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# A novel semi-analytical method for solutions of two dimensional fuzzy fractional wave equation using natural transform

• * Corresponding author: Ali Ahmadian
• In this research article, the techniques for computing an analytical solution of 2D fuzzy wave equation with some affecting term of force has been provided. Such type of achievement for the aforesaid solution is obtained by applying the notions of a Caputo non-integer derivative in the vague or uncertainty form. At the first attempt the fuzzy natural transform is applied for obtaining the series solution. Secondly the homotopy perturbation (HPM) technique is used, for the analysis of the proposed result by comparing the co-efficient of homotopy parameter $q$ to get hierarchy of equation of different order for $q$. For this purpose, some new results about Natural transform of an arbitrary derivative under uncertainty are established, for the first time in the literature. The solution has been assumed in term of infinite series, which break the problem to a small number of equations, for the respective investigation. The required results are then determined in a series solution form which goes rapidly towards the analytical result. The solution has two parts or branches in fuzzy form, one is lower branch and the other is upper branch. To illustrate the ability of the considered approach, we have proved some test problems.

Mathematics Subject Classification: Primary: 26A33, 34A07; Secondary: 35R13.

 Citation: • • Figure 1.  Representation of three dimensional(3D) graph of fuzzy solution at four different fractional order of $\theta$ with $\texttt{y} = 0.5, \texttt{t} = 0.5,$ of example 1. The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 2.  Representation of three dimensional(3D) graph of fuzzy solution at four other different fractional order of $\theta$ with $\texttt{y} = 0.5, \texttt{t} = 0.5$ of example 1. The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 3.  Representation of two dimensional(2D) graph of fuzzy solution at four different fractional order of $\theta$ with $\texttt{x} = 0.5, \texttt{y} = 0.5, \texttt{t} = 0.5$ of example 1. The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 4.  Representation of two dimensional(2D) graph of fuzzy solution at four other different fractional order of $\theta$ with $\texttt{x} = 0.5, \texttt{y} = 0.5, \texttt{t} = 0.5$ of example 1. The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 5.  Representation of comparison of fuzzy solution for upper and lower branches of example 1 by LADM and NTHPM in three dimension(3D). The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 6.  Representation of comparison of fuzzy solution for upper and lower branches of example 1 by LADM and NTHPM in two dimension(2D). The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 7.  Representation of three dimensional(3D) graph of fuzzy solution at four different fractional order of $\theta$ with $\texttt{y} = 0.5, \texttt{t} = 0.5$ of example 2. The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 8.  Representation of three dimensional(3D) graph of fuzzy solution at four other different fractional order of $\theta$ with $\texttt{y} = 0.5, \texttt{t} = 0.5$ of example 2. The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 9.  Representation of two dimensional(2D) graph of fuzzy solution at four different fractional order of $\theta$ with $\texttt{x} = 0.5, \texttt{y} = 0.5, \texttt{t} = 0.5$ of example 2. The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 10.  Representation of two dimensional(2D) graph of fuzzy solution at four other different fractional order of $\theta$ with $\texttt{x} = 0.5, \texttt{y} = 0.5, \texttt{t} = 0.5$ of example 2. The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 11.  Representation of comparison of fuzzy solution for upper and lower branches of example 2 by LADM and NTHPM in three dimension(3D). The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 12.  Representation of comparison of fuzzy solution for upper and lower branches of example 2 by LADM and NTHPM in two dimension(2D). The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 13.  Representation of three dimensional(3D) graph of fuzzy solution at four different fractional order of $\theta$ with $\texttt{y} = 0.5, \texttt{t} = 0.5,$ of example 3. The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 14.  Representation of three dimensional(3D) graph of fuzzy solution at four other different fractional order of $\theta$ with $\texttt{y} = 0.5, \texttt{t} = 0.5$ of example 3. The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 15.  Representation of two dimensional(2D) graph of fuzzy solution at four different fractional order of $\theta$ with $\texttt{x} = 0.5, \texttt{y} = 0.5, \texttt{t} = 0.5$ of example 3. The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 16.  Representation of two dimensional(2D) graph of fuzzy solution at four other different fractional order of $\theta$ with $\texttt{x} = 0.5, \texttt{y} = 0.5, \texttt{t} = 0.5$ of example 3. The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 17.  Representation of comparison of fuzzy solution for upper and lower branches of example 3 by LADM and NTHPM in three dimension(3D). The two similar color legends represents upper and lower portion of fuzzy solution respectively

Figure 18.  Representation of comparison of fuzzy solution for upper and lower branches of example 3 by LADM and NTHPM in two dimension(2D). The two similar color legends represents upper and lower portion of fuzzy solution respectively

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