Non-Euclidean distance fundamental solution of Hausdorff derivative partial differential equations

The Hausdorff derivative partial differential equations have in recent years been found to be capable of describing complex mechanics and physics behaviors such as anomalous diffusion, creep and relaxation in fractal media. But most research is concerned with time Hausdorff derivative models, and little has been reported on the numerical solution of spatial Hausdorff derivative partial differential equations. In this study, we derive the fundamental solutions of the Hausdorff derivative Laplace, Helmholtz, modified Hemholtz, and convection-diffusion equations via a non-Euclidean metric, called the Hausdorff fractal distance. And then the singular boundary method is used to numerically simulate the steady heat transfer governed by the Hausdorff Laplace equation in comparison with the corresponding fractional Laplacian models. Numerical experiments show the validity and applicability of the derived fundamental solution of the Hausdorff Laplace equation.