Home Login  |   Contact  |   About Us       Wednesday, September 28, 2022   

j0110924 - Back to Home
   Skip Navigation LinksHOME ›  AREAS OF EXPERTISE ›   Differential Equations ›  ~ Higher-Order



Skip Navigation Links.




DISCUSSION
Discussion of Runge-Kutta for Systems

The methods discussed in previous tabs (refer to menu)  "ODE - Euler Method" and   "ODE - 2nd-Order Runge-Kutta" and   "ODE - 4th-Order Runge-Kutta" apply to only a single first-order ordinary differential equation as described in these tabs.

However, most problems in engineering governed by differential equations are either high-order equations or coupled differential equation systems.

A high-order differential equation can always be transformed into a coupled first-order system of equations. The trick is to expand higher-order derivatives into a series of first-order equations.

A very common example described in technical literature applies to model a spring-mass system with damping, which describes the use and calculation of second-order differential equations. We followed very closely these techniques for our own implementation.

m d2x / d t2 = -kx - b dx/dt

where k is the spring constant and b is the damping coefficient. Since the velocity:

v = dx/dt

the equation of motion for a spring-mass system can be rewritten in terms of two first-order differential equations:

dv/dt = -k/m x - b/m v
dx/dt = v

In the above equation, the derivative of v is a function of v and x, and the derivative of x is a function of v. Since the solution of v as a function of time depends on x and the solution of x as a function of time depends on v, the two equations are coupled and must be solved simultaneously.

Most of the differential equations in engineering are higher-order equations. This means that they must be expanded into a series of first-order differential equations before they can be solved using numerical methods.

The next tab shows extending the fourth-order Runge-Kutta method discussed previously to a system of ordinary differential equations.






Other Applications...


Object-Oriented Implementation
Graphics and Animation
Sample Applications
Ore Extraction Optimization
Vectors and Matrices
Complex Numbers and Functions
Ordinary Differential Equations - Euler Method
Ordinary Differential Equations 2nd-Order Runge-Kutta
Ordinary Differential Equations 4th-Order Runge-Kutta
Higher Order Differential Equations
Nonlinear Systems
Numerical Integration
Numerical Differentiation







Consulting Services - Back to Home
Home

Home Math, Analysis,
  expertise..."

EIGENVALUE
SOLUTIONS...


> Rayleigh-Quotient Method

> Cubic Spline Method

 

Applied Mathematical Algorithms

Home

ComplexFunctions

Home

NonLinear
Home

Differentiation
Home

Integration
About Us


KMP Software Engineering is an independent multidisciplinary engineering consulting company specializing in mathematical algorithms.
Areas of
Expertise


SpecialFunctions
VectorsMatrices
OptimizationMethods
ComplexNumbers
Interpolation
CurveFitting
NonLinearSystems
LinearEquations
DistributionFunctions
NumericalDifferentiation
NumericalIntegration
DifferentialEquations
Smalltalk
FiniteBoundary
Eigenvalue
Graphics
Understanding
Mining


MiningMastery
MineralNews
MineralCommodities
MineralForum
Crystallography
Services


NumericalModeling
WebServices
MainframeServices
OutsourceServices

LINKED IN
KMP ARTICLES
Brand





Home

Login

Contact
Since 2006 All Rights Reserved  © KMP Software Engineering LINKS | PRIVACY POLICY | LEGAL NOTICE