# A spline collocation method for a fractional mobile–immobile equation with variable coefficients

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## Abstract

The Crank–Nicolson orthogonal spline collocation (OSC) methods are considered for approximate solution of the variable coefficient fractional mobile–immobile equation. The convection, diffusion, and reaction coefficients can depend on both the spatial and temporal variables, simultaneously. Combining with Crank–Nicolson scheme and weighted and shifted Grünwald difference approximation in time, we establish OSC method in space. It is proved that our proposed fully methods are of optimal order in certain \(H_j\) (\(j=0,1\)) norms. Moreover, we derive \(L^{\infty }\) estimates in space. Numerical results are also provided to verify our proposed algorithm.

## Keywords

Fractional convection diffusion equation Collocation method Variable coefficient Finite-difference method Stability and convergence## Mathematics Subject Classification

65M60 26A33## Notes

### Acknowledgements

Many thanks to Prof. Graeme Fairweather for stimulating discussions and for his constant encouragement and support.

### Compliance with ethical standards

### Conflict of interest

The authors declare to have no conflict of interest.

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