I need help with solving this difficult fluid dynamic expression. I have tried using rules of logs, symbolab algebra calculator and Wolfram Alpha calculator, and I have got no solution.
How would you solve the following expression USING the NEWTON-RAPHSON method for $x$?
$$e^{\frac{x^2}{4vt}} = 1+\frac{x^2}{2vt}$$
When solving this USING the NEWTON-RAPHSON method, the solution is: $x=2.2418\sqrt{vt}$
I want to know how you could solve the first expression using the NEWTON-RAPHSON method to get the solution. So could someone please provide a step-by-step solution, by using this method please?
Note: This question was answered, however it was NOT answered using NEWTON-RAPHSON method. It was answered using the Lambert W function, which is a very long and complicated process as compared to the Newton-Raphson method.
Best Answer
Well, we have:
$$\exp\left(\frac{x^2}{4\cdot\text{v}\cdot t}\right)=1+\frac{x^2}{2\cdot\text{v}\cdot t}\tag1$$
Now, we know that we can write:
$$\exp\left(\alpha\right)=\sum_{\text{n}=0}^\infty\frac{\alpha^\text{n}}{\text{n}!}=\frac{\alpha^0}{0!}+\frac{\alpha^1}{1!}+\frac{\alpha^2}{2!}+\dots=$$ $$1+\alpha+\frac{\alpha^2}{2}+\dots\tag2$$
So, for equation $(1)$ we can write:
$$1+\frac{x^2}{4\cdot\text{v}\cdot t}+\frac{1}{2}\cdot\left(\frac{x^2}{4\cdot\text{v}\cdot t}\right)^2+\dots=1+\frac{x^2}{2\cdot\text{v}\cdot t}\tag3$$
Using the aproximation of three terms we have:
$$1+\frac{x^2}{4\cdot\text{v}\cdot t}+\frac{1}{2}\cdot\left(\frac{x^2}{4\cdot\text{v}\cdot t}\right)^2\approx1+\frac{x^2}{2\cdot\text{v}\cdot t}\space\Longleftrightarrow\space$$ $$x\approx0\space\vee\space x\approx\pm2\sqrt{2}\cdot\sqrt{\text{v}\cdot\text{t}}\tag4$$