A new class of travelling wave solutions for local fractional diffusion differential equations

Ziyue Shi,Wei Qi,Jing Fan

doi:10.1186/s13662-020-2500-y

Ziyue Shi, Wei Qi + Show 1 more

Open Access

https://doi.org/10.1186/s13662-020-2500-y

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Abstract

In this paper, we investigate a 3-D diffusion equation within the scope of the local fractional derivative. For this model, we establish local fractional vector operators and a local fractional Laplace operator defined on Cantor-type cylindrical coordinate and Cantor-type spherical coordinate, respectively. With the help of the spherical symmetry method based on those operators, we obtain exact traveling wave solutions of the 3-D diffusion equation. The results reveal that the proposed schemes are very effective for obtaining nondifferentiable solutions of fractional diffusion problems.

Highlights

In recent years, classical mathematics has been greatly enriched because of the advancement and application of fractional calculus
5 Conclusion In this paper, we investigated a 3-D fractional diffusion equation defined on Cantor sets with local fractional derivative, which is quite useful in solving nondifferentiable problems in fractal time-space
We have derived transformations of local fractional vector operators and the local fractional Laplace operator in the Cantor-type cylindrical and spherical coordinates, and those operators are little different from the existing research results

Summary

Introduction

Classical mathematics has been greatly enriched because of the advancement and application of fractional calculus. Local fractional calculus has become increasingly popular and gained important advancement due mainly to its outstanding properties in modeling complex nonlinear dynamical systems in different branches of mathematical physics, such as nanoscale flows [19], heat transmission [36], diffusion on Cantor sets [34], and others. Several analytical methods, such as the local fractional homotopy perturbation Sumudu transform method [26], local fractional variational iteration algorithm [37] and local fractional Fourier series [33], have been proposed to address local fractional partial differential equations. The local fractional gradient operator in the Cantor-type cylindrical coordinates reads as

Rα eαθ

Conclusion