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The function in example 1, a removable discontinuity. Consider the piecewise function = {< = >. The point = is a removable discontinuity.For this kind of discontinuity: The one-sided limit from the negative direction: = and the one-sided limit from the positive direction: + = + at both exist, are finite, and are equal to = = +.
The above example simply states that the function takes the value () for all x values larger than a. With this, all the forces acting on a beam can be added, with their respective points of action being the value of a. A particular case is the unit step function,
A graph of a parabola with a removable singularity at x = 2. In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.
The Riemann problem is very useful for the understanding of equations like Euler conservation equations because all properties, such as shocks and rarefaction waves, appear as characteristics in the solution. It also gives an exact solution to some complex nonlinear equations, such as the Euler equations.
Conversely, by a differentiation theorem of Lebesgue, the jump function f is uniquely determined by the properties: [14] (1) being non-decreasing and non-positive; (2) having given jump data at its points of discontinuity x n; (3) satisfying the boundary condition f (a) = 0; and (4) having zero derivative almost everywhere.
The starting point is the relation from Euler-Bernoulli beam theory = Where is the deflection and is the bending moment. This equation [7] is simpler than the fourth-order beam equation and can be integrated twice to find if the value of as a function of is known.
In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by Sergei Godunov in 1959, [1] for solving partial differential equations. One can think of this method as a conservative finite volume method which solves exact, or approximate Riemann problems at each inter-cell boundary.
In computational fluid dynamics, the MacCormack method (/məˈkɔːrmæk ˈmɛθəd/) is a widely used discretization scheme for the numerical solution of hyperbolic partial differential equations. This second-order finite difference method was introduced by Robert W. MacCormack in 1969. [ 1 ]