Euler Method:
In mathematics and computational science, the Euler method is a first-order numerical procedure for solving ordinary differential equations with a given initial value. It is the most basic explicit method for the numerical integration of ordinary differential equations and is the simplest Runge–Kutta method.
Applications of Euler Method:
- Konigsberg Bridge problem, Euler circuits and paths are also useful to painters, garbage collectors, airplane pilots, and all world navigators.
- The Euler Method is used in solving ODEs, estimating the Force-Deformation behavior in the non-linear range, Euler applications in the stability of structures and earthquake engineering and structural dynamics.
Taylor Method:
In calculus, Taylor’s theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial.
Application of Taylor Method:
- The Galerkin Computational Fluid Dynamics, The cumulants (some expectation value or variance of statistical data), The current-mode pseudo-exponential circuit, the Dirichlet-Neumann operator (used in analyzing boundary conditions e.g fluid dynamics and crystal growth).
- The Taylor Series is used in power flow analysis of electrical power systems, Generating functions.
Comparison between Euler and Taylor Method:
Euler |
Taylor |
1. A linear system of ODE can be solved analytically and/or also numerically (e.g. by Euler explicit method). |
1. Taylor cannot solve this. |
2. If the matrix elements are non-stationary (e.g. time-dependent ones) then the system of equations cannot be solved by Euler implicit method. |
2. If the matrix elements are non-stationary (e.g. time-dependent ones) then the system of equations cannot be solved by the Taylor expansion method. |
3. All discrete equations carried-out by Euler explicit-, implicit- |
3. Taylor expansion methods are easily solvable by numerical computing because their modified (fictitious) matrices are stationary ones. |
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