# Finite-difference method for fuzzy singular integro-differential equation deriving from fuzzy non-linear differential equation

## Abstract

In this article, first time, a well-known equation, which can be used to analyze the behavior of the bubble formulation in the mixture of gas and liquid in fluid dynamics, has been considered in the form of a non-linear ordinary differential equation in fuzzy environment. In addition, the fuzzy non-linear ordinary differential equation has been presented in the sense of $$\alpha$$-cut. Then, the left and right branches of ordinary differential equations have been converted into Volterra integro-differential equations with singular kernels. A numerical method based on the numerical approximation of first-order derivative by finite difference has been modified to solve the equation after converting it into the non-linear singular integro-differential equations. In the first-order method, the right rectangles rules have been used to compute the integrals in the converted equations. In addition, in the second-order method, the integrals have been analyzed with the help of trapezoidal rule instead of right rectangles rule. After that, the iterative sequence has been defined to find the approximate solution of the given equation for both first- and second-order methods. A convergence analysis has been developed for both first- and second-order methods in the form of different types of lemmas and theorems for different cases. Some of these theorems have been shown that the equations are uniquely solvable for both first- and second-order methods. These results have been shown with the help of Banach fixed-point theorem. A brief comparison of our method with other existing methods has been presented to show the efficiency and reliability of our proposed method. In addition, a brief discussion about the advantages and disadvantages of our method has been discussed. Some numerical examples have been chosen and examined to show the validation of our proposed method. In addition, some error analysis have been examined in the form of different types of tables and figures which shows the effectiveness of our method.

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## Data availability statement

Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.

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## Acknowledgements

In this article, the study of Sandip Moi is funded by Council of Scientific and Industrial Research (CSIR) Government of India (File No.- 08/003(0135)/2019-EMR-I).

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Correspondence to Suvankar Biswas.

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