A Meshfree Approach for Solving Fractional Galilei Invariant Advection–Diffusion Equation through Weighted–Shifted Grünwald Operator

Abstract

Fractional Galilei invariant advection–diffusion (GIADE) equation, along with its more general version that is the GIADE equation with nonlinear source term, is discretized by coupling weighted and shifted Grünwald difference approximation formulae and Crank–Nicolson technique. The new version of the backward substitution method, a well-established class of meshfree methods, is proposed for a numerical approximation of the consequent equation. In the present approach, the final approximation is given by the summation of the radial basis functions, the primary approximation, and the related correcting functions. Then, the approximation is substituted back to the governing equations where the unknown parameters can be determined. The polynomials, trigonometric functions, multiquadric, or the Gaussian radial basis functions are used in the approximation of the GIADE. Moreover, a quasilinearization technique is employed to transform a nonlinear source term into a linear source term. Finally, three numerical experiments in one and two dimensions are presented to support the method.

Approximation of three‐dimensional nonlinear wave equations by fundamental solutions and weighted residuals process

“…, an iterative approach in this case a predictor-corrector scheme combined with one-step time discretization and CN technique as [27][28][29]…”

“…To eliminate the nonlinearity $$$ \left(g(C)\frac{\partial C}{\partial t}\right) $$$, an iterative approach in this case a predictor–corrector scheme combined with one‐step time discretization and CN technique as [27–29] $$$$ g\left({C}^{l+1}\right)\frac{\partial {C}^{l+1}}{\partial t}=\frac{1}{3}\left[g\left({\tilde{C}}^{l+1}\right)\frac{{\tilde{C}}^{l+1}-{\tilde{C}}^l}{h_t}+g\left({\tilde{C}}^l\right)\frac{{\tilde{C}}^{l+1}-{\tilde{C}}^{l-1}}{2{h}_t}+g\left({\tilde{C}}^{l-1}\right)\frac{{\tilde{C}}^l-{\tilde{C}}^{l-1}}{h_t}\right], $$$$ where $$$ {\tilde{C}}^l $$$ is the previously determined approximation of $$$ {C}&#x00...$…”