in ordinary differential equations we use shooting method. I don't know any methods in fractional calculus for solving boundary value problems.
There are many methods for example:
Homotopy analysis method
Decomposition method
Modi ed Laplace decomposition method
Variational iteration method,
Modi ed variational iteration method
Finite Difference Method
Newton-Raphson method
Chebyshev and Legendre Collocation Methods...
If you write differentiation operator in Finite Difference Method, like forward operator, center operator (2nd order) and backward operator. You may use that with operator calculus for building other differentiation numerically:
For example D(.) := d(.)/dxwith central discrete differencing, D acting on D is D^2 , you can use that to make D to the power of alpha which is a non integer (What you see in fractional PDE)
I know some deep literature on Ft-ADE (Time-Fractional Advection Diffusion Equation) from groundwater transport researchers, but I do not know the Fx-ADE
See the works by Kavvas:
ML Kavvas, A Ercan, J Polsinelli (2017), Governing equations of transient soil water flow and soil water flux in multi-dimensional fractional anisotropic media and fractional time, Hydrology and Earth System Sciences, 21 (3), 1547
ML Kavvas, T Tu, A Ercan, J Polsinelli (2017), Fractional governing equations of transient groundwater flow in confined aquifers with multi-fractional dimensions in fractional time, Earth System Dynamics, 8 (4), 921
The Blue Book by Randall J. LeVeque on Finite Difference
Hope it helps,
Kaveh
There are many methods for example:
Homotopy analysis method
Decomposition method
Modi ed Laplace decomposition method
Variational iteration method,
Modi ed variational iteration method
Finite Difference Method
Newton-Raphson method
Chebyshev and Legendre Collocation Methods...
You may choose the "Laplace decomposition method" for fractional differential equations.
Alal
Wavelet based numerical methods, hopefully efficient scheme is other option.
@ Ahmed A. Hamoud, Thankfully, could you please recommend an efficient method from the above mentioned ones in order to be compared analytically with RK4 method,.. e.t.c,
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