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Euler's method: General exploration

1.3.1 to 1.3.2 Euler's Method for Solving ODEs Numerically, Euler's method using Matlab

Script

#----------------------------------- # Define ODE y' = f #----------------------------------- f(t,y) = sin(1-t)*(y-1)*(y+1) #----------------------------------- # Define increment #----------------------------------- Δt = Slider(0.01, 1, 0.01, 1, 400, false, true, false, false) SetValue(Δt, 0.05) #----------------------------------- # Euler's method #----------------------------------- n = 50 Execute( Join( {"T0 = 0"}, Zip( "T"+ (k+1) +" = T"+ k +" + Δt", k, 0..n ) ) ) Execute( Join( {"Y0 = 0.4"}, Zip( "Y"+ (k+1) +" = Y"+ k +"+ f(T"+ k +", Y"+ k +") * Δt", k, 0..n ) ) ) Execute( Join( {"P0 = (T0, Y0)"}, Zip( "P"+ (k+1) +" = (T"+ (k+1) +", Y"+ (k+1) +")", k, 0..n ) ) ) # 4th Degree Runge-Kutta solution with step = 0.001 sol = SolveODE(f, T0, Y0, 3, 0.001) #----------------------------------- # Settings #----------------------------------- Execute( Join( {"X0 = (T0, 0)"}, Zip( "X"+ (k+1) +" = (T"+ (k+1) +", 0)", k, 0..n ) ) ) # Texts T = Zip("y_{"+k"}", k, 0..n) U = Zip("t_{"+k"}", k, 0..n) Execute(Zip("SetCaption(P"+ k +", Element(T, "+ (k+1) +"))", k, 0..n)) Execute(Zip("SetCaption(X"+ k +", Element(U, "+ (k+1) +"))", k, 0..n)) b = "Blue" r = "Red" Execute(Zip("SetColor(X"+ k +", b)", k, 0..n)) Execute(Zip("SetColor(P"+ k +", r)", k, 0..n)) Execute(Zip("SetPointSize(X"+ k +", 5)", k, 0..n)) Execute(Zip("SetPointSize(P"+ k +", 5)", k, 0..n)) Execute(Zip("SetPointStyle(X"+ k +", 10)", k, 0..n)) Execute(Zip("SetPointStyle(P"+ k +", 10)", k, 0..n)) Execute(Zip("S"+ k +" = Segment(P"+ k +", X"+ k +")", k, 0..n)) Execute(Zip("SetLineStyle(S"+ k +", 2)", k, 0..n)) Execute(Zip("SetLineThickness(S"+ k +", 2)", k, 0..n)) SL = Join({P0}, CellRange(P1, P50)) appx = Polyline(SL) #----------------------------------------------- Finally add in Update this script to Δt #----------------------------------------------- If( Δt <= 0.04, Execute(Zip("ShowLabel(X"+ k +", false)", k, 1..n)), Execute(Zip("ShowLabel(X"+ k +", true)", k, 1..n))) If( Δt <= 0.04, Execute(Zip("ShowLabel(P"+ k +", false)", k, 1..n)), Execute(Zip("ShowLabel(P"+ k +", true)", k, 1..n)))