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NUMERICAL ANALYSIS SECOND EDITION TIMOTHY SAUER
Final lecture D.E. gundy/d² усх) ул finite difference approach. +auty/dx + y = fox) Yn y (x₁) = y₁ •Y 3 TOT α X₁ X₂ X3 1₂ Х, Хін Б fist order derivative. dy₁_lim y(x+E) — y(x) dx x Ex. Є X AX= X₂=a X₁ = b_a_h n =X₂+ isx x₁ = a + Ax x₂=a+24x x₂ =A+NAX =a+nb-a = b = dx ¹x₁ n 1 Y(X+ DX)_Y(X₂) DX
zo| dy dX lai R_P da ² = = Yiti -Yi дл --!'h !h dy da - дл ++ -yi ! R_!+! R dy/dxlx dy/data₁ + dy/dalzi DX initial value Boundary Value Forward difference AX backward difference dy - y² dr 1- ! f - 16 AX Yitl 01/2₁) 2 === V₁+1 = 28/₁ + Y₁-1 - dx ² x i 2 (AX) AX IC BC 1
a(x) y'(x) + you = f(x) y(x₁) = x=y₂ усх. IC. how many Solve the X₁ = IC or BC differential equation numericaly the rank or degree of the D.E. X₁ tiAx-ation at x₁ = a(x) y'ox) + you! - fox) → arx;) y'(x₂) + you;)=f(x;) yiti Q₁ Xit Yi AX a; (y₁ - y₁) = (f; _y₁) Ax Y₁H₁_y₁ = (fi-y₁) AX Yi tı-yi + Y . = we need to fi y(x) = ? x₂ =α AX 2 b-a n -a (x₁) = a₁ f(x₁) = f₁
Ax - Y₁ - f; sx -y; a₁ + Y₁ di tl a. Y₁ = α y₁ - AX i-. y₁= f. sx_ y. sx + y₂ A₂ 1=1 /₂² 1818² (1) f₁ 4 _asx + x адх X₂ a = f₁ Ax a₁ हृ - Уг 7 Уз 2 x₁₂=b n at x₁ : acx;) y₁ ах + y. a(x) d²y acx d'y + body + qui yox = fox) dx² g(x) I.C. I need two Conditions - Уп y (x₂) усказк → sy = y(x₂) = α Zy₁- ५- ५, AX 2 (₁) dy | + b(x₁) dd 1/₁ + geox, yox;) = f(x₂) fi ✓ = B a₁ Yitl - 2y it Yi+ +b; Yi+ı-Y; + 9₁; Y; ditı-Y₁ (AX) ² AX 3
а; (уін -27; +у;+) + axb; (уін-У;) теход, у; =fiex² 2 Уін (а; + Axb;) = 20; у; -а, У,- нахв, у ; - (Ax)'9; y; +f; (Ax)2 а таны з у, 120, +axb, -кизді) tarbi - Qiyi thaxses а, Уін - = уін - бі Уі тді Уі У-уск) - 0 a у У-у-в N = Дх Yi+1 i=| =x (У.-а у-у-бар = 5 гу, = Дар та охр в а, таль, 7, (29, +ань, -озд) -9,8% +fox33 4
izz y = a +Axl {1/2₂ (20₂ + Axb₂ = (x³q? -221₁ + {₂01³} a X₂ 2 X4 X5 x 26 an d'y + boody + gas yoxs = f(x) a(x) dx² g(x) dx B.C. yo = x 2 a; (y₁ -2y; +y₁; I+Axb; (Y₁+1=Y₁ ) + (x³ g₁Y; itl ditl yn = = B 2 = f₁ (AxU² đi cái thời ty (20-bi trường thi + (DX) =f₁5x₁² 5 Yox) = a(x) + Neboo + T0 = -2000) - Ax bi to gan 800 = f(xXx(Ax) ²
Yі Уіні тоту, + a, yi, - Y;-, = itl i=1, 2, 3, __(n-1) (n-1) I=I: X,Y, тау, тa, y = Уг dо i=2; 1283 +1282+агу, узун тоз уз тазуг 1= 3: 3 2 I 1 n-i=i - бз = бз бі Ү, Уін + оту; +9; Ун-бі В to write + Alter by = + уп там a n- n-2 ntl У-уп Ў, Уп- n-
- Хі Уіні тау +Q, Уі-ї ді itl i=1,2,3, ---(n-1) 1=!: 8,Уг I (n-1) тату Уг тау, тa, y i=2: Y283 +12Yz+ агу, уз уч тоз уз тазуг 1=3: 3 2 1 n-i=i Vn. - Fu. В - Y; Уін + оту; +9,7; б простков = + он упитат Уп-ч a n-2 - бз бі ntl У -- Уп У, Уп- n-