Uncertainty Modeling and Differential Equation Applications

Uncertainty modelling provides a rigorous framework for representing imprecise or indeterminate information in dynamical systems, distinguishing itself from classical probabilistic approaches by adhering to axioms of uncertainty theory. Differential equations enriched with uncertainty constructs—ranging from ordinary and delay forms to fractional‐order variants—enable the description of complex phenomena where data are imprecise or subject to expert judgement. Analytical techniques often invoke fixed‐point theorems, Picard iteration or comparison principles to establish existence, uniqueness and stability of solutions, while numerical schemes such as Adams–Bashforth–Moulton or predictor–corrector methods support practical computation. Applications span financial engineering, where uncertain differential equations driven by Liu processes yield novel option‐pricing formulas; control theory, where uncertain switched systems are optimally regulated; and biological or ecological contexts, where delay or memory effects are crucial. Recent advances emphasise fractional operators to capture long‐range dependence, tailored stability criteria for delay systems under uncertainty, and efficient parameter‐estimation algorithms that combine classical collocation with modern optimisation. Overall, the interplay between uncertainty quantification and differential‐equation modelling offers robust tools for decision support in engineering, finance, epidemiology and environmental science, ensuring that solutions remain informative even when empirical data are scarce or noisy.

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One study develops a novel uncertain differential‐equation model of stock prices influenced by a Liu process, deriving closed‐form American barrier option-pricing formulas. Four types of barrier options (up-and-in, down-and-out calls and puts) are analysed, leading to explicit pricing expressions and illustrative numerical examples that demonstrate model flexibility under varying uncertainty levels.

In the realm of fractional dynamics, a new approach combines fractional uncertain differential equations with the Adams numerical method to estimate system parameters. By formulating an optimisation problem for the fractional order and coefficients, then deploying an Adam algorithm within a predictor–corrector scheme, the authors achieve improved forecasting accuracy and offer guidance on selecting α-paths for complex time‐series data.

A separate work addresses stability of uncertain delay differential equations, introducing the concept of almost sure stability under uncertainty theory. The authors establish three sufficient conditions for stability, explore relationships with stability in measure, and illustrate results through examples. This framework extends classical delay‐system analysis to accommodate imprecise delays and state dependencies.

Uncertainty Modeling and Differential Equation Applications publication trend

The graph below shows the total number of articles in uncertainty modeling and differential equation applications across all publications each year (not limited to Nature Index journals).

Technical terms

Uncertain differential equation: A dynamic equation in which one or more terms are modelled as uncertain variables following uncertainty theory rather than random variables.

Fractional differential equation: A generalisation of classical differential equations involving derivatives of non-integer order, used to capture memory and hereditary properties in systems.

Liu process: An uncertain process characterised by stationary and independent increments, employed to model time-varying uncertainty in place of stochastic noise.

Delay differential equation: A differential equation in which the rate of change depends on past states, incorporating time delays explicitly into the model.

References

1. American Barrier Option Pricing Formulas for Stock Model in Uncertain Environment. IEEE Access (2019).
2. A linear-quadratic control problem of uncertain discrete-time switched systems. Journal of Industrial and Management Optimization (2017).
3. Parameter estimation of fractional uncertain differential equations via Adams method. Nonlinear Analysis Modelling and Control (2022).
4. A New Stability Analysis of Uncertain Delay Differential Equations. Mathematical Problems in Engineering (2019).