Non-probabilistic solutions of imprecisely defined fractional-orderdiffusion equations

doi:10.1088/1674-1056/23/12/120202

摘要/Abstract

摘要： The fractional diffusion equation is one of the most important partial differential equations (PDEs) to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0< α ≤ 1 with fuzzy initial condition is taken into consideration. Fuzziness appearing in the initial conditions is modelled through convex normalized triangular and Gaussian fuzzy numbers. A new computational technique is proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy diffusion equation is converted first into an interval-based fuzzy differential equation. Next, this equation is transformed into crisp form by using the proposed double parametric form of fuzzy numbers. Finally, the same is solved by Adomian decomposition method (ADM) symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases.

关键词: double parametric form of fuzzy number, fuzzy fractional diffusion equation, ADM

Abstract: The fractional diffusion equation is one of the most important partial differential equations (PDEs) to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0< α ≤ 1 with fuzzy initial condition is taken into consideration. Fuzziness appearing in the initial conditions is modelled through convex normalized triangular and Gaussian fuzzy numbers. A new computational technique is proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy diffusion equation is converted first into an interval-based fuzzy differential equation. Next, this equation is transformed into crisp form by using the proposed double parametric form of fuzzy numbers. Finally, the same is solved by Adomian decomposition method (ADM) symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases.

Key words: double parametric form of fuzzy number, fuzzy fractional diffusion equation, ADM

中图分类号: (Numerical simulation; solution of equations)

02.60.Cb

02.60.-x (Numerical approximation and analysis) 02.90.+p (Other topics in mathematical methods in physics) 07.05.Mh (Neural networks, fuzzy logic, artificial intelligence)

S. Chakraverty, Smita Tapaswini. Non-probabilistic solutions of imprecisely defined fractional-orderdiffusion equations[J]. , 2014, 23(12): 120202-120202.

S. Chakraverty, Smita Tapaswini. Non-probabilistic solutions of imprecisely defined fractional-orderdiffusion equations[J]. Chin. Phys. B, 2014, 23(12): 120202-120202.

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