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Some solutions of a differential equation having a regular singular point with indicial roots = and .. In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form ″ + ′ + = with ′ and ″.
This project work also aims at determining the correct value of density by clearing the objects touching the borders of the image. In this project three applications are taken into account and using Matlab with image processing toolbox the count and density values are calculated for each.
MATLAB (an abbreviation of "MATrix LABoratory" [18]) is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks.MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages.
An autonomous system is a system of ordinary differential equations of the form = (()) where x takes values in n-dimensional Euclidean space; t is often interpreted as time. ...
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. [1] In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
For an arbitrary system of ODEs, a set of solutions (), …, are said to be linearly-independent if: + … + = is satisfied only for = … = =.A second-order differential equation ¨ = (,, ˙) may be converted into a system of first order linear differential equations by defining = ˙, which gives us the first-order system:
For example, a naive choice of =, =, would correspond to a so-called synchronous coordinate system: one where t-coordinate coincides with proper time for any comoving observer (particle that moves along a fixed trajectory.)
The flexible version is also shown [15] to be robust even if the preconditioner is not symmetric positive definite (SPD). The implementation of the flexible version requires storing an extra vector. For a fixed SPD preconditioner, r k + 1 T z k = 0 , {\displaystyle \mathbf {r} _{k+1}^{\mathsf {T}}\mathbf {z} _{k}=0,} so both formulas for β k ...