Week - 2 - Explicit and Implicit Analysis

Objective: Implicit and Explicit methods for solving F(u) = u³+9u²+4u

Introduction:

Numerical methods play a crucial role in solving equations where analytical solutions are dufficult or impossible to obtain. Here, we explore two numerical techniques - explicit and implicit methos - for solving the given equation. We'll compare their performance in terms of accuracy and computational efficiency.

Explicit methods are numerical techniques where the value of the function at a certain point is expressed explicity in terms of the values of the function at previous points. In other words, the y ionvolve computing the solution directly from known quanities at earlier time steps.

Implicit methods, on the other hand , involve computing the solution through an implicit that relates the value of the function at the current time step to its value at previous time steps. These methods typically require solving a system of equations, often through iterative techniques.

Implicit Method vs Explicit Method

	IMPLICIT METHOD	EXPLICIT METHOD
Deformation	small	Large
What cause large deformation	Force and dynamic	Velocity, acceleration with mass and inertia
K matrix inversion	yes	No
Time step size	Large	Small
What limits the time step	Accuracy	Stability
Cost per time step	large	Small
What dominate the cost	Matrix inversion	Element process
Parallel process	Possible	Not possible

Solution:

By Explicit Method

F(u) = u³+9u²+4u

K(u) = df / du = 2u² +18u + 4

Let ΔF = 1

	F(u) = u3 + 9u2 + 4u
	K(u) = dF/du = 3u2 + 18u + 4
	ΔF = k(un).Δu
	Δu = ΔF/k(un)
	Δu = un+1 - un
	un	F(un)	K(un)	ΔF	Δu	un+1
Step 1	0	0	4	1	0.25	0.25
Step 2	0.25	1.58	8.69	2.00	0.23	0.48
Step 3	0.48	4.11	13.34	3.00	0.22	0.71
Step 4	0.71	7.65	18.18	4.00	0.22	0.93

By Implicit Method

Conclusion

Explicit and implicit analysis is done for the given equation, implicit method is more accurate than the explicit but the time consumption is more in umplicit method.