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Partial differential equation. Nonlinear partial differential equation. list of nonlinear partial differential equations; Boundary condition; Boundary value problem. Dirichlet problem, Dirichlet boundary condition; Neumann boundary condition; Stefan problem; Wiener–Hopf problem; Separation of variables; Green's function; Elliptic partial ...
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.
In the mathematical study of partial differential equations, Lewy's example is a celebrated example, due to Hans Lewy, of a linear partial differential equation with no solutions. It shows that the analog of the Cauchy–Kovalevskaya theorem does not hold in the smooth category.
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. [1] The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the ...
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.
Partial derivatives appear in any calculus-based optimization problem with more than one choice variable. For example, in economics a firm may wish to maximize profit π( x , y ) with respect to the choice of the quantities x and y of two different types of output.
Often a partial differential equation can be reduced to a simpler form with a known solution by a suitable change of variables. The article discusses change of variable for PDEs below in two ways: by example; by giving the theory of the method.
In finite-element analysis, the essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation. [2] The dependent unknown u in the same form as the weight function w appearing in the boundary expression is termed a primary variable , and its specification constitutes the essential or Dirichlet boundary ...