Naive regular perturbation

[sam_0017]

can anyone hlep me with this qustion ?

Consider the equation

ε x^3 + x^2 - x - 6 = 0 ,ε > 0. (1)

1. Apply a naive regular perturbation of the form

x~[itex]^{0}_{∞}Ʃ[/itex] x_n εⁿ as ε→0⁺

do derive a three-term approximation to the solutions of (1).

2. The above perturbation expansion should only give you an approximation for 2 of the roots.
Apply a leading order balance argument to device suitable expansions for the other root, again
in the limit ε → 0⁺. Again, derive a three-term approximation this third case.

3. Solve (1) numerically for ε = 0.01 ?

 

[jvicens]

Hello there,

I would be happy to help you with this question. Let's start by breaking down the problem and discussing the steps we can take to solve it.

1. To begin, we can apply a naive regular perturbation of the form x~^{0}_{∞}Ʃ xn εn as ε→0+ to the given equation (1). This means we will be expanding the solution of x in powers of ε. We can then use this expansion to derive a three-term approximation of the solutions to the equation.

2. However, this perturbation expansion will only give us approximations for two of the roots. To find the third root, we can use a leading order balance argument. This involves balancing the leading order terms of the equation to determine suitable expansions for the third root. Once we have these expansions, we can again derive a three-term approximation for this case.

3. Finally, we can solve the equation (1) numerically for ε = 0.01. This will give us an accurate solution to the equation, which we can compare to our approximations from the previous steps.

I hope this helps to clarify the steps needed to solve this problem. Let me know if you have any further questions or need more assistance. Good luck!