Examples are provided as scripts that you can run by choosing the Example Scripts... item from the File menu. Some of the current examples are described briefly below.
Solves the equations of motion in a plane for a small body under the influence of two large bodies of masses a and b = 1 - a. With u1, u2, u3, and u4 as unknowns, the differential equations are du1/dt = u3, du2/dt = u4, du3/dt = u1 + 2·u4 + b·(u1 + a)/D1 - a·(u1 - b)/D2, and du4/dt = u2 - 2·u3 - b·u2/D1 - a·u2/D2, where D1 and D2 are both functions of u1 and u2.
Solves a reactor dynamics problem involving an autocatalytic reaction scheme. With y1, y2, and y3 as unknowns, the differential equations are dy1/dt = a - y1 - b·y1·y3, dy2/dt = y1 - y2·y3, and y3/dt + y2·y3 - b·y1·y3. When a = 3, the problem has a Hopf bifurcation at b = 1.30176.
One pendulum attached to another pendulum. A famous system illustrating chaotic behavior, formulated here as a high index problem.
Illustrates the dangers of applying numerical methods to the solution of differential equations. Using x and y as unknowns, it solves the equations dx/dt = -y and dy/dt = x. With a constant step size of 0.02, the Implicit Midpoint method generates a circle, as expected, while the Backward Euler method generates a spiral.
A small circuit containing resistors, transistors, a capacitor, and a power source. It is described by a set of four algebraic equations and one differential equation.
Simple example with only one unknown, y. It solves the differential equation dy/dt = -y.
Index 3 problem in Hessenberg form for x, y and z as the unknowns. It solves the differential equations dx/dt = y and dy/dt = z subject to the algebraic constraint x = 1 - exp(-2·t).
Index 2 problem in Hessenberg form for x and y as the unknowns. It solves the differential equation dx/dt = -y subject to the algebraic constraint x = sin(2·pi·t).
With y1 and y2 as unknowns, it solves the differential equations dy1/dt = alpha - y1 - 4·y1·y2 / (1 + y1·y1) and dy2/dt = beta * y1 * (1 - y2 / (1 + y1·y1)). The problem is solved twice, for values of beta lower and higher than the critical value beta = 3·alpha/5 - 25/alpha.
Solves Kepler's Equation, M = E - e·sin(E) for an orbit with eccentricity e = 0.7 and period T = 365.
Illustrates the dangers of applying numerical methods to the solution of differential equations. Using y as unknown, it solves the equation dy/dt = y·cos(t), whose exact solution is y = exp(sin(t)). The solution obtained using the Forward Euler method with a constant step size of 0.1 deviates significantly from this exact solution.
Solves the famous problem illustrating chaotic behavior. As expected for chaotic systems, this problem is very sensitive to the choices you make of error tolerance, step size, and solution method.
Solves the single nonlinear equation: 2 = (x2 + y2) (1 + 2x + 5x2 + 6x3 + 6x4 + 4x5 + x6 - 3y2 - 2xy2 + 8x2y2 + 8x3y2 + 3x4y2 + 2y4 + 4xy4 + 3x2y4 + y6)
Reactions in Series
Solves the mass balances in a batch reactor where a first order reaction turns A into B and another first order reaction turns B into C. With Ca, Cb, and Cc as unknowns, the equations are dCa/dt = -k1·Ca, dCb/dt = k1·Ca - k2·Cb, and dCc/dt = k2·Cb.
A high index problem. The robotic system is made up of two arms connected to each other, the first one fixed at the origin, while the second one is subject to the vertical constraint: y = sin2(wt). Fast changes in constraint forces make this a very difficult problem to solve.
One of the best known index 3 problems, solves the equations of motion for a mass attached to a rigid rod. With q1, q2, v1, v2, and z as unknowns, the differential equations are dq1/dt = v1, dq2/dt = v2, dv1/dt = -z·q1, and dv2/dt = -z·q2 - g, subject to the algebraic constraint 1 = q1·q1 + q2·q2.
When the rigid rod in the simple pendulum is replaced with a spring, the index of the problem is lowered to 1. With q1, q2, v1, v2, and z as unknowns, the differential equations are dq1/dt = v1, dq2/dt = v2, dv1/dt = -z·q1/r, dv2/dt = -z·q2/r - g, subject to the algebraic constraint e·z = r - 1, where r is the length of the spring and 1/e is the spring constant.
With y as the unknown, solves the equation dy/dt = -100·(y - sin(t)), whose solution varies rapidly initially but slows afterwards.