assuming

Anne Wacker
2022-01-20
Answered

Solve fractional differential equation?

$\frac{{d}^{2}}{{dx}^{2}}u\left(x\right)+b\frac{{d}^{\frac{1}{k}}}{{dx}^{\frac{1}{k}}}u\left(x\right)+cu\left(x\right)=0$

assuming$(a,b,c)=const$ and k a parameter?

assuming

You can still ask an expert for help

Kindlein6h

Answered 2022-01-20
Author has **27** answers

Since some of the solutions (from integral transforms) are already listed I'll discuss some of the other approaches to this.

Looking at this:

One thing to note is that a solution is likely anylytic (since x is a real variable), so a polynomial series solution is an option.

Assume u(x) is an analytic function with series expansion (I'll use alpha in the series to avoid confusion with your a):

Which then gives an equation of the form:

So something like:

will give a recurrance relation of the second order for the

Also, if you don't need an exact solution as k grows large the solution will be approximated by the solution to:

due to the order of differentiation approaching 0 (you can see this by taking the limit of the series representation of the fractional part for

and for

The error will grow in factorial order from the points where these O.D.E. approximations exist, but for finding particular points (or a general form to start with variation of prarameters) they're useful.

Mary Nicholson

Answered 2022-01-21
Author has **38** answers

You can use the formula obtained via Laplace Transforms, which works as far as I know for $n\in Q$

${D}^{-n}\left\{f\left(t\right)\right\}=f\left(t\right)\cdot \frac{{t}^{n-1}}{\Gamma \left(n\right)}={\int}_{0}^{t}\frac{{(t-u)}^{n-1}}{\Gamma \left(n\right)}f\left(u\right)du$

So for example, setting$n=\frac{1}{2}$ and multiplying by D you get an expression for the half derivative of f(t)

${D}^{\frac{1}{2}}f\left(t\right)=\frac{1}{\sqrt{\pi}}\frac{d}{dt}{\int}_{0}^{t}\frac{f\left(u\right)}{\sqrt{t-u}}du$

So maybe you can multiply your equation by D to get$D}^{\frac{k+1}{k}$ and then use the equation I give you with appropiate n.

So for example, setting

So maybe you can multiply your equation by D to get

RizerMix

Answered 2022-01-27
Author has **438** answers

it depends on the function u(x). you can try Laplace Transform or Fourier Transform or some other approaches for sufficiently suitable functions to the chosen method.

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Solve the differential equation and obtain the output of the system x(t) as afunction of t.

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to

More specific, I am curious to why we get an 4 and an 8. What formula or equation are being used to get those numbers?I know that the Laplace Transform calculates it for us, however, I want to calculate it "by hand" and show why we get the answer. Any feedback would be gladly appreciated!

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