Key to "Solve the ode using fixed-point iteration−[(1+u4)ux]x=sin⁡(x)+sin⁡(5x), where the..."

sempteim245

Answered question

2022-03-20

Solve the ode using fixed-point iteration

- {[(1 + u^{4}) u_{x}]}_{x} = \sin (x) + \sin (5 x)

, where the domain is

[0, 2 π] u (0) = u (2 π) = 0

for boundaries.

membatas0v2v

Beginner2022-03-21Added 19 answers

Step 1
The equation can be written in the form

(1 + u^{4}) u_{\times} + 4 u^{3} {(u_{x})}^{2} = f (x) .

Using centred finite differences, we get

(1 + u_{i}^{4}) (u_{i + 1} - 2 u_{i} + u_{i - 1}) + u_{i}^{3} {(u_{i + 1} - u_{i - 1})}^{2} = h^{2} f_{i} .

Finally, you must solve this nonlinear system using a fixed point method. One possibility could be

u_{i}^{(k + 1)} = \frac{u_{i + 1}^{(k)} + u_{i - 1}^{(k)}}{2} + \frac{{(u_{i}^{(k)})}^{3} {(u_{i + 1}^{(k)} - u_{i - 1}^{(k)})}^{2}}{2 (1 + {(u_{i}^{(k)})}^{4})} - \frac{h^{2} f_{i}}{2 (1 + {(u_{i}^{(k)})}^{4})}

Now, you need to check if it converges and, if it does not converge, rewrite the nonlinear system into another fixed point problem.

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