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The purpose of this paper is to apply a new modified homotopy perturbation method, which is effective to solve multi-order fractional equations with non-polynomial initial and boundary conditions.

The proposed algorithm is tested on multi-order fractional advection-dispersion equations. The fractional derivatives described in this paper are in the Caputo sense.

Approximate results explicitly reveal the complete reliability, efficiency and accuracy of the new modified technique.

It is observed that the approach may be implemented to other multi-fractional models with non-polynomial initial and boundary conditions.

The purpose of this article is to understand the current research status and future development trends in the field of numerical simulation on rock mass grouting.

This article first searched the literature database (EI, Web of Science, CNKI, etc.) for keywords related to the numerical simulation of rock mass grouting to obtain the initial literature database. Then, from the initial database, several documents with strong relevance to the numerical simulation theme of rock mass grouting and high citation rate were selected; some documents from the references were selected as supplements, forming the sample database of this review study (a total of 90 articles). Finally, through sorting out the relationship among the literature, this literature review was carried out.

The numerical simulation of rock mass grouting is mainly based on the porous media model and the fractured media model. It has experienced the development process from Newtonian fluid to non-Newtonian fluid, from time-invariant viscosity to time-varying viscosity, and from generalized theoretical model to engineering application model. Based on this, this article summarizes four scientific problems that need to be solved in the future in this research field: the law of grout distribution at the cross fissures, the grout diffusion mechanism under multi-field coupling, more accurate grouting theoretical model and simulation technology with strong engineering applicability.

This research systematically analyzes the current research status and shortcomings of numerical simulation on rock mass grouting, summarizes four key issues in the future development of this research field and provides new ideas for the future research on numerical simulation on rock mass grouting.