# On random fuzzy fractional partial integro-differential equations under Caputo generalized Hukuhara differentiability

## Abstract

This paper is devoted to the study of the solvability of random fuzzy fractional partial integro-differential equations under Caputo generalized Hukuhara differentiability. The notions of random fuzzy variables and fuzzy stochastic processes are developed for multivariable functions. The existence and uniqueness of two types of integral solutions generated from Darboux problem for nonlinear wave equations are proved using the successive approximations method and Gronwall’s inequality for stochastic processes. The continuous dependence on the data, the boundedness, and the stability with probability one of integral solutions are established to confirm the well posedness of our model. Some computational examples are presented to illustrate the theoretical results.

## Keywords

Random fuzzy variable Random fuzzy fractional partial integro-differential equations Caputo gH derivatives Gronwall’s inequality Integral solution## Mathematics Subject Classification

35R13 35R60 45R05 26A33 26E50## Notes

### Acknowledgements

The author is greatly indebted to Editor-in-Chiefs (Prof. Jose E. Souza de Cursi), Associate Editor (Prof. José Tenreiro Machado), and anonymous reviewers for their comments and valuable suggestions that greatly improve the quality and clarity of the paper.

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