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In this article, some properties of neutrosophic derivative and neutrosophic numbers have been presented. This properties have been used to develop the neutrosophic differential calculus. By considering different types of first- and second-order derivatives, different kind of systems of derivatives have been developed. This is the first time where a second-order neutrosophic boundary-value problem has been introduced with different types of first- and second-order derivatives. Some numerical examples have been examined to explain different systems of neutrosophic differential equation.

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In this article, different types of Gaussian quadrature methods have been presented to find the numerical integration of a neutrosophic valued function. A new definition of the distance between two neutrosophic number has been defined and it has been proved that the distance and the set of all neutrosophic number form a complete metric space. Also, the definition of neutrosophic continuity on a closed-bounded interval has been defined in the sense of $$(\alpha ,\beta ,\gamma )$$ ( α , β , γ ) -cut. This is the first time, when the Gauss–Legendre integration, Gauss–Chebyshev integration and Gauss–Laguerre integration rule have been discussed in neutrosophic environment. In the first test example, the comparison between one-point, two-point and three-point Gauss–Legendre integration rules have been presented in terms of tables and figures. Also, in the second example, the comparison between one-point, two-point Gauss–Chebyshev and Gauss–Laguerre integration rules have been discussed.