A numerical method for solving the time variable fractional order mobile–immobile advection–dispersion model

2017 ◽  
Vol 119 ◽  
pp. 18-32 ◽  
Author(s):  
Wei Jiang ◽  
Na Liu
Keyword(s):  
Numerical Method ◽  
Fractional Order ◽  
Dispersion Model ◽  
Time Variable ◽  
Advection Dispersion

Jacobi Spectral Collocation Method for the Time Variable-Order Fractional Mobile-Immobile Advection-Dispersion Solute Transport Model

2016 ◽  
Vol 6 (3) ◽  
pp. 337-352 ◽  
Author(s):  
Heping Ma ◽  
Yubo Yang
Keyword(s):  
Fractional Derivative ◽  
Solute Transport ◽  
Collocation Method ◽  
Dispersion Model ◽  
Time Variable ◽  
Spectral Collocation Method ◽  
Variable Order ◽  
Spectral Collocation ◽  
Advection Dispersion ◽  
Variable Order Fractional Derivative

AbstractAn efficient high order numerical method is presented to solve the mobile-immobile advection-dispersion model with the Coimbra time variable-order fractional derivative, which is used to simulate solute transport in watershed catchments and rivers. On establishing an efficient recursive algorithm based on the properties of Jacobi polynomials to approximate the Coimbra variable-order fractional derivative operator, we use spectral collocation method with both temporal and spatial discretisation to solve the time variable-order fractional mobile-immobile advection-dispersion model. Numerical examples then illustrate the effectiveness and high order convergence of our approach.

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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