Approximation Solution for Fuzzy Fractional-Order Partial Differential Equations

Abstract

In this article, the authors study the comparison of the generalization differential transform method (DTM) and fuzzy variational iteration method (VIM) applied to determining the approximate analytic solutions of fuzzy fractional KdV, K(2,2) and mKdV equations. Furthermore, we establish the approximation solution two-and three-dimensional fuzzy time-fractional telegraphic equations via the fuzzy reduced differential transform method (RDTM). Finding an exact or closed-approximation solution to a differential equation is possible via fuzzy RDTM. Finally, we present the fuzzy fractional variational homotopy perturbation iteration method (VHPIM) with a modified Riemann-Liouville derivative to solve the fuzzy fractional diffusion equation (FDE). Using this approach achieves a rapidly convergent sequence that approaches the exact solution of the equation. The proposed methods are investigated based on fuzzy fractional derivatives with some illustrative examples. The results reveal that the schemes are highly effective for obtaining the solutions to fuzzy fractional partial differential equations.

Numerical investigation of two-dimensional fuzzy fractional heat problem with an external source variable

“…Fuzzy calculus involves the study of fuzzy sets and fuzzy numbers, which enable the representation and manipulation of uncertain and imprecise numbers. Thus, numerous scholars have studied FDEs [10][11][12].…”

“…This method was discovered to be a direct and effective approach for getting analytical-numerical results. Osman et al [ 22 ] introduced some analytical schemes for obtaining the solution of fuzzy fractional PDEs.…”

Local Fuzzy Fractional Partial Differential Equations in the Realm of Fractal Calculus with Local Fractional Derivatives

“…where L β = ∂ 2β ∂θ 2β specifies the linear local fuzzy fractional differential operator of order 2β, U β is called the linear local fuzzy fractional derivative operator of order less than in L β , N β is represent the nonlinear local fractional operator, and h(ϑ, θ) is a non-differentiable source term. Using the LFFST of Equation ( 40), we obtain (41) Applying the property of the LFFST, we obtain…”

“…Recently, scientists and engineers have applied the ADM in order to solve linear, nonlinear differential, and integral problems ( [38][39][40][41][42]). Moreover, some researchers have considered the ADM with the Sumudu transform.…”

Analytical solution of fuzzy heat problem in two-dimensional case under Caputo-type fractional derivative

“…Arfan et al [18] developed an algorithm based on the HPS to analyze the analytical results for two dimensional fuzzy fractional heat problem consisting of external source term. [19] introduced two different schemes to find out the approximate and analytical results of fuzzy fractional problems. Hamoud and Ghadle [20] considered the homotopy analysis strategy and obtained the solution of the first order fuzzy Volterra-Fredholm integro-differential equations.…”