Heun method example to solve equations, Exercises of Mathematics

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Example: Use the non-self-starting Heun method to integrate 0.82 y = 4e""" — 0.5y using a step size of h. = 1.0 and an initial condition of y = 2 at x = 0. Additional information is required for the multistep method: y = —0.3929953 at « = —1. Solution: x_, = —1, y_) = —0.3929953; xy = 0, yy = 2. Step 1: 23 =29+h=1. The predictor is used to extrapolate linearly from x_; to x1: y} = y-1 + f (x0, yo)2h = —0.3929953 + (4e28*? — 0.5 x 2) x 2 x 1 = 5.607005 The corrector is then used to compute the value. When j = 1, a F (0, Yo) * F(x1, vw), ft de? 0.5 x 24 i 8x9 _ 0.5 x 5.607005 = 6.519331 The approximate percentage relative error is éa= ui : vi x 100% = 14.39% n When j = 2, yh = yp + Leow) * flrwl), — Fy. 4e98*0 0.5 x 2+ — — 0.5 x 6.549331 = 6.313749 The approximate percentage relative error is — ii2 3 ny x 100% = 3.73% The iteration can be repeated until €, is below a prespecified value of €,. The itera- tions converge on a value of 6.360865. Sei 2: te — 4) +R — 7. The predictor is: yd = yo + fai, y1)2h = 2 + (49° — 0.5 x 6.360865) x 2 x 1 = 13.44346 The correctors can be calculated similarly as in Step 1.