Fractional calculus can provide an accurate model of many dynamical systems, which leads to a set of partial differential equations (PDE). Fisher's equation is one of these PDEs. This article focuses on a new method that is used for the analytical solution of Fuzzy nonlinear time fractional generalized Fisher's equation (FNLTFGFE) with a source term. While the uncertainty is considered in the initial condition, the proposed technique supports the process of the solution commencing from the parametric form (double parametric form) of a fuzzy number. Next, a joint mechanism of natural transform (NT) coupled with Adomian decomposition method (ADM) is utilized, and the nonlinear term is calculated through ADM. The obtained solution of the unknown function is written in infinite series form. It has been observed that the solution obtained is rapid and accurate. The result proved that this method is more efficient and less time-consuming in comparison with all other methods. Three examples are presented to show the efficiency of the proposed techniques. The result shows that uncertainty plays an important role in analytical sense. i.e., as the uncertainty decreases, the solution approaches a classical solution. Hence, this method makes a very useful contribution towards the solution of the fuzzy nonlinear time fractional generalized Fisher's equation. Moreover, the matlab (2015) software has been used to draw the graphs.
Citation: Muhammad Usman, Hidayat Ullah Khan, Zareen A Khan, Hussam Alrabaiah. Study of nonlinear generalized Fisher equation under fractional fuzzy concept[J]. AIMS Mathematics, 2023, 8(7): 16479-16493. doi: 10.3934/math.2023842
Fractional calculus can provide an accurate model of many dynamical systems, which leads to a set of partial differential equations (PDE). Fisher's equation is one of these PDEs. This article focuses on a new method that is used for the analytical solution of Fuzzy nonlinear time fractional generalized Fisher's equation (FNLTFGFE) with a source term. While the uncertainty is considered in the initial condition, the proposed technique supports the process of the solution commencing from the parametric form (double parametric form) of a fuzzy number. Next, a joint mechanism of natural transform (NT) coupled with Adomian decomposition method (ADM) is utilized, and the nonlinear term is calculated through ADM. The obtained solution of the unknown function is written in infinite series form. It has been observed that the solution obtained is rapid and accurate. The result proved that this method is more efficient and less time-consuming in comparison with all other methods. Three examples are presented to show the efficiency of the proposed techniques. The result shows that uncertainty plays an important role in analytical sense. i.e., as the uncertainty decreases, the solution approaches a classical solution. Hence, this method makes a very useful contribution towards the solution of the fuzzy nonlinear time fractional generalized Fisher's equation. Moreover, the matlab (2015) software has been used to draw the graphs.
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Muhammad Usman, Hidayat Ullah Khan, Zareen A Khan, Hussam Alrabaiah. Study of nonlinear generalized Fisher equation under fractional fuzzy concept[J]. AIMS Mathematics, 2023, 8(7): 16479-16493. doi: 10.3934/math.2023842
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