Abstract
This article discusses an approximate scheme for solving one-dimensional heat-like and wave-like equations in fuzzy environment based on the homotopy perturbation method (HPM). The concept of topology in homotopy is used to create a convergent series solution of the fuzzy equations. The objective of the study is to formulate the double parametric fuzzy HPM to obtain approximate solutions of fuzzy heat-like and fuzzy wave-like equations. The fuzzification and the defuzzification analysis for the double parametric form of fuzzy numbers of the fuzzy heat-like and the fuzzy wave-like equations is carried out. The proof of convergence of the solution under the developed approximate scheme is provided. The effectiveness of the proposed method is tested by numerically solving examples of fuzzy heat-like and wave-like equations where results indicate that the approach is efficient not only in terms of accuracy but also with respect to CPU time consumption.
Highlights
The principles of fuzzy sets theory have been illustrated and applied in many fields [1,2].Over the last years, the use of fuzzy sets in computational mathematics has gained particular attention, in order to properly revise existing techniques for solving differential equations from a fuzzy theory standpoint e.g., [3,4]
For the purpose of this paper, we only provide the definition of double parametric form of fuzzy numbers and its properties, while for the single parametric form, the reader can refer to [26] for further details
We successfully developed an approximate scheme for solving one-dimensional heat-like and wave-like equations in a fuzzy environment based on the Homotopy Perturbation Method (HPM)
Summary
The principles of fuzzy sets theory have been illustrated and applied in many fields [1,2].Over the last years, the use of fuzzy sets in computational mathematics has gained particular attention, in order to properly revise existing techniques for solving differential equations from a fuzzy theory standpoint e.g., [3,4]. Fuzzy partial differential equations (FPDEs) are known to be useful in modeling a dynamic system with inadequate knowledge about the behavior of the system, where they incorporate uncertainty characteristics into the model [5,6]. Due to their frequent role in the design and simulation of many technological applications, such as heat transmission and mass transfer, electromagnetic fields, static and dynamic structure, meteorology, and biomechanics, FPDEs have gained great interest among scientists and engineers [7,8,9]. Homotopy Perturbation Method (HPM) is Mathematics 2020, 8, 1737; doi:10.3390/math8101737 www.mdpi.com/journal/mathematics
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