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In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.. The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation like x 2 − 3x + 2 = 0.
e. In mathematics, the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski theorem) is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A special case was proven by Augustin Cauchy (1842), and the full result by Sofya Kovalevskaya (1874).
Lectures on Theoretical Physics is a six-volume series of physics textbooks translated from Arnold Sommerfeld 's classic German texts Vorlesungen über Theoretische Physik. The series includes the volumes Mechanics, Mechanics of Deformable Bodies, Electrodynamics, Optics, Thermodynamics and Statistical Mechanics, and Partial Differential ...
In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of n variables. The equation takes the form. Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations ...
In higher dimensions. A general second-order partial differential equation in n variables takes the form. This equation is considered elliptic if there are no characteristic surfaces, i.e. surfaces along which it is not possible to eliminate at least one second derivative of u from the conditions of the Cauchy problem. [1]
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. Simplified forms are studied in more detail in the literature.
Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic and parabolic partial differential equation. The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data ...
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena in, i.a., engineering science, quantum mechanics and financial mathematics. Examples include the heat equation, time-dependent Schrödinger equation and the Black–Scholes ...