Fuzzy-Based Analysis And Numerical Approximation Of Parabolic Partial Differential Equations With Uncertain Parameters
Keywords:
Fuzzy set theory, Parabolic partial differential equations, Uncertain parameters, Numerical approximation, Finite difference method, α-cut, Stability, Convergence.Abstract
This study focuses on the fuzzy-based analysis and numerical approximation of parabolic partial differential equations with uncertain parameters. In many practical problems, the parameters of parabolic partial differential equations cannot be determined precisely due to measurement limitations, incomplete information, and variations in physical conditions. To address this issue, fuzzy set theory will be incorporated into the mathematical formulation to represent uncertain parameters using fuzzy numbers and membership functions. The α-cut technique will be employed to transform the fuzzy problem into a family of interval-based problems, while the finite difference method will be used to obtain numerical approximations of the solutions. The proposed approach will be evaluated through stability, convergence, and error analysis under different spatial and temporal discretizations. The fuzzy solutions will also be compared with corresponding crisp solutions to examine the effect of parameter uncertainty. The study is expected to demonstrate that the proposed framework can provide reliable numerical solutions while effectively representing uncertainty through lower and upper solution bounds. The methodology will offer a useful mathematical and computational framework for the analysis of uncertain parabolic partial differential equations with potential applications in physical, engineering, and scientific modelling.
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