Numerical Treatment of Neutrosophic Differential Equations via (α,β,γ)-Cut Decomposition in a Functional-Analytic Framework

In this study, the numerical comparison is performed in solving first-order neutrosophic initial value problems employing two algorithms, namely, the Euler and fourth-order Runge–Kutta (RK4) methods in the neutrosophic environment. As an example of such a problem, the model y′(t) = 0.5∙y(t) on [0, 2] is chosen, but in the case of the additional neutrosophic initial condition y(0)=⟨(0.8, 1.0, 1.3)⟩. The truth, indeterminacy, and falsity functions of the initial datum are given distinct supports ([0.8, 1.3], [0.7, 1.45], and [0.75, 1.4]) so that the three uncertainty components vary independently. This problem is represented as crisp-endpoint systems by means of (α,β,γ)-cut approach, and then it is solved numerically at each cut level while using the exact solution as a reference. In accordance with the iteration results, the maximum absolute error of the Euler algorithm at h = 0.2 is 1.619 × 10−1 at point t = 2, whereas for the RK4 approach, this value is 2.7 × 10−6, meaning that the improvement degree reaches about five orders of magnitude. The analysis of step refinement confirms the convergence orders of the methods that are 1.00 and 4.00 for the Euler and RK4 approaches, respectively. At equal computational work (40 evaluations of the right-hand side per endpoint), the RK4 error is still 1.6 × 104 times smaller than the Euler error. The neutrosophic solution is defined precisely as the solution set of the crisp problems issuing from the cuts of the initial datum, and a theorem in the complete metric space of cut-represented neutrosophic numbers establishes its existence, uniqueness, nested-interval structure, and continuous dependence under a Lipschitz condition only. To validate the findings beyond the benchmark, the framework is applied to a first-order pharmacokinetic elimination model (decay) and to the nonlinear logistic growth model; the observed orders again approach 1.00 and 4.00. A linear stability analysis identifies the step-size thresholds h < 1/|k| (ordering of the cut endpoints) and h < 2/|k| (Euler), h < 2.785/|k| (RK4) (boundedness), and the experiments confirm that beyond these thresholds the computed object ceases to be a neutrosophic number. The claims are limited to the problems tested; the novelty relative to earlier (α,β,γ)-cut studies is stated explicitly, and the approach is placed in context with three related paradigms: the convergence merits of the variational iteration method for neutrosophic problems are discussed, step-wise solvers are compared with physics-informed neural networks, and a combined two-scale fractal–neutrosophic treatment of imperfect real-world data is outlined.