Non-probabilistic solutions of imprecisely defined fractional-order diffusion equations

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.

Efficient Numerical Solutions for Fuzzy Time Fractional Convection Diffusion Equations Using Two Explicit Finite Difference Methods

“…where c 1 (x), c 2 (x ), c 3 (x), and c 4 (x) are the crisp functions, with According to the r-cut technique discussed in [30], the defuzzification of Equation ( 1) can be presented for all r ∈ [0, 1] as follows:…”

Efficient Numerical Solutions for Fuzzy Time Fractional Diffusion Equations Using Two Explicit Compact Finite Difference Methods

“…In this section, we present the related theorems and definitions that are used further in this paper. Definition 1. r-level set [31].…”

“…Salah et al [30] introduced the homotopy analysis transform method (HATM) to address fuzzy fractional heat and wave partial differential equations. In a separate approach, Chakraverty and Tampaswini [31] proposed a novel computational technique to solve the time fractional diffusion equation with uncertainties in the initial conditions. Their method employed a single-parametric form of fuzzy numbers to transform the fuzzy diffusion equation into an interval-based fuzzy differential equation, which was then converted into a crisp form using a double-parametric form of fuzzy numbers.…”

Numerical solutions of fuzzy time fractional advection‐diffusion equations in double parametric form of fuzzy number

“…The single parametric form was used in the development of a finite difference schemes for solving the fuzzy time fractional diffusion equation by the present authors. 31 Chakraverty and Tampaswini 32 proposed a new computational technique to solve the time fractional diffusion equation with uncertainties in the initial conditions. The approach used single parametric form of fuzzy numbers to convert the fuzzy diffusion equation into an interval-based fuzzy differential equation and then transform the obtained equation into crisp form based on double parametric form of fuzzy numbers.…”