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In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation, is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of its particularly simple equidimensional structure, the differential equation can be solved ...
In the special case when Ulam's problem accepts a solution for Cauchy's functional equation f(x + y) = f(x) + f(y), the equation E is said to satisfy the Cauchy–Rassias stability. The name is referred to Augustin-Louis Cauchy and Themistocles M. Rassias .
If the eigenvalues (the case of Euler equations) are all real the system is defined hyperbolic, and physically eigenvalues represent the speeds of propagation of information. [12] If they are all distinguished, the system is defined strictly hyperbolic (it will be proved to be the case of one-dimensional Euler equations). Furthermore ...
Cauchy's functional equation is the functional equation: (+) = + (). A function that solves this equation is called an additive function.Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely : for any rational constant .
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.
Since the parameter is usually time, Cauchy conditions can also be called initial value conditions or initial value data or simply Cauchy data. An example of such a situation is Newton's laws of motion, where the acceleration ″ depends on position , velocity ′, and the time ; here, Cauchy data corresponds to knowing the initial position and ...
An example, which comes from a solution of the Euler–Tricomi equation of transonic gas dynamics, [61] is the rescaled Airy function / (/). Although using the Fourier transform, it is easy to see that this generates a semigroup in some sense—it is not absolutely integrable and so cannot define a semigroup in the above strong sense.
For example, the function f(z) = i − iz has real part Re f(z) = Im z. On the unit circle this can be written i / z − iz / 2 . Using the Möbius transformation and the Stieltjes formula we construct the function inside the circle. The i / z term makes no contribution, and we find the function −iz. This has the ...