Week - 2 - Explicit and Implicit Analysis

Explicit Calculations

Stepi	∆Fi	∆ui	ui	Fexti	finti	Fint-Fexti
1	1	0.25	0.25	1	1.578	0.5781
2	1	0.1758	0.365	2	2.709	0.709
3	1	0.1	0.465	3	3.8467	0.8647

F(u)=u³+9u²+4u K(u)=3u²+18u+4

Step1

U₀=0, k(u₀)=4, ∆F=1, ∆F=1, ∆u₁=∆F/K(u₀)=1/4=0.25 u₁=u₀+∆u₁=0.25

F_int1=1.578

Step2

K(u₁) = 8.6875 ∆u₂=∆F/k(u₁)=0.115

U₂=u₁+∆u₂=0.3651

F_int2=2.709

Step3

K(u₂) = 11, ∆u₃=∆F/K(u₂)=0.1, u₃=u₂+∆u₂=0.4651

F_int3=3.8467

Implcit Calculations

Stepi	∆Fi	ui	Fexti	finti	finti - Fext
1	1	0.17665	1	1.000206	0.000206
2	1	0.296	2	2	0
3	1	0.3911	3	3.0008	0.0008

F(u)=u³+9u²+4u K(u)=3u²+18u+4

Step1

U₀=0, k(u₀)=4, ∆F=1, ∆F=1, ∆u₁=∆F/K(u₀)=1/4=0.25 u₁=u₀+∆u₁=0.25

F_int1=1.578

R⁽⁰⁾=0.5781 > 0.001

δu¹= -[K(u₁⁽⁰⁾]^-1R⁽⁰⁾=-0.066546

u₁⁽¹⁾=u₁⁽⁰⁾ + δu¹ = 0.1834

F_ext=1, f_int = 1.042884, R⁽¹⁾=0.042884 >0.001

U₁= u₁⁽⁰⁾=0.25

δu⁽²⁾ = δu⁽¹⁾ – [K(u₁⁽¹⁾]^-1R⁽¹⁾ = 0.07325

u₁⁽²⁾=u₁⁽⁰⁾ + δu⁽²⁾ = 0.17665 F_ext = 1, f_int=1.000206, R⁽²⁾ = 0.000206 < 0.01

Step 2

F(u)=u³+9u²+4u K(u)=3u²+18u+4, u₁ = 0.17665

K(u₁) = 7.3, ∆u_{2 =} ∆F/K(u­₁)=0.137, u₂ = u₁ + ∆u₂ = 0.3146

F_ext = ∆F=1, f_int­ = 2.18 R⁽⁰⁾ = f_int – F_ext =2.18 – 2 =0.18 >0.001

U₂ = u₂⁽⁰⁾=0.3146, δu⁽¹⁾ = -[K(u₂⁽⁰⁾]^-1R⁽⁰⁾

K[u₂^(0)] =9.9628

δu⁽¹⁾ = -0.01806, u₂⁽¹⁾=u₂⁽⁰⁾ +δu⁽¹⁾ = 0.296

F_int = 2, R⁽¹⁾ = f_int - F_ext = 2 -2 =0 < 0.001

Step 3

F(u)=u³+9u²+4u K(u)=3u²+18u+4, u₁ = 0.17665

U₂ = 0.296

K(u₂) = 9.6

∆u₃ = ∆F/K(u₂) = 0.1042 u₃=u₂₊∆u₃ =0.4

F_int = 3.104, R⁽⁰⁾ =3.104 – 3 = 0.104 > 0.001

U₃= u₃⁽⁰⁾=0.4, K[u₃⁽⁰⁾] = 11.68

Δu⁽¹⁾ = -[K(u₃⁽⁰⁾]^-1R⁽⁰⁾ = 0.0089, u₃⁽¹⁾= u₃⁽⁰⁾ + δu⁽¹⁾ = 0.3911

F_int = 3.0008 F­­_ext = 3, R⁽¹⁾ = 3.008 -3 = 0.0008 <0.001

Also find in the attached word document the above calculations