Numerical Solution of Neutrosophic Fuzzy Schrödinger Equation Using Modified Finite Difference Method

In this paper, a comprehensive study of the fuzzification and defuzzification of the Schrödinger equation in fuzzy neutrosophic environment is presented involving fuzzy truth (T) component, an indeterminacy (I) component, and a falsehood (F) component. Moreover, a modified numerical method of second-order accuracy in both time and space based on Crank–Nicolson method is reformulated and implemented to solve the neutrosophic Schrödinger equation under both amplitude and phase terms. The triangular neutrosophic number t was used. The stability of the modified Crank–Nicolson method has been investigated using the von Neumann method to show that the proposed approach is unconditionally stable. A numerical experiment is carried out, and the results obtained indicate the effectiveness and reliability of the proposed modified approach. Finally, an accuracy study comparing the neutrosophic exact and numerical results at the α,β,γ-cut level set is conducted. The mathematical novelty of this work lies in reformulating the classical Crank–Nicolson finite difference scheme, for the first time, into six coupled systems that govern the lower and upper bounds of all three neutrosophic components of the neutrosophic fuzzy Schrödinger equation (NFSHE) simultaneously, and in extending the von Neumann stability criterion to prove unconditional stability of the NFSHE across this coupled neutrosophic system, a result with no counterpart in the crisp or single-valued fuzzy literature.