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The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice—once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale (also known as a time-set ...
In power engineering, the power-flow study, or load-flow study, is a numerical analysis of the flow of electric power in an interconnected system. A power-flow study usually uses simplified notations such as a one-line diagram and per-unit system, and focuses on various aspects of AC power parameters, such as Voltage, voltage angles, real power and reactive power.
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
The parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters. An example is the well-studied logistic map , x n + 1 = r x n ( 1 − x n ) {\displaystyle x_{n+1}=rx_{n}(1-x_{n})} , whose basins of attraction for various values of the parameter r {\displaystyle r} are shown ...
William Feller, in 1949, used the names "forward equation" and "backward equation" for his more general version of the Kolmogorov's pair, in both jump and diffusion processes. [1] Much later, in 1956, he referred to the equations for the jump process as "Kolmogorov forward equations" and "Kolmogorov backward equations". [3]
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f ′(x + h / 2 ) and f ′(x − h / 2 ) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f:
We have taken our HoopsHype 75 list and split it into five positions: point guards, shooting guards, small forwards, power forwards and centers. Through these rankings, so you can see who nearly ...
For very small problems, the spectral method is unique in that solutions may be written out symbolically, yielding a practical alternative to series solutions for differential equations. Spectral methods can be computationally less expensive and easier to implement than finite element methods; they shine best when high accuracy is sought in ...