1 Introduction

1.1 What it does

Runge is an Interactive Solver for Systems of Ordinary Differential Equations. It solves initial value problem (aka Cauchy problem) defined as the following: for a given system of ordinary differential equations

˙x = F(t,x),  (t,x) ∈ ℝn+ 1

and given initial values

x(t0)=  x0

find solution

x(tk)∈ ℝn

at a given point of “time” i.e. for a given value tk of independent variable t . Actually Runge produces solutions set

x(t0),x(t1),x(t2),..., x(tk)

where k is the number of steps taken. This allows to build trajectories of solutions.

1.2 Why Runge?

1.3 System Types and Solvers

Runge comes with pre-installed solvers optimized for solving differential equations of different types:

The solvers (algorithms) coming in standard package are:

1.4 Expressions and Functions

The following functions and operators are supported for programming the systems mentioned above.

+ - * / ^ arithmetic operators: add, subtract, multiply, divide, power

exp(x) ex

sqrt(x) √x--

log(x) natural logarithm of x

log10(x) common (base 10) logarithm of x

sin(x) sine of x

cos(x) cosine of x

tan(x) tangent of x

asin(x) arc sine of x

acos(x) arc cosine of x

atan(x) arc tangent of x

sinh(x) hyperbolic sine of x

cosh(x) hyperbolic cosine of x

tanh(x) hyperbolic tangent of x

sinint(x) sine integral of x
∫x sin t
 0  t

cosint(x) cosine integral of x
    ∞ cost-
-      t  dt

sign(x) sign of x
|{ - 1 if x < 0
    0 if x = 0
    1 if x > 0

abs(x) |x|

iif(x,expr1,expr2) immediate if
  expr1  if x < 0
  expr2  if x ≥ 0

sat(x,y) satellite function of x and y
|{   1 if x > |y |
    0 if -  |y| ≤ x ≤ |y|
  - 1 if x < - |y |

i 1 (one)

0 (empty field means zero)

Examples: 2*sin(t-1)+cos(t)-x^2, sqrt(abs(x)), iif(t,sin(x),cos(x)) etc.