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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.
z is a set of dependent variables for which no partial derivatives are defined. The relationship between a PDAE and a partial differential equation (PDE) is analogous to the relationship between an ordinary differential equation (ODE) and a differential algebraic equation (DAE). PDAEs of this general form are challenging to solve.
In many practical partial differential equations, one has a term that involves derivatives (such as a kinetic energy contribution), and a multiplication with a function (for example, a potential). In the spectral method, the solution ψ {\displaystyle \psi } is expanded in a suitable set of basis functions, for example plane waves,
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
The mathematical analysis of partial differential equations uses analytical techniques to study partial differential equations. The subject has connections to and motivations from physics and differential geometry, the latter through the branches of global and geometric analysis .
Numerical Methods for Partial Differential Equations is a bimonthly peer-reviewed scientific journal covering the development and analysis of new methods for the numerical solution of partial differential equations. It was established in 1985 and is published by John Wiley & Sons.
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 ...
In numerical analysis, the Lax equivalence theorem is a fundamental theorem in the analysis of finite difference methods for the numerical solution of partial differential equations. It states that for a consistent finite difference method for a well-posed linear initial value problem, the method is convergent if and only if it is stable. [1]
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