Client Account:   Login
Home Site Statistics   Contact   About Us   Wednesday, May 24, 2017

users on-line: 2 | Forum entries: 6   
j0110924 - Back to Home
   Skip Navigation LinksHOME › AREAS OF EXPERTISE › Differential Equations Methods › ~ Higher-Order Differential Equations


      Skip Navigation Links
   DISCUSSION   
   MULTIRUNGE-KUTTA4 METHOD   
   IMPLEMENTATION   
   OUR SOLUTIONS   
 

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.






Skip Navigation Links.

Home

Home Math, Analysis & More,
  our established expertise..."

EIGENVALUE SOLUTIONS...
  Eigen Inverse Iteration
  Rayleigh-Quotient Method

INTERPOLATION APPLICATIONS...
  Cubic Spline Method
  Newton Divided Difference

 

Applied Mathematical Algorithms

    Home Complex Functions
A complex number z = x + iy, where...
     Home Non-Linear Systems
Non-linear system methods...
     Home Numerical Differentiation
Construction of numerical differentiation...
     Home Numerical Integration
Consider the function I = ah f(x)dx where...
 

2006-2017 © Keystone Mining Post  |   2461 E. Orangethorpe Av., Fullerton, CA 92631 USA  |   info@keystoneminingpost.com