Hi, I have a nonlinear ODE of third order, I like to either transform into a solvable form, or solve analytically , however it appears limited using Wolfram Alpha or MATLAB.
Are there any methods of transform (i.e. Darboux ) to solve this?
y'''+(x^2-1)y=0
I.C.: y(0)=0, y'(0)=0, y''(0)=0
Thanks!
With the help of Haar wavelet you can solve linear or nonlinear ODE equation. In this method you can convert your ODE in algebraic equations (n algebraic) and with the help of Matlab software (fsolve function) you can find solution of nonlinear ODE.
As a linear equation (with non-constant coefficients) the eqation is uniquely solvable for arbitrary initial values y(0), y'(0), y'''(0). y(t)=0 solves this equation for y(0)=y'(0)=y''(0)=0, therefore it is the unique solution. I guess, you are interested in the solutions with other (more interesting) intitial values. Applying integral transformations (like Fourier-transform) will _not_ yield an algebraic equation, as the coefficients are not constant, but instead an integral equation (involving a convolution integral).
Fundamental solutions of this equation exist according to theory, but appear to be inexpressible by elementary functions. However by making a power series ansatz for an analytical solution you will obtain a recursion formula for the coefficients of the taylor series of the solutions (developed at 0). Wolfram Alpha (e.g.) can easily give you the Taylor expansion of a certain degree. (This is, in a sense, another transformation functions -> coefficients of Power Series.)
Is there any chance the procedure of the Haar wavelet can be done numerically via MATLAB or Wolfram?
When it comes to numerical solutions, there are many solvers for initial value problems both in Matlab and in Wolfram/Mathematica. Using wavelet transform for this purpose is not among the standard methods. It can only make sense if you know beforehand on which interval you want to solve the equation and with which resolutions (in order to chose a Haar-basis with respect to which you can transform). I'm not sure this is a good idea, but it has been done (e.g. Article Haar wavelet collocation method for the numerical solution o...
). One would transform the highest derivative occurring in the equation (y''') and express the other derivatives as integrals of this highest derivative. The integrals of the elements of the wavelet basis can be expressed as a linear combination of wavelets, so you obtain a linear equation. (This would happen with ANY other linear discretization.) The transformation itself is a matlab function (https://de.mathworks.com/help/wavelet/ref/haart.html). Maybe someone has written code to assemble the matrices that describe the integration, otherwise you would have to do it yourself.
It is noteworthy to say that one is looking for an analytical solution here, in this thread. One said that when the initial conditions are given,
y(0)=1, y'(0)=0, y''(0)=0
they can be used to simplify the ODE from the start. Is this correct? If so , how?