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Differential Equations: Half-Increment Solution, Numerical Methods - Eddie W. Shore - 11-27-2016 06:41 PM
Introduction The program HALFSTEP solves the numerical differential equation d^2y/dt^2 = f(dy/dt, y, t) given the initial conditions y(t0) = y0 and dy/dt (t0) = dy0 In this notation, y is the independent variable and t is the dependent variable. The Method Let C = f(dy/dt, y, t). Give the change of t as Δt. First Step: With t = t0: h_1/2 = dy0 + C * Δt/2 y1 = y0 + dy0 * Δt Loop: t = t0 + Δt h_I+1/2 = h_I-1/2 + C * Δt y_I+1 = y_I + h_I+1/2 * Δt Repeat as many steps as desired. This method was presented by Robert M. Eisberg in his 1976 calculator programming book (see source below). Variables The program uses the following variables: C: d^2y/dt^2. Represent dy/dt as the variable A, y as the variable Y, and t as the variable T. The program will always designate Y as the independent variable and T as the dependent variable. Examples: Free-Fall d^2y/dt^2 = g Set C as “9.80665” (SI) or “32.1740468” (US) Free-Fall with Friction d^2y/dt^2 = g - α (dy/dt)^2 (α = F/m) Set C as “g - α * A^2” (sub numeric values for g, α) Spring d^2x/dt = -k/m * x Set C as “-k/m * T” (sub numeric values for k, m) Pendulum d^2θ/dt = -α*sin(θ) (α = -g/l) Set C as “-α * sin(Y)” (sub numeric values for α) Damped, Driven Oscillations d^2x/dt = -α*x – β*dx/dt + γ * sin(ω*t) Set C as “-α*Y-β*A+γ*sin(ω*T)” (sub numeric values for α, β, γ) HP Prime Program HALFSTEP Input: C. Use single quotes to enclose d^2y/dt^2. Represent dy/dt as A, y as Y, and t as T. Output: A matrix of two columns, t and y. Code: `EXPORT HALFSTEP(c,A,Y,D,tmax)` Source: Eiseberg, Robert M. Applied Mathematical Physics with Programmable Pocket Calculators McGraw-Hill, Inc: New York. 1976. ISBN 0-07-019109-3 |