International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(4): 409-417
Published online December 25, 2023
https://doi.org/10.5391/IJFIS.2023.23.4.409
© The Korean Institute of Intelligent Systems
Hamzeh Husin Zureigat
Department of Mathematics, Faculty of Science and Technology, Jadara University, Irbid, Jordan
Correspondence to :
Hamzeh Husin Zureigat (hamzeh.zu@jadara.edu.jo)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper discusses fractional partial differential equations that provide a more accurate representation of complex phenomena than classical partial differential equations. Specifically, the natural Daftardar-Jafari method is developed and applied for the first time to solving the fuzzy time-fractional wave equation, where the fractional order is 1 < α ≤ 2. Fuzziness is presented through convex normalized triangular fuzzy numbers under the initial and boundary conditions. The time-fractional derivative is defined using the Caputo formula, and the convergence of the proposed approach is also discussed. An example with numerical values is provided to demonstrate the practicality of the method, and it is shown that the outcomes obtained corresponded to those produced by other established, well-known methods used to solve identical issues.
Keywords: Fuzzy time-fractional wave equation, Natural Daftardar-Jafari Method, Fuzzy number, Caputo formula
In recent years, fractional partial differential equations (FPDEs) have attracted significant interest because of their applicability in diverse fields, including engineering, science, and medicine [1–5]. Owing to their importance, researchers have focused on developing efficient and accurate methods for their solutions. Despite considerable progress in this area, there is currently no consensus on universally accepted techniques for solving FPDEs. The time-fractional wave equation (TFWE) is considered one of the important FPDEs used to describe wave behavior in a medium by replacing the time derivative with a fractional derivative of order
The TFWE has been studied extensively [11–16]. Jafari and Daftardar-Gejji [11] used the Adomian decomposition method (ADM). Similarly, Jafari and Momani [12] used the homotopy perturbation method to solve linear and nonlinear fractional wave equations. In another study, Odibat and Momani [13] applied the ADM with Caputo sense to describe the fractional derivative when solving a TFWE with boundary conditions. They found that the ADM is effective and convenient for solving TFWEs. The fractional diffusion and wave equations were solved by Jafari et al. [14] using the Laplace decomposition method, which employs a combination of the Laplace transform and ADM. The natural decomposition method presented by Abdel-Rady et al. [15], which uses a combination of the natural transform and ADM, was utilized to solve fractional models. A new mathematical approach, called the natural Daftardar-Jafari method, was recently proposed by Jafari et al. [16] to solve TFWEs. This method combines a natural transform and an iterative technique that considers the Caputo sense of the fractional derivative. Their study revealed that the natural Daftardar-Jafari method yields outcomes identical to those of previously established methods.
The conventional analysis of wave phenomena assumes that the parameters and variables involved are clear and precise. However, in reality, they may be uncertain and imprecise owing to errors in measurement and experimentation. To address this issue, mathematicians have identified the need to use a fuzzy fractional wave equation instead of a fractional wave equation. Fuzzy time-fractional wave equations (FTFWE) can be used to model real-world data in various fields such as physics, engineering, finance, and biology. These equations are dependent on the physical system being modeled, and can be used to simulate seismic wave propagation, heat transfer in fractal geometry, stock price dynamics, biological diffusion, and quantum mechanics. The initial and boundary conditions of the equations depend on the specific problem being solved and physical quantities involved. FTFWEs provide a powerful tool for modeling complex phenomena and can help understand the behavior of real-world systems. FTFWEs have been studied by many authors [17–19]. Khan and Ghadle [17] developed and applied the variational iteration method (VIM) to solve FTFWE. The results obtained by the VIM are expressed in the form of an infinite series. Subsequently, the differential transform method (DTM) was proposed by Osman et al. [18] to solve FTFWEs. Coputo’s derivative was used to calculate fractional derivatives, and the findings demonstrate that the utilization of the DTM is an exceedingly efficient approach for acquiring analytical solutions of the fuzzy fractional wave equation. Recently, Arfan et al. [19] applied a fuzzy natural transform to solve one and two-dimensional FTFWEs. The Caputo non-integer derivative was used to handle the fractional derivative, and the problem was simplified to a smaller number of equations by assuming a solution in terms of an infinite series. Thus, the fuzzy natural transform was found to be an efficient and effective approach for solving FTFWE.
The motivation for using the fuzzy natural Daftardar-Jafari method (FNDJM) to solve FTFWEs is its ability to handle both fractional derivatives and fuzzy set theory simultaneously. This approach provides a powerful tool for researchers to investigate the behavior of a system more accurately and efficiently. After conducting a review of existing literature, we observed that only a few articles explored the utilization of approximate analytical methods to resolve FTFWE.
The aim of this study was to solve the FTFWE problem using transform iterative methods. Specifically, we discuss, develop, and apply the natural Daftardar-Jafari method for the first time to solve FTFWEs.
Consider the one-dimensional FTFWE with the initial and boundary conditions [20]
where
where
In this section, the FTFWE is represented using the r-level cut approach in a single parametric form of fuzzy numbers under Hukuhara derivatives. We can write
where
The membership function is defined using using the fuzzy extension principle [23] as
Now,
In this section, the natural Daftardar-Jafari method is developed and applied to obtain a numerical solution of the FTFWE.
First, the FTFWE in
with the fuzzy initial conditions
where
In the first step, we performed a natural transform on both sides of
As defined in [16] the natural transform of the fuzzy Caputo derivative
Simplifying
In the second step, we consider the inverse natural transform on both sides of
Then,
where
In the final step, an iterative method known as the fuzzy Daftardar-Jafari method [11] is applied, and the solution of
Substituting
The nonlinear term is decomposed as [11]
Substituting
The following iteration is then deduced:
where
In this section, the fuzzy convergence condition of the FNDJM is discussed based on the following two lemmas:
If N is
If
Consider the one-dimensional of FTFWE with the initial and boundary conditions [16]
subject to the fuzzy boundary conditions
where
Now, let
We consider the natural transform of both sides of
Now, we consider the inverse natural transform of
The, we implement the FNDJM to obtain the following terms:
By simplifying
We obtained the same solution as in
From Tables 1, 2, and Figure 1, it can be seen that the fuzzy solution of the FTFWE obtained using the FNDJM satisfies the properties of fuzzy numbers by attaining a triangular fuzzy number shape. Furthermore, as seen in Figure 2, the fuzzy solution of the FTFWE by FNDJM at different values of
In this paper, the FNDJM has been presented and applied for the first time to obtain the fuzzy solution of FTFWEs. The fractional derivative was considered in the Caputo sense, and Wolfram Mathematica was used to obtain the fuzzy results. The approach presented in this study yielded results that adhered to the characteristics of fuzzy numbers using a triangular fuzzy number shape. Moreover, we show that the proposed method achieves convergence. The results obtained correspond to those from other established and well-known methods used to solve identical issues. This approach can be used for both integer-order ordinary and partial differential equations, and will be investigated in detail at a later stage.
No potential conflict of interest relevant to this article was reported.
Table 1. Fuzzy solutions of Eq. (28) by the FNDJM with
Fuzzy lower solution by FNDJM | Fuzzy upper solution by FNDJM | |
---|---|---|
0.1 | 0.056787587174806295 | 0.09464597862467715 |
0.2 | 0.11556554008029116 | 0.19260923346715192 |
0.3 | 0.17728599167107556 | 0.2954766527851259 |
0.4 | 0.24274043049430702 | 0.40456738415717836 |
0.5 | 0.3126795754594954 | 0.5211326257658256 |
0.6 | 0.38785646753289166 | 0.6464274458881527 |
0.7 | 0.4690506460321175 | 0.7817510767201958 |
0.8 | 0.557085080619679 | 0.9284751343661315 |
0.9 | 0.6528399072789791 | 1.0880665121316317 |
1 | 0.7572647600414578 | 1.2621079334024297 |
Table 2. Fuzzy solutions of Eq. (28) by the FNDJM with
Fuzzy lower solution by | Fuzzy upper solution by | |
---|---|---|
0 | 0.3126795754594954 | 0.5211326257658256 |
0.2 | 0.33352488049012846 | 0.5002873207351926 |
0.4 | 0.3543701855207614 | 0.4794420157045595 |
0.6 | 0.37521549055139447 | 0.4585967106739266 |
0.8 | 0.39606079558202745 | 0.43775140564329357 |
1 | 0.41690610061266054 | 0.41690610061266054 |
E-mail: hamzeh.zu@jadara.edu.jo
International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(4): 409-417
Published online December 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.4.409
Copyright © The Korean Institute of Intelligent Systems.
Hamzeh Husin Zureigat
Department of Mathematics, Faculty of Science and Technology, Jadara University, Irbid, Jordan
Correspondence to:Hamzeh Husin Zureigat (hamzeh.zu@jadara.edu.jo)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper discusses fractional partial differential equations that provide a more accurate representation of complex phenomena than classical partial differential equations. Specifically, the natural Daftardar-Jafari method is developed and applied for the first time to solving the fuzzy time-fractional wave equation, where the fractional order is 1 < α ≤ 2. Fuzziness is presented through convex normalized triangular fuzzy numbers under the initial and boundary conditions. The time-fractional derivative is defined using the Caputo formula, and the convergence of the proposed approach is also discussed. An example with numerical values is provided to demonstrate the practicality of the method, and it is shown that the outcomes obtained corresponded to those produced by other established, well-known methods used to solve identical issues.
Keywords: Fuzzy time-fractional wave equation, Natural Daftardar-Jafari Method, Fuzzy number, Caputo formula
In recent years, fractional partial differential equations (FPDEs) have attracted significant interest because of their applicability in diverse fields, including engineering, science, and medicine [1–5]. Owing to their importance, researchers have focused on developing efficient and accurate methods for their solutions. Despite considerable progress in this area, there is currently no consensus on universally accepted techniques for solving FPDEs. The time-fractional wave equation (TFWE) is considered one of the important FPDEs used to describe wave behavior in a medium by replacing the time derivative with a fractional derivative of order
The TFWE has been studied extensively [11–16]. Jafari and Daftardar-Gejji [11] used the Adomian decomposition method (ADM). Similarly, Jafari and Momani [12] used the homotopy perturbation method to solve linear and nonlinear fractional wave equations. In another study, Odibat and Momani [13] applied the ADM with Caputo sense to describe the fractional derivative when solving a TFWE with boundary conditions. They found that the ADM is effective and convenient for solving TFWEs. The fractional diffusion and wave equations were solved by Jafari et al. [14] using the Laplace decomposition method, which employs a combination of the Laplace transform and ADM. The natural decomposition method presented by Abdel-Rady et al. [15], which uses a combination of the natural transform and ADM, was utilized to solve fractional models. A new mathematical approach, called the natural Daftardar-Jafari method, was recently proposed by Jafari et al. [16] to solve TFWEs. This method combines a natural transform and an iterative technique that considers the Caputo sense of the fractional derivative. Their study revealed that the natural Daftardar-Jafari method yields outcomes identical to those of previously established methods.
The conventional analysis of wave phenomena assumes that the parameters and variables involved are clear and precise. However, in reality, they may be uncertain and imprecise owing to errors in measurement and experimentation. To address this issue, mathematicians have identified the need to use a fuzzy fractional wave equation instead of a fractional wave equation. Fuzzy time-fractional wave equations (FTFWE) can be used to model real-world data in various fields such as physics, engineering, finance, and biology. These equations are dependent on the physical system being modeled, and can be used to simulate seismic wave propagation, heat transfer in fractal geometry, stock price dynamics, biological diffusion, and quantum mechanics. The initial and boundary conditions of the equations depend on the specific problem being solved and physical quantities involved. FTFWEs provide a powerful tool for modeling complex phenomena and can help understand the behavior of real-world systems. FTFWEs have been studied by many authors [17–19]. Khan and Ghadle [17] developed and applied the variational iteration method (VIM) to solve FTFWE. The results obtained by the VIM are expressed in the form of an infinite series. Subsequently, the differential transform method (DTM) was proposed by Osman et al. [18] to solve FTFWEs. Coputo’s derivative was used to calculate fractional derivatives, and the findings demonstrate that the utilization of the DTM is an exceedingly efficient approach for acquiring analytical solutions of the fuzzy fractional wave equation. Recently, Arfan et al. [19] applied a fuzzy natural transform to solve one and two-dimensional FTFWEs. The Caputo non-integer derivative was used to handle the fractional derivative, and the problem was simplified to a smaller number of equations by assuming a solution in terms of an infinite series. Thus, the fuzzy natural transform was found to be an efficient and effective approach for solving FTFWE.
The motivation for using the fuzzy natural Daftardar-Jafari method (FNDJM) to solve FTFWEs is its ability to handle both fractional derivatives and fuzzy set theory simultaneously. This approach provides a powerful tool for researchers to investigate the behavior of a system more accurately and efficiently. After conducting a review of existing literature, we observed that only a few articles explored the utilization of approximate analytical methods to resolve FTFWE.
The aim of this study was to solve the FTFWE problem using transform iterative methods. Specifically, we discuss, develop, and apply the natural Daftardar-Jafari method for the first time to solve FTFWEs.
Consider the one-dimensional FTFWE with the initial and boundary conditions [20]
where
where
In this section, the FTFWE is represented using the r-level cut approach in a single parametric form of fuzzy numbers under Hukuhara derivatives. We can write
where
The membership function is defined using using the fuzzy extension principle [23] as
Now,
In this section, the natural Daftardar-Jafari method is developed and applied to obtain a numerical solution of the FTFWE.
First, the FTFWE in
with the fuzzy initial conditions
where
In the first step, we performed a natural transform on both sides of
As defined in [16] the natural transform of the fuzzy Caputo derivative
Simplifying
In the second step, we consider the inverse natural transform on both sides of
Then,
where
In the final step, an iterative method known as the fuzzy Daftardar-Jafari method [11] is applied, and the solution of
Substituting
The nonlinear term is decomposed as [11]
Substituting
The following iteration is then deduced:
where
In this section, the fuzzy convergence condition of the FNDJM is discussed based on the following two lemmas:
If N is
If
Consider the one-dimensional of FTFWE with the initial and boundary conditions [16]
subject to the fuzzy boundary conditions
where
Now, let
We consider the natural transform of both sides of
Now, we consider the inverse natural transform of
The, we implement the FNDJM to obtain the following terms:
By simplifying
We obtained the same solution as in
From Tables 1, 2, and Figure 1, it can be seen that the fuzzy solution of the FTFWE obtained using the FNDJM satisfies the properties of fuzzy numbers by attaining a triangular fuzzy number shape. Furthermore, as seen in Figure 2, the fuzzy solution of the FTFWE by FNDJM at different values of
In this paper, the FNDJM has been presented and applied for the first time to obtain the fuzzy solution of FTFWEs. The fractional derivative was considered in the Caputo sense, and Wolfram Mathematica was used to obtain the fuzzy results. The approach presented in this study yielded results that adhered to the characteristics of fuzzy numbers using a triangular fuzzy number shape. Moreover, we show that the proposed method achieves convergence. The results obtained correspond to those from other established and well-known methods used to solve identical issues. This approach can be used for both integer-order ordinary and partial differential equations, and will be investigated in detail at a later stage.
Fuzzy solution of
Fuzzy solution of
Table 1 . Fuzzy solutions of Eq. (28) by the FNDJM with
Fuzzy lower solution by FNDJM | Fuzzy upper solution by FNDJM | |
---|---|---|
0.1 | 0.056787587174806295 | 0.09464597862467715 |
0.2 | 0.11556554008029116 | 0.19260923346715192 |
0.3 | 0.17728599167107556 | 0.2954766527851259 |
0.4 | 0.24274043049430702 | 0.40456738415717836 |
0.5 | 0.3126795754594954 | 0.5211326257658256 |
0.6 | 0.38785646753289166 | 0.6464274458881527 |
0.7 | 0.4690506460321175 | 0.7817510767201958 |
0.8 | 0.557085080619679 | 0.9284751343661315 |
0.9 | 0.6528399072789791 | 1.0880665121316317 |
1 | 0.7572647600414578 | 1.2621079334024297 |
Table 2 . Fuzzy solutions of Eq. (28) by the FNDJM with
Fuzzy lower solution by | Fuzzy upper solution by | |
---|---|---|
0 | 0.3126795754594954 | 0.5211326257658256 |
0.2 | 0.33352488049012846 | 0.5002873207351926 |
0.4 | 0.3543701855207614 | 0.4794420157045595 |
0.6 | 0.37521549055139447 | 0.4585967106739266 |
0.8 | 0.39606079558202745 | 0.43775140564329357 |
1 | 0.41690610061266054 | 0.41690610061266054 |
Diptiranjan Behera
International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(2): 173-180 https://doi.org/10.5391/IJFIS.2023.23.2.173Diptiranjan Behera and S. Chakraverty
International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(3): 252-260 https://doi.org/10.5391/IJFIS.2022.22.3.252Woo-Joo Lee, Hye-Young Jung, Jin Hee Yoon, and Seung Hoe Choi
Int. J. Fuzzy Log. Intell. Syst. 2017; 17(1): 43-50 https://doi.org/10.5391/IJFIS.2017.17.1.43Fuzzy solution of
Fuzzy solution of