An Approximate Solution of the Space Fractional-Order Heat Equation by the Non-Polynomial Spline Functions

Abstract

The linear non-polynomial spline is used here to solve the fractional partial differential equation (FPDE). The fractional derivatives are described in the Caputo sense. The tensor products are given for extending the one-dimensional linear non-polynomial spline to a two-dimensional spline to solve the heat equation. In this paper, the convergence theorem of the method used to the exact solution is proved and the numerical examples show the validity of the method. All computations are implemented by Mathcad15.

Transient and passage to steady state in fluid flow and heat transfer within fractional models

“…Anomalous sub diffusion wherein the spread of particles occurs slower than the classical diffusion was numerically solved via compact finite differences in space and Grn¨wald–Letnikov scheme in fractional time in Al-Shibani et al (2013). A few more such recent attempts are variational iteration method (Ates and Yıldırım, 2010), operational Tau method (Vanani and Aminataei, 2012), homotopy perturbation method (Liu and Dong, 2015), reproducing kernel (Arqub, 2018), H2 matrix representation (Boukaram et al , 2020), nonpolynomial spline functions (Hasan and Salim, 2021), Legendre spectral method (Liu and Lu, 2021), Sinc and B-Spline interpolation (Adibmanesha and Rashidiniab, 2021), Chebyshev wavelets (Dincel and Polat, 2021), meshless methods (Nikan et al , 2021; Bavi et al , 2022), Crank–Nicholson method (Xie and Fang, 2022), neural network method with shifted Legendre orthogonal polynomials (Qu et al , 2022), compact implicit difference method (Ali et al , 2022), finite volume method (Fang et al , 2022) and boundary integral equation method (Yao and Wang, 2022).…”