Homotopy Analysis Method - Science topic

Does someone have HAM code for fractional PDES?

such as homotopy analysis method

in the method (HPM) I do not know how to construct the homotopy convex in many papers i find different of type homotopy (is it modification of these method or what ?)

and what is the conditions for construct the correct homotopyconvex

#applied mathematics #Neumerical analysis #Homotopy perturbation method

I am modeling flow of fluid through elastic tubes.

homotopy perturbation method

Interested in using HAM method to find solution of flow and heat transfer problems.

I have learnt to solve fractional order differential equation using Homotopy Analysis Method, I am trying to learn Adomian Decomposition Method, please share MATLAB codes to implement HAM or ADM......

Homotopy Methods are claimed to solve a system of nonlinear algebraic equations (NAEs) and to find solution of nonlinear PDEs without having to invert Jacobian (tangent stiffness) matrix like Newton methods.

http://techscience.com/CMES/v98n6/27167

http://www.techscience.com/cmc/v23n1/22627

https://www.depts.ttu.edu/coe/CARES/pdf/570.pdf

However, in the mentioned papers the Homotopy Methods are used for abstract NAEs or by deriving tangent stiffness matrix explicitly (analytically). So I am not clear whether it is possible to apply the Homotopy Methods to ordinary FEM process in which the tangent stiffness matrix is assembled by looping through elements. I would like to know if there is a possibility to replace Newton iteration with such methods in nonlinear FEM and how one might implement them.

Thank you in advance.

Numerical optimization

Fore more explanation :

I am searching for an analytical solution/method to solve the equation of moving medium with space dependent coefficients. In fact, the separation variable method is used to express the unknown displacement, then the initial wave equation of a moving medium (string/cable/beam...) is reformulated such as the new unknowns are the modal shapes (since the time dependent function/modal coordinates are assumed to describe simple harmonics; an assumption generally adopted in linear dynamics as I think!!). The obtained differential equation has a space dependent coefficient (the tension/stress in the medium as well as the travelling velocity (not the wave celerity!! ) are considered as space dependent functions for sake of generality). Moreover, as it is well known, this differential equation contains complex coefficient (also space-dependent) due to the travelling velocity.

So, my question will be :

How can I solve this differential equation ? It is a differential equation with space dependent coefficients in the most general term.

P.S : I am not interested in the determination of time contributions/modal coordinates but I am searching for an analytical method to find these modal shapes in this particular case (not another one) or an analytical solution if it exists!! (I did some analytical development using the Homotopy Analysis Method but I obtained a not very "elegant"/"beautiful" expression of the solution , especially if I want to compute it, I think it is a little bit complicated as expression!! )

I am not also interested also in simple sinusoids solutions (the classic procedure to approach modal shapes).

Thank you.

(Attached the equation with the variables in a pdf file : equation to solve.pdf)

Official page of BVPh 2.0 project does not respond.

Thank you!

Could anyone please provide me with Mathematics Subject Classification 2010 codes for the following methods applied to PDE:

1- Approximate-Analytical methods;

2- Optimal Homotopy Asymptotic Method;

3- Homotopy Analysis Method;

4- Homotopy Perturbation Method;

Many Thanks

Regards.

\frac{dx}{dt}-a_1x+ \frac{x}{a}\left[\frac{dx}{dt}+a_2x+a_3x^2+a_4y\right]=0\\

\frac{dy}{dt}-b_1y+ \frac{x}{b}\left[\frac{dy}{dt}-b_1y\right]+b_2y^2=0

Homotopy analysis method.

Dear researchers

I can not understand what happen in homotopy analysis method when it used for PDEs. could you help me by simple examples in details, particularly to show how R_(m-1) is calculated and how to plot the solution?

We want to use HAM Method, a numerical method of solution to solve flow and heat transfer problems under certain boundary conditions.