Abstract
Since the concept of uncertain partial differential equations (UPDEs) was proposed, it has been developed significantly and led us to study parameter estimation for UPDEs. This paper proposes a concept of residual of a class of UPDEs, which follows a linear uncertainty distribution. Afterwards, an \(\alpha\)-path of a class of UPDEs is introduced and the important result that the inverse uncertainty distribution of solution of a class of UPDEs is just the \(\alpha\)-path of the corresponding UPDEs is reached. And a numerical method is designed to obtain the inverse uncertainty distribution of solution of UPDEs. In addition, based on the \(\alpha\)-path and the inverse uncertainty distribution, an algorithm is designed for calculating the residuals of UPDEs corresponding to the observed data. Then a method of moments to estimate unknown parameters in UPDEs is provided. Furthermore, uncertain hypothesis test is recast to evaluate whether an uncertain partial differential equation fits the observed data. Finally, the method of moments is applied to modeling China’s population and the fitness of the estimated parameters is verified by using uncertain hypothesis test.
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Notes
(Comparison Theorem Imbert (2005)) Assume that U(t, y) and V(t, y) are bounded and continuously differentiable functions in \([0,+\infty )\times [0,+\infty )\), satisfying
$$\begin{aligned} \frac{\partial U}{\partial t}(t,y)+\frac{\partial U}{\partial y}(t,y)\le F\left( t,y,U(t,y)\right) ,\\ \frac{\partial V}{\partial t}(t,y)+\frac{\partial V}{\partial y}(t,y)\ge F\left( t,y,V(t,y)\right) , \end{aligned}$$respectively, where F(t, y, p) is Lipschitz continuous with respect to y and p. If
$$\begin{aligned} U(0,y)\le V(0,y), \end{aligned}$$then
$$\begin{aligned} U(t,y)\le V(t,y), ~~~(t,y)\in [0,+\infty )\times [0,+\infty ). \end{aligned}$$China Population and Employment Statistics Yearbook 2010–2017.
MATLAB R2022a, 9.12.0.1884302, maci64, Optimization Toolbox, “fminsearch" function.
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Acknowledgements
This work was supported by Natural Science Foundation of Shaanxi Province of China (No.2022JQ-042), Key Program of National Statistical Science Research (No.2021LZ28) and the Young Talent Support Program of Xi’an University of Finance and Economics.
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Yang, L., Liu, Y. Solution method and parameter estimation of uncertain partial differential equation with application to China’s population. Fuzzy Optim Decis Making 23, 155–177 (2024). https://doi.org/10.1007/s10700-023-09415-5
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DOI: https://doi.org/10.1007/s10700-023-09415-5