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Explicit numerical schemes for fractional partial differential equations.

I want to know how to build the fast decaying solution for 

(-\Delta)^\alpha u=u^p   in   R^N\setminus {0}

where p>\frac{N}{N-2\alpha}.

Here fast decay means the solution hjas decaying rate |x|^{2\alpha-N}.

For \alpha=1, Dynamic method was applied. But for fractional laplacian, how

can we build the fast decaying singuar solutions?

Question: Are there studies or findings on the relationship (asymptotic behavior, ...) between U and Laplace of U?

Thank you.

Hello Dear Researchers and Scientific Community:

I would like to know which are the emerging topics in the field of Fractional Partial Differential Equations, it is, topics or subfields where I can perform research and obtain interesting contributions.

Thank you in advance.

Luis

I would like to get a reference containing the existence result of Fractional laplacian Poisson type equation, where the boundary of the domain is lipschitz . To be precise I added the question in pdf format. Please look after it .

How to evaluate the integral term in fractional differential equations.

Dear Research Community devoted to Fractional Calculus and Fractional Differential Equations:

Based in your own teaching and research experience, please mention all the mathematical solution methods (analytical, numerical and analytical combined with numerical and viceversa) you know for Fractional Differential Equations, both Ordinary and Partial, both also linear and nonlinear.

Thank you very much in advance for your kind contributions.

Sincerely

Luis

Can anyone suggest material regarding Von Neuman stability for fractional finite difference techniques?

My answer yes, if we use Complex Fractional Transform with He's Fractional derivative.

Definitely, I ask myself whether it was useful to introduce fractional partial differential equations.

As based on fractional differential operators this class of equations also surfers from the same drawbacks as the one described in:

Fractional partial differential equations were introduced to model anomalous diffusion, i.e. phenomena that exhibits power law behaviours other than 1/2. But it was shown recently that these kinds of behaviours can also be obtained with classical partial differential equations with spatially variables coefficients:

Where it is difficult to propose a physical interpretation with the fractional partial differential equations, classical partial differential equations with spatially variables coefficients allow interpretations relating to the geometry of the systems studied.

What is your opinion ?