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The Diffusion of Passive Tracers in Laminar Shear Flow

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Solving Cauchy reaction-diffusion equation by using Picard method.

Shadan Sadigh Behzadi¹

¹Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.

|August 21, 2013

Summary

This study introduces the Picard method to solve fuzzy reaction-diffusion equations, proving its solution

Keywords:

Cauchy reaction-diffusion equation Fuzzy number Fuzzy-valued function Generalized differentiability Picard method h-difference

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Area of Science:

Numerical Analysis
Partial Differential Equations
Fuzzy Mathematics

Background:

Reaction-diffusion equations model complex phenomena.
Fuzzy initial conditions introduce uncertainty.
Generalized H-differentiability extends calculus to fuzzy sets.

Purpose of the Study:

To propose and analyze the Picard method for fuzzy Cauchy reaction-diffusion equations.
To establish the existence, uniqueness, and convergence of the proposed numerical method.
To demonstrate the method's efficacy through illustrative examples.

Main Methods:

Application of the Picard iteration method.
Analysis under generalized H-differentiability.
Theoretical proofs for existence, uniqueness, and convergence.

Main Results:

The Picard method successfully solves the fuzzy Cauchy reaction-diffusion equation.
Existence and uniqueness of the solution are rigorously proven.
Convergence of the Picard method is demonstrated theoretically and numerically.
Switching points in example cases were identified.

Conclusions:

The Picard method is an effective tool for solving fuzzy reaction-diffusion problems.
The theoretical framework ensures reliable and accurate solutions.
The method's efficiency is validated by numerical examples.