scholarly journals Application of a Legendre collocation method to the space–time variable fractional-order advection–dispersion equation

2019 ◽  
Vol 13 (1) ◽  
pp. 324-330 ◽  
Author(s):  
F. Mallawi ◽  
J. F. Alzaidy ◽  
R. M. Hafez
Keyword(s):  
Dispersion Equation ◽  
Collocation Method ◽  
Fractional Order ◽  
Space Time ◽  
Time Variable ◽  
Advection Dispersion ◽  
Legendre Collocation Method

Ulam–Hyers stability and analytical approach for m-dimensional Caputo space-time variable fractional order advection–dispersion equation

2021 ◽  
pp. 2250004
Author(s):  
Pratibha Verma ◽  
Manoj Kumar ◽  
Anand Shukla
Keyword(s):  
Numerical Methods ◽  
Dispersion Equation ◽  
Fractional Order ◽  
Analytical Approach ◽  
Dimensional Space ◽  
Adomian Decomposition Method ◽  
Space Time ◽  
Time Variable ◽  
Advection Dispersion ◽  
Adomian Decomposition

This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomian decomposition method and obtain the exact solution in just one iteration. Moreover, with the help of fixed point theory, we study the existence and uniqueness conditions for the positive solution and prove some new results. Also, obtain the Ulam–Hyers stabilities for the proposed problem. Two generalized examples are considered to show the method’s applicability and compared with other existing numerical methods. The present method performs exceptionally well in terms of efficiency and simplicity. Further, we solved both examples using the two most well-known numerical methods and compared them with the TSADM solution.

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