Fourth-Order Numerical Solutions for a Fuzzy Time-Fractional Convection–Diffusion Equation under Caputo Generalized Hukuhara Derivative

Abstract

The fuzzy fractional differential equation explains more complex real-world phenomena than the fractional differential equation does. Therefore, numerous techniques have been timely derived to solve various fractional time-dependent models. In this paper, we develop two compact finite difference schemes and employ the resulting schemes to obtain a certain solution for the fuzzy time-fractional convection–diffusion equation. Then, by making use of the Caputo fractional derivative, we provide new fuzzy analysis relying on the concept of fuzzy numbers. Further, we approximate the time-fractional derivative by using a fuzzy Caputo generalized Hukuhara derivative under the double-parametric form of fuzzy numbers. Furthermore, we introduce new computational techniques, based on fuzzy double-parametric form, to shift the given problem from one fuzzy domain to another crisp domain. Moreover, we discuss some stability and error analysis for the proposed techniques by using the Fourier method. Over and above, we derive several numerical experiments to illustrate reliability and feasibility of our proposed approach. It was found that the fuzzy fourth-order compact implicit scheme produces slightly better results than the fourth-order compact FTCS scheme. Furthermore, the proposed methods were found to be feasible, appropriate, and accurate, as demonstrated by a comparison of analytical and numerical solutions at various fuzzy values.

A Solution of the Complex Fuzzy Heat Equation in Terms of Complex Dirichlet Conditions Using a Modified Crank–Nicolson Method

“…Now substitute Equations ( 17)- (20) into Equations ( 15) and ( 16), respectively, to obtain the following:…”

“…The fuzzy partial differential equation is commonly utilized to explain the behavior of dynamic phenomena in which imprecision or indeterminacy is present. This includes fuzzy heat conduction and fuzzy particle diffusion, with the fuzzy heat equation being one of the most important fuzzy parabolic partial differential equations for describing how a fuzzy quantity such as heat diffuses through a given area [9][10][11][12][13][14][15][16][17][18][19][20][21]. While exact analytical solutions for fuzzy heat equations may be challenging to obtain, numerical techniques are needed to achieve the solution.…”

Numerical Solution for Fuzzy Time-Fractional Cancer Tumor Model with a Time-Dependent Net Killing Rate of Cancer Cells

“…This led to a fuzzy fractional diffusion equation. As discussed by many researchers, the cancer tumor model can be represented by a fractional diffusion equation [ 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 ]. However, in reality, the crisp quantities of the cancer tumor model are deemed uncertain.…”

Numerical Solution of Fuzzy Heat Equation with Complex Dirichlet Conditions

“…A fuzzy partial differential equation has been used to describe the behavior of many time-dependent phenomena, including fuzzy heat conduction and fuzzy particle diffusion, in which uncertainty or vagueness exists. The fuzzy heat equation is considered one of the most significant fuzzy parabolic partial differential equations used to describe how a fuzzy quantity, such as heat, diffuses through a given region [7][8][9][10][11][12][13][14][15][16][17][18][19]. In general, the exact analytical solution for the fuzzy heat equations is difficult to obtain.…”