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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 ...
The Euler equations are quasilinear hyperbolic equations and their general solutions are waves. Under certain assumptions they can be simplified leading to Burgers equation . Much like the familiar oceanic waves , waves described by the Euler Equations 'break' and so-called shock waves are formed; this is a nonlinear effect and represents the ...
1824 - Augustin Louis Cauchy proves convergence of the Euler method. In this proof, Cauchy uses the implicit Euler method. 1855 - First mention of the multistep methods of John Couch Adams in a letter written by Francis Bashforth. 1895 - Carl Runge publishes the first Runge–Kutta method.
Blue: the Euler method, green: the midpoint method, red: the exact solution, =. The step size is = The same illustration for = It is seen that the midpoint method converges faster than the Euler method. The midpoint method is a refinement of the Euler method
The black curve shows the exact solution. The Euler method can also be numerically unstable, especially for stiff equations, meaning that the numerical solution grows very large for equations where the exact solution does not. This can be illustrated using the linear equation
It is named after Karl Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. Both variants can be seen as extensions of the Euler method into two-stage second-order Runge–Kutta methods. The procedure for calculating the numerical solution to the initial value problem:
In mathematics, a Cauchy (French:) boundary condition augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so as to ensure that a unique solution exists. A Cauchy boundary condition specifies both the function value and normal derivative on the ...
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.