Dgl-numerisch und analytisch

Heun Start bei A?

C

Euler

Euler modifiziert

Heun 

m=\frac{m_3+m_Z}{2}

A mit Pfeiltasten bewegen

Heun-Verfahren

Rechts:
Visualisierung der Formel

L = (2.63793, 1.365)
O = (1.9004393047775028, 1.3650907463781723)
c = Segment(O, L)
N: (2, 0)
l = OrthogonalLine(N, xAxis)
n = 1 (slider from -1 to 3, step 1)
h: 2^((-n))
Heun = Sequence((Cell[2, i], Cell[3, i]), i, 1, 1 + 8 / h)
Heunzug = Sequence(Segment[Element[Heun, i], Element[Heun, i + 1]], i, 1, 8 / h)
AgleichC = true
E = (1, 0)
A = (0, 0.7)
α = 1 (slider from 0 to 1.5, step 0.1)
b = If(α ≟ 0, 0, 1 / 2 / α)
a = If(α ≟ 0, 1, 1 - b)
C = If(AgleichC, (x(A) + 0.25, y(A)), E)
C1: y(C)
g: x = x(A)
w(x, y) = y^(2) - x
Steigungsfeld2 = SlopeField(w(x, y))
A1 = 0 (slider from -5 to 5)
A2 = 1 (slider from -5 to 5)
A3 = 2 (slider from -5 to 5)
A4 = 3 (slider from -5 to 5)
A5 = 4 (slider from -5 to 5)
A6 = 5 (slider from -5 to 5)
A7 = 6 (slider from -5 to 10)
A8 = 7 (slider from -5 to 10)
A9 = 8 (slider from -5 to 10)
A10 = 9 (slider from -5 to 10)
A11 = 10 (slider from -5 to 20)
A12 = 11 (slider from -5 to 20)
A13 = 12 (slider from -5 to 20)
A14 = 13 (slider from -5 to 20)
A15 = 14 (slider from -5 to 20)
A16 = 15 (slider from -5 to 20)
A17 = 16 (slider from -5 to 20)
A18 = 17 (slider from -5 to 20)
A19 = 18 (slider from -5 to 20)
A20 = 19 (slider from -5 to 20)
A21 = 20 (slider from -5 to 20)
A22 = 21 (slider from -5 to 50)
A23 = 22 (slider from -5 to 50)
A24 = 23 (slider from -5 to 50)
A25 = 24 (slider from -5 to 50)
A26 = 25 (slider from -5 to 50)
A27 = 26 (slider from -5 to 50)
A28 = 27 (slider from -5 to 50)
A29 = 28 (slider from -5 to 50)
A30 = 29 (slider from -5 to 50)
A31 = 30 (slider from -5 to 50)
A32 = 31 (slider from -5 to 50)
A33 = 32 (slider from -5 to 50)
B1: 0
B2: B1 + h
B3: B2 + h
B4: B3 + h
B5: B4 + h
B6: B5 + h
B7: B6 + h
B8: B7 + h
B9: B8 + h
B10: B9 + h
B11: B10 + h
B12: B11 + h
B13: B12 + h
B14: B13 + h
B15: B14 + h
B16: B15 + h
B17: B16 + h
B18: B17 + h
B19: B18 + h
B20: B19 + h
B21: B20 + h
B22: B21 + h
B23: B22 + h
B24: B23 + h
B25: B24 + h
B26: B25 + h
B27: B26 + h
B28: B27 + h
B29: B28 + h
B30: B29 + h
B31: B30 + h
B32: B31 + h
B33: B32 + h
yheun(x, y) = y + ((a * h) * w(x, y)) + ((b * h) * w(x + (α * h), y + ((α * h) * w(x, y))))
C2: yheun(B1, C1)
C3: yheun(B2, C2)
C4: yheun(B3, C3)
C5: yheun(B4, C4)
C6: yheun(B5, C5)
C7: yheun(B6, C6)
C8: yheun(B7, C7)
C9: yheun(B8, C8)
C10: yheun(B9, C9)
C11: yheun(B10, C10)
C12: yheun(B11, C11)
C13: yheun(B12, C12)
C14: yheun(B13, C13)
C15: yheun(B14, C14)
C16: yheun(B15, C15)
C17: yheun(B16, C16)
C18: yheun(B17, C17)
C19: yheun(B18, C18)
C20: yheun(B19, C19)
C21: yheun(B20, C20)
C22: yheun(B21, C21)
C23: yheun(B22, C22)
C24: yheun(B23, C23)
C25: yheun(B24, C24)
C26: yheun(B25, C25)
C27: yheun(B26, C26)
C28: yheun(B27, C27)
C29: yheun(B28, C28)
C30: yheun(B29, C29)
C31: yheun(B30, C30)
C32: yheun(B31, C31)
C33: yheun(B32, C32)
B34: B33 + h
C34: yheun(B33, C33)
B35: B34 + h
C35: yheun(B34, C34)
B36: B35 + h
C36: yheun(B35, C35)
B37: B36 + h
C37: yheun(B36, C36)
B38: B37 + h
C38: yheun(B37, C37)
B39: B38 + h
C39: yheun(B38, C38)
B40: B39 + h
C40: yheun(B39, C39)
B41: B40 + h
C41: yheun(B40, C40)
B42: B41 + h
C42: yheun(B41, C41)
B43: B42 + h
C43: yheun(B42, C42)
B44: B43 + h
C44: yheun(B43, C43)
B45: B44 + h
C45: yheun(B44, C44)
B46: B45 + h
C46: yheun(B45, C45)
B47: B46 + h
C47: yheun(B46, C46)
B48: B47 + h
C48: yheun(B47, C47)
B49: B48 + h
C49: yheun(B48, C48)
B50: B49 + h
C50: yheun(B49, C49)
B51: B50 + h
C51: yheun(B50, C50)
B52: B51 + h
C52: yheun(B51, C51)
B53: B52 + h
C53: yheun(B52, C52)
B54: B53 + h
C54: yheun(B53, C53)
B55: B54 + h
C55: yheun(B54, C54)
B56: B55 + h
C56: yheun(B55, C55)
B57: B56 + h
C57: yheun(B56, C56)
B58: B57 + h
C58: yheun(B57, C57)
B59: B58 + h
C59: yheun(B58, C58)
B60: B59 + h
C60: yheun(B59, C59)
B61: B60 + h
C61: yheun(B60, C60)
B62: B61 + h
C62: yheun(B61, C61)
B63: B62 + h
C63: yheun(B62, C62)
B64: B63 + h
C64: yheun(B63, C63)
B65: B64 + h
C65: yheun(B64, C64)
A34 = 33
A35 = 34
A36 = 35
A37 = 36
A38 = 37
A39 = 38
A40 = 39
A41 = 40
A42 = 41
A43 = 42
A44 = 43
A45 = 44
A46 = 45
A47 = 46
A48 = 47
A49 = 48
A50 = 49
A51 = 50
A52 = 51
A53 = 52
A54 = 53
A55 = 54
A56 = 55
A57 = 56
A58 = 57
A59 = 58
A60 = 59
A61 = 60
A62 = 61
A63 = 62
A64 = 63
A65 = 64
Text1: "h = " + (LaTeX[h]) + ""
H = (-2.5, 3.5)
P_3: ((3 * h), Cell[3, 4])
f = SolveODE(w(x, y), x(A), y(A), 8, 0.05)
j: x = 2
k = Line(P_3, j)
g_0: y = (-0.453 * (x - x(P_3))) + y(P_3)
ym3: y = (w(x(P_3), y(P_3)) * (x - x(P_3))) + y(P_3)
Z = Intersect(j, ym3)
r: x = 2.5
P_4: ((4 * h), Cell[3, 5])
ymz: y = (w(x(Z), y(Z)) * (x - x(Z))) + y(Z)
K = Intersect(k, ymz)
M = Midpoint(K, P_3)
i = Line(M, Z)
m = Line(P_3, i)
p = Segment(K, P_3)
e = Vector(P_3, Z)
P3 = NaN
u = If(α ≟ 1, Vector[P_3, P_4])
B = Intersect(r, ymz)
I = Point(c)
J = Point(c)
F = Point(c)
G = Point(c)
s = Segment(G, I)
t = Segment(J, L)
q = Segment(O, F)
v = Vector(Z, B)
D = (1.268304691619222, 0.3797044376314397)
P = (1.2565296350996067, 0.04566467636446564)
S = (1.6277537853758912, 0.04194623746353455)