus of variations - Wikipedia

    en.wikipedia.org/wiki/Calculus_of_Variations

    Hence, solving the associated partial differential equation of first order is equivalent to finding families of solutions of the variational problem. This is the essential content of the Hamilton–Jacobi theory , which applies to more general variational problems.

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