Neutrosophic Logic is a tool based on non-standard analysis to represent mathematical model of uncertainty, vagueness, ambiguity, incompleteness, and inconsistency. In Neutrosophic set, indeterminacy is quantified explicitly whereas the truth membership, indeterminacy membership, and falsity membership are independent. This plays a vital role in many situations when we handle inconsistent and incomplete information. In modeling problems, differential equations have major applications in the field of science and engineering and the study of differential equation with uncertainty is one of emerging field of research. In this paper, the differential equations in neutrosophic environment are explored, also the solution of second-order linear differential equation with trapezoidal neutrosophic numbers as boundary conditions is discussed. Furthermore, the numerical example is given to demonstrate the solution with different values of (α, β, γ)-cut of trapezoidal neutrosophic number.

In this paper the Neutrosophic ordinary differential equation of first order via neutrosophic numbers is epitomized. We also intend to define the neutrosophic numbers and their (α, β, γ)-cut. Finally a numerical example is given to demonstrate its practicality and             effectiveness of the differential equation involving neutrosophic numbers.

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This chapter devises a new concept of priority weighted neutrosophic refined soft set (PWNRSS) by combining neutrosophic refined sets and soft sets implemented with prioritized universal elements and weightage-imposed parameters. The concepts of PWNRS subset, PWNRS null set, and PWNRS universal set are defined. Based on the definitions of n-norm and n-co-norm, the theoretical operations of PWNRS sets such as union, intersection, and complement are defined. AND-product and OR-product between two priority-weighted neutrosophic refined soft sets are introduced. Furthermore, priority weights neutrosophic refined soft set is expanded to MCDM technique to handle decision-making issues. TOPSIS has been examined more thoroughly for the PWNRSS decision-making issue. This proposed method might be extremely valuable in large-scale decision-making situations. Numerical examples are also provided to demonstrate the methodologies' dependability and applicability.