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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

Modified Natural Daftardar-Jafari Method for Solving Fuzzy Time-Fractional Wave Equation

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)

Received: March 5, 2023; Revised: May 12, 2023; Accepted: October 16, 2023

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.

Abstract

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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

1. Introduction

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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 α, where α ranges from 0 < α ≤ 2. This substitution incorporates nonlocality and memory effects into the equation, resulting in intriguing phenomena, such as anomalous dispersion, non-exponential decay, and subdiffusive behavior. Applications of the fractional wave equation span multiple fields, including physics, biology, and finance, because it is particularly useful for modelling nonlocal and non-Markovian behavior [5–10]. Solutions of the FPDEs have been analyzed using various techniques, such as fractional calculus, Green’s functions, and numerical simulations.

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.

2. Time-Fractional Wave Equation in Fuzzy Environment

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Consider the one-dimensional FTFWE with the initial and boundary conditions [20]

∂αu˜(x,t,α)∂αt=C˜2∂2u˜(x,t)∂x2+q˜(x,t),   1<α≤2,         (x,t)∈Ω=[0, L]×[0, T],u˜(x,0)=f˜1(x),∂u˜∂t(x,0)=f˜2(x),u˜(0,t)=v˜, u˜(l,t)=y˜,

where ũ(x, t, α) is a fuzzy function of the crisp independent variables t and x and α is fractional order. The nonhomogeneous term q̃(x, t) is a fuzzy function of the crisp variables t and x. The ∂αu˜(x,t,α)∂αt is the fuzzy time-fractional derivative of order α [20]. ∂2U˜(x,t)∂x2 is a second order fuzzy partial Hukuhara derivative with respect to x. Furthermore, in Eq. (1) the fuzzy initial conditions are ũ(0, x), ∂u∂t(x,0) and f̃1(x) and f̃2(x) are fuzzy functions of x. The boundary conditions in the fuzzy form are ũ(0, t) and ũ(l, 0) and are equal to the fuzzy convex numbers ṽ and ỹ, respectively. Finally in Eq. (1) the fuzzy functions f̃(x) are defined as follows [21]:

{b˜(x,t)=ω˜1s1(x),f1˜(x,t)=ω˜2s2(x),f2˜(x)=ω˜3s3(x),

where s1(x), s2(x) and s3(x) are the crisp functions of the crisp variable x with ω̃1, ω̃2, ω̃3 and ω̃4 being the fuzzy convex numbers.

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 Eq. (1) in a single parametric form of the fuzzy number under the Hukuhara derivatives. Eq. (1) is rewritten for all r ∈ [0, 1] is as follows [22]:

[u˜(x,t)]r=u_(x,t;r),u¯(x,t;r),[∂αu˜(x,t,α)∂αt]r=∂αu_(x,t,α;r)∂αt,∂αu¯(x,t,α;r)∂αt,[∂2u˜(x,t)∂x2]r=∂2u_(x,t;r)∂x2,∂αu¯(x,t;r)∂x2,[c2˜]r=c2_,c2¯,[q˜(x,t)]r=q_ (x,t;r),q¯(x,t;r),[u˜(x,0)]r=u_ (x,0;r),u¯(x,0;r),[∂u˜∂t(x,0)]r=∂u_∂t(x,0;r), ∂u¯∂t(x,0;r),[u˜(0,t)]r=u_ (0,t;r),u¯(0,t;r),[u˜(l,t)]r=u_ (l,t;r),u¯(l,t;r),[f˜(x)]r=f_(x;r),f¯(x;r),{[v˜]r=v_ (r),v¯(r),[y˜]r=y_(r),y¯(r),

where

{[q˜(x,t)]r=[ω_ (r)1,ω¯1 (r)] s1(x),[f1˜(x)]r=[ω_ (r)2, ω¯2 (r)] s2(x),[f2˜(x)]r=[ω_ (r)3, ω¯3(r)] s3(x).

The membership function is defined using using the fuzzy extension principle [23] as

{u_ (x,t;r)=min {u˜ (μ˜(r),t))|μ˜(r)∈u˜(x,t;r)},u¯ (x,t;r)=max {u˜ (μ˜(r),t)|μ˜(r)∈u˜(x,t;r)}.

Now, Eqs. (4)–(14) for 0 < x ≤ l, t > 0 and r ∈ [0, 1] are rewritten to obtain the general FTFWE equation as follows:

{∂αu_(x,t,α)∂αt=c_2(x)∂2u_(x,t;r)∂x2+[ω_ (r)1] s1(x),u_ (x,0;r)=ω_ (r)2s2(x),∂u_∂t(x,0;r)=ω_ (r)3s3(x),u_ (0,t;r)=v_ (r),u_ (l,t;r)=y_ (r),{∂αu¯(x,t,α)∂αt=c¯2∂2u¯(x,t;r)∂x2+[ω¯1 (r)] s1(x),u¯ (x,0;r)=ω¯ (r)2s2(x),∂u¯∂t(x,0;r)=ω¯ (r)3s3(x),u¯ (0,t;r)=v¯ (r),u¯ (l,t;r)=y¯ (r).

Eqs. (15) and (16) present the lower and upper bounds, respectively, of the general form of the FTFWE.

3. The Fuzzy Natural Daftardar-Jafari Method for the Solution of FTFWE

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In this section, the natural Daftardar-Jafari method is developed and applied to obtain a numerical solution of the FTFWE.

First, the FTFWE in Eqs. (15) and (16) can be represented in the following general form:

{Dtαu_(x,t)+R(u_(x,t))+F(u_(x,t))=q_(x,t),Dtαu¯(x,t)+R(u¯(x,t))+F(u¯(x,t))=q¯(x,t),,

with the fuzzy initial conditions

{u_(i)(x,0)=∂iu_(x,0)∂ti,u¯(i)(x,0)=∂iu¯(x,0)∂ti,i=0,1,2,…,p-1,

where D˜tα=∂α∂tα is the fuzzy Caputo time-fractional derivative, R(ũ(x, t)) is the fuzzy linear partial differential operator, F(ũ(x, t)) represent the nonlinear partial terms, and q̃(x, t) is a fuzzy source term.

In the first step, we performed a natural transform on both sides of Eq. (17)

{N[Dtαu_(x,t)]+N[R(u_(x,t))]+N[F(u_(x,t))]=N[q_(x,t)],N[Dtαu¯(x,t)]+N[R(u¯(x,t))]+N[F(u¯(x,t))]=N[q¯(x,t)].

As defined in [16] the natural transform of the fuzzy Caputo derivative N[Dtαu˜(x,t)] is defined as follows:

N [Dtαu˜(x,t)]=(sm)αψ˜(s,m)-∑i=0p-11s(sm)α-iy(i)(0),α∈(p-1;p].

Simplifying Eq. (18) and applying the initial conditions, we obtain

{ψ_(x,s,u)   =(ms)α∑i=0p-11s(sm)α-iy(i)(0)+(ms)αN [q_(x,t)]         -(ms)α[N[R(u_(x,t))]+N[F(u_(x,t))]],ψ¯(x,s,u)   =(ms)α∑i=0p-11s(sm)α-iy(i)(0)+(ms)αN [q¯(x,t)]         -(ms)α[N[R(u¯(x,t))]+N[F(u¯(x,t))]].

In the second step, we consider the inverse natural transform on both sides of Eq. (20) to obtain

{u_(x,t)   =N-1 [(ms)α∑i=0p-11s(sm)α-iy(i)(0)+(ms)αN [q_(x,t)]]         -N-1[(ms)μ[N[R(u_(x,t))]+N[F(u_(x,t))]]],u¯(x,t)   =N-1 [(ms)α∑i=0p-11s(sm)α-iy(i)(0)+(ms)αN [q¯(x,t)]]         -N-1[(ms)μ[N[R(u¯(x,t))]+N[F(u¯(x,t))]]].

Then, Eq. (21) can be rewritten as follows:

{u_(x,t)   =Q_(x,t)-N-1[(ms)α[N[R(u_(x,t))]+N[F(u_(x,t))]]],u¯(x,t)   =Q¯(x,t)-N-1[(ms)α[N[R(u¯(x,t))]+N[F(u¯(x,t))]]],

where Q(x, t), Q̄(x, t) are the terms corresponding to the lower and upper initial conditions, respectively.

In the final step, an iterative method known as the fuzzy Daftardar-Jafari method [11] is applied, and the solution of Eq. (17) is written as an infinite series as follows:

{u_(x,t)=∑n=0∞un_(x,t),u¯(x,t)=∑n=0∞un¯(x,t).

Substituting Eq. (23) into Eq. (22), we obtain

{∑n=0∞un_(x,t)   =Q_(x,t)-N-1[(ms)α[ N [R (∑n=0∞un_)]         +N [F(∑n=0∞un_)]]],∑n=0∞un¯(x,t)   =Q¯(x,t)-N-1[(ms)α[ N [R (∑n=0∞un¯)]         +N [F(∑n=0∞un¯)]]].

The nonlinear term is decomposed as [11]

{F (∑n=0∞u_(x,t))   =F (u0_(x,t))+∑n=1∞ [F (∑k=0nuk_)-F (∑k=0n-1uk_)],F (∑n=0∞u¯(x,t))   =F (u0¯(x,t))+∑n=1∞ [F (∑k=0nuk¯)-F (∑k=0n-1uk¯)].

Substituting Eq. (25) into Eq. (24), we get

{∑n=0∞un_(x,t)=Q_(x,t)-N-1[(ms)αN [R∑n=0∞un_(x,t)]]         -N-1 [(ms)αN [F (u0_(x,t))         +∑n=1∞[F (∑k=0nuk_)-F (∑k=0n-1uk_)]]],∑n=0∞un¯(x,t)=Q¯(x,t)-N-1[(ms)αN [R∑n=0∞un¯(x,t)]]         -N-1 [(ms)αN [F (u0¯(x,t))         +∑n=1∞[F (∑k=0nuk¯)-F (∑k=0n-1uk¯)]]].

The following iteration is then deduced:

u0_(x,t)=Q_(x,t),u1_(x,t)=-N-1[(ms)αN [R (u0_(x,t))] -N-1[(ms)αN [F (u0_(x,t))]]],u2_(x,t)=-N-1[(ms)αN [R (u1_(x,t))] -N-1[(ms)αN [F (u1_+α0)-F (u0_)]]],un_(x,t)=-N-1[(ms)αN [R (un-1_)]] -N-1[(ms)αN [F (u0_+⋯+un-1_) -F (u0_+…+un-2_)]], n=3,4,…,u0¯(x,t)=Q¯(x,t),u1¯(x,t)=-N-1[(ms)αN [R (u0¯(x,t))] -N-1[(ms)αN [F (u0¯(x,t))]]],u2¯(x,t)=-N-1[(ms)αN [R (u1¯(x,t))] -N-1[(ms)αN [F (u1¯+α0)-F (u0¯)]]],un¯(x,t)=-N-1[(ms)αN [R (un-1¯)]] -N-1[(ms)αN [F (u0¯+⋯+un-1¯) -F (u0¯+…+un-2¯)]], n=3,4,…,

where Eqs. (26) and (27), respectively represent the fuzzy lower and upper approximation solutions of Eq. (17). The n + 1 term approximate solution of Eq. (17) is given by

{u_(x,t)=u_0+u_1+⋯+u_n,u¯(x,t)=u¯0+u¯1+⋯+u¯n.

4. Convergence of the Fuzzy Natural Daftardar-Jafari Method

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In this section, the fuzzy convergence condition of the FNDJM is discussed based on the following two lemmas:

Lemma 1 [15]

If N is C∞ in a neighborhood of ũ0 and ||Nn(ũ0)|| ≤ L for any n and for some real L > 0 and||ũi|| < M <e−1, i = 1, 2, . . . , then the series ∑n=0∞H˜n is absolutely convergent and

‖H˜n‖≤L Mnen-1(e-1), n=1,2,….

Lemma 2 [16]

If N is C∞ and ||Nn(u0˜)|| ≤M<e-1 ∀n, then the series ∑n=0∞H˜n is absolutely convergent.

5. Example

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Consider the one-dimensional of FTFWE with the initial and boundary conditions [16]

∂αu˜(x,t,α)∂αt=12x2∂2u˜(x,t)∂x2,1<α≤2, (x,t)∈Ω=[0, L]×[0, T],

subject to the fuzzy boundary conditions ũ(0, t) = 0, ũ(l, t) = 0, and fuzzy initial conditions

u˜(x,0)=k˜(r)x, ∂u˜∂t(x,0)=k˜(r)x2,

where k̃(r) = [0.75 + 0.25r, 1.25 − 0.25r] for all r ∈ [0, 1].

Now, let R˜(u(x,t))=12 x2∂2u˜(x,t)∂x2, F̃(u(x, t)) = 0 and q̃(x, t) = 0.

We consider the natural transform of both sides of Eq. (28) and use the initial conditions to yield

ψ_(x,s,u)=(0.75+0.25r) (xs+mx2s2)+(ms)α [N [12x2∂2u˜(x,t)∂x2]],ψ¯(x,s,u)=(1.25-0.25r) (xs+mx2s2)+(ms)α [N [12x2∂2u˜(x,t)∂x2]].

Now, we consider the inverse natural transform of Eq. (15) to get

u_(x,t)=(0.75+0.25r) (x+tx2)+N-1[(ms)α [N [12x2∂2u˜(x,t)∂x2]]],u¯(x,t)=(1.25-0.25r) (x+tx2)+N-1[(ms)α [N [12x2∂2u˜(x,t)∂x2]]].

The, we implement the FNDJM to obtain the following terms:

u0_(x,t)=(0.75+0.25r)(x+tx2),u1_(x,t)=N-1 [(ms)α [N [12x2∂2u0_(x,t)∂x2]]]=(r-1)x2tα+1Γ(α+2),u2_(x,t)=N-1 [(ms)α [N [12x2∂2u1_(x,t)∂x2]]]=(r-1)x2t2α+1Γ(2α+2),un_(x,t)=N-1 [(ms)α [N [12x2∂2un-1_(x,t)∂x2]]]=(r-1)x2tnα+1Γ (nα+2),u0¯(x,t)=(1.25-0.25r)(x+tx2),u1¯(x,t)=N-1 [(ms)α [N [12x2∂2u0_(x,t)∂x2]]]=(r-1)x2tα+1Γ(α+2),u2¯(x,t)=N-1 [(ms)α [N [12x2∂2u1_(x,t)∂x2]]]=(r-1)x2t2α+1Γ(2α+2),un¯(x,t)=N-1 [(ms)α [N [12x2∂2un-1_(x,t)∂x2]]]=(r-1)x2tnα+1Γ (nα+2).

By simplifying Eqs. (31) and (32), we obtain the general solution of the FTFWE

{u_(x,t)=[(0.75+0.25r)(x+x2)]∑n=0∞tnα+1Γ(nα+2),u¯(x,t)=[(1.25-0.25r)(x+x2)]∑n=0∞tnα+1Γ(nα+2).

We obtained the same solution as in Eq. (24), where the Laplace homotopy perturbation method was used to solve the problem.

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 α indicates that the FNDJM is feasible, accurate, and satisfies the fuzzy properties and theories. Furthermore, the obtained results agree with the theoretical results.

6. Conclusion

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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.

Figures

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Fig. 1.

Fuzzy solution of Eq. (28) by FNDJM at t = 0.5, x = 0.5 and α = 1.5 for all r ∈ [0, 1].


Fig. 2.

Fuzzy solution of Eq. (28) by FNDJM at t = 0.5, x = 0.5 and α = 1.2, 1.4, 1.5, 1.8 for all r ∈ [0, 1].


Tables

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Table. 1.

Table 1. Fuzzy solutions of Eq. (28) by the FNDJM with α = 1.5, x = 0.5, r = 0 for all t ∈ [0, 1].

tFuzzy lower solution by FNDJMFuzzy upper solution by FNDJM
0.10.0567875871748062950.09464597862467715
0.20.115565540080291160.19260923346715192
0.30.177285991671075560.2954766527851259
0.40.242740430494307020.40456738415717836
0.50.31267957545949540.5211326257658256
0.60.387856467532891660.6464274458881527
0.70.46905064603211750.7817510767201958
0.80.5570850806196790.9284751343661315
0.90.65283990727897911.0880665121316317
10.75726476004145781.2621079334024297

Table. 2.

Table 2. Fuzzy solutions of Eq. (28) by the FNDJM with x = 0.5, t = 0.5 and α = 1.5 for all r ∈ [0, 1].

rFuzzy lower solution by FNDJMFuzzy upper solution by FNDJM
00.31267957545949540.5211326257658256
0.20.333524880490128460.5002873207351926
0.40.35437018552076140.4794420157045595
0.60.375215490551394470.4585967106739266
0.80.396060795582027450.43775140564329357
10.416906100612660540.41690610061266054

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  10. Al-khateeb, A, Zureigat, H, Ala’yed, O, and Bawaneh, S (2021). Ulam–Hyers stability and uniqueness for nonlinear sequential fractional differential equations involving integral boundary conditions. Fractal and Fractional. 5, article no 235. https://doi.org/10.3390/fractalfract5040235
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  11. Jafari, H, and Daftardar-Gejji, V (2016). Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition. Applied Mathematics and Computation. 180, 488-497. https://doi.org/10.1016/j.amc.2005.12.031
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  12. Jafari, H, and Momani, S (2007). Solving fractional diffusion and wave equations by modified homotopy perturbation method. Physics Letters A. 370, 388-396. https://doi.org/10.1016/j.physleta.2007.05.118
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  13. Odibat, ZM, and Momani, S (2006). Approximate solutions for boundary value problems of time-fractional wave equation. Applied Mathematics and Computation. 181, 767-774. https://doi.org/10.1016/j.amc.2006.02.004
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  14. Jafari, H, Khalique, CM, and Nazari, M (2011). Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion–wave equations. Applied Mathematics Letters. 24, 1799-1805. https://doi.org/10.1016/j.aml.2011.04.037
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  15. Abdel-Rady, AS, Rida, SZ, Arafa, AAM, and Abedl-Rahim, HR (2015). Natural transform for solving fractional models. Journal of Applied Mathematics and Physics. 3. article no 62184. https://doi.org/10.4236/jamp.2015.312188
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  16. Jafari, H, Ncube, MN, Moshokoa, SP, and Makhubela, L (2020). Natural Daftardar-Jafari method for solving fractional partial differential equations. Nonlinear Dynamics and Systems Theory. 20, 299-306.
  17. Khan, F, and Ghadle, KP (2019). Solving fuzzy fractional wave equation by the variational iteration method in fluid mechanics. Journal of the Korean Society for Industrial and Applied Mathematics. 23, 381-394. https://doi.org/10.12941/jksiam.2019.23.381
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  20. Khan, F, and Ghadle, KP (2019). Solving fuzzy fractional wave equation by the variational iteration method in fluid mechanics. Journal of the Korean Society for Industrial and Applied Mathematics. 23, 381-394. https://doi.org/10.12941/jksiam.2019.23.381
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  21. Jameel, AF, Saaban, A, and Zureigat, HH (2018). Numerical solution of second-order fuzzy nonlinear two-point boundary value problems using combination of finite difference and Newton’s methods. Neural Computing and Applications. 30, 3167-3175. https://doi.org/10.1007/s00521-017-2893-z
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  23. Zureigat, HH, and Ismail, AIM . Numerical solution of fuzzy heat equation with two different fuzzifications., Proceedings of 2016 SAI Computing Conference (SAI), 2016, London, UK, Array, pp.85-90. https://doi.org/10.1109/SAI.2016.7555966
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Biography

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Hamzeh Husin Zureigat is an assistant professor in the Department of Mathematics at Jadara University in Jordan. He received his M.A. and Ph.D. from the University of Science, Malaysia (USM). His research interests focus on numerical analysis and computational optimization fields in fuzzy fractional partial differential equations.

E-mail: hamzeh.zu@jadara.edu.jo

Article

Original Article

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.

Modified Natural Daftardar-Jafari Method for Solving Fuzzy Time-Fractional Wave Equation

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)

Received: March 5, 2023; Revised: May 12, 2023; Accepted: October 16, 2023

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.

Abstract

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

1. Introduction

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 α, where α ranges from 0 < α ≤ 2. This substitution incorporates nonlocality and memory effects into the equation, resulting in intriguing phenomena, such as anomalous dispersion, non-exponential decay, and subdiffusive behavior. Applications of the fractional wave equation span multiple fields, including physics, biology, and finance, because it is particularly useful for modelling nonlocal and non-Markovian behavior [5–10]. Solutions of the FPDEs have been analyzed using various techniques, such as fractional calculus, Green’s functions, and numerical simulations.

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.

2. Time-Fractional Wave Equation in Fuzzy Environment

Consider the one-dimensional FTFWE with the initial and boundary conditions [20]

∂αu˜(x,t,α)∂αt=C˜2∂2u˜(x,t)∂x2+q˜(x,t),   1<α≤2,         (x,t)∈Ω=[0, L]×[0, T],u˜(x,0)=f˜1(x),∂u˜∂t(x,0)=f˜2(x),u˜(0,t)=v˜, u˜(l,t)=y˜,

where ũ(x, t, α) is a fuzzy function of the crisp independent variables t and x and α is fractional order. The nonhomogeneous term q̃(x, t) is a fuzzy function of the crisp variables t and x. The ∂αu˜(x,t,α)∂αt is the fuzzy time-fractional derivative of order α [20]. ∂2U˜(x,t)∂x2 is a second order fuzzy partial Hukuhara derivative with respect to x. Furthermore, in Eq. (1) the fuzzy initial conditions are ũ(0, x), ∂u∂t(x,0) and f̃1(x) and f̃2(x) are fuzzy functions of x. The boundary conditions in the fuzzy form are ũ(0, t) and ũ(l, 0) and are equal to the fuzzy convex numbers ṽ and ỹ, respectively. Finally in Eq. (1) the fuzzy functions f̃(x) are defined as follows [21]:

{b˜(x,t)=ω˜1s1(x),f1˜(x,t)=ω˜2s2(x),f2˜(x)=ω˜3s3(x),

where s1(x), s2(x) and s3(x) are the crisp functions of the crisp variable x with ω̃1, ω̃2, ω̃3 and ω̃4 being the fuzzy convex numbers.

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 Eq. (1) in a single parametric form of the fuzzy number under the Hukuhara derivatives. Eq. (1) is rewritten for all r ∈ [0, 1] is as follows [22]:

[u˜(x,t)]r=u_(x,t;r),u¯(x,t;r),[∂αu˜(x,t,α)∂αt]r=∂αu_(x,t,α;r)∂αt,∂αu¯(x,t,α;r)∂αt,[∂2u˜(x,t)∂x2]r=∂2u_(x,t;r)∂x2,∂αu¯(x,t;r)∂x2,[c2˜]r=c2_,c2¯,[q˜(x,t)]r=q_ (x,t;r),q¯(x,t;r),[u˜(x,0)]r=u_ (x,0;r),u¯(x,0;r),[∂u˜∂t(x,0)]r=∂u_∂t(x,0;r), ∂u¯∂t(x,0;r),[u˜(0,t)]r=u_ (0,t;r),u¯(0,t;r),[u˜(l,t)]r=u_ (l,t;r),u¯(l,t;r),[f˜(x)]r=f_(x;r),f¯(x;r),{[v˜]r=v_ (r),v¯(r),[y˜]r=y_(r),y¯(r),

where

{[q˜(x,t)]r=[ω_ (r)1,ω¯1 (r)] s1(x),[f1˜(x)]r=[ω_ (r)2, ω¯2 (r)] s2(x),[f2˜(x)]r=[ω_ (r)3, ω¯3(r)] s3(x).

The membership function is defined using using the fuzzy extension principle [23] as

{u_ (x,t;r)=min {u˜ (μ˜(r),t))|μ˜(r)∈u˜(x,t;r)},u¯ (x,t;r)=max {u˜ (μ˜(r),t)|μ˜(r)∈u˜(x,t;r)}.

Now, Eqs. (4)–(14) for 0 < x ≤ l, t > 0 and r ∈ [0, 1] are rewritten to obtain the general FTFWE equation as follows:

{∂αu_(x,t,α)∂αt=c_2(x)∂2u_(x,t;r)∂x2+[ω_ (r)1] s1(x),u_ (x,0;r)=ω_ (r)2s2(x),∂u_∂t(x,0;r)=ω_ (r)3s3(x),u_ (0,t;r)=v_ (r),u_ (l,t;r)=y_ (r),{∂αu¯(x,t,α)∂αt=c¯2∂2u¯(x,t;r)∂x2+[ω¯1 (r)] s1(x),u¯ (x,0;r)=ω¯ (r)2s2(x),∂u¯∂t(x,0;r)=ω¯ (r)3s3(x),u¯ (0,t;r)=v¯ (r),u¯ (l,t;r)=y¯ (r).

Eqs. (15) and (16) present the lower and upper bounds, respectively, of the general form of the FTFWE.

3. The Fuzzy Natural Daftardar-Jafari Method for the Solution of FTFWE

In this section, the natural Daftardar-Jafari method is developed and applied to obtain a numerical solution of the FTFWE.

First, the FTFWE in Eqs. (15) and (16) can be represented in the following general form:

{Dtαu_(x,t)+R(u_(x,t))+F(u_(x,t))=q_(x,t),Dtαu¯(x,t)+R(u¯(x,t))+F(u¯(x,t))=q¯(x,t),,

with the fuzzy initial conditions

{u_(i)(x,0)=∂iu_(x,0)∂ti,u¯(i)(x,0)=∂iu¯(x,0)∂ti,i=0,1,2,…,p-1,

where D˜tα=∂α∂tα is the fuzzy Caputo time-fractional derivative, R(ũ(x, t)) is the fuzzy linear partial differential operator, F(ũ(x, t)) represent the nonlinear partial terms, and q̃(x, t) is a fuzzy source term.

In the first step, we performed a natural transform on both sides of Eq. (17)

{N[Dtαu_(x,t)]+N[R(u_(x,t))]+N[F(u_(x,t))]=N[q_(x,t)],N[Dtαu¯(x,t)]+N[R(u¯(x,t))]+N[F(u¯(x,t))]=N[q¯(x,t)].

As defined in [16] the natural transform of the fuzzy Caputo derivative N[Dtαu˜(x,t)] is defined as follows:

N [Dtαu˜(x,t)]=(sm)αψ˜(s,m)-∑i=0p-11s(sm)α-iy(i)(0),α∈(p-1;p].

Simplifying Eq. (18) and applying the initial conditions, we obtain

{ψ_(x,s,u)   =(ms)α∑i=0p-11s(sm)α-iy(i)(0)+(ms)αN [q_(x,t)]         -(ms)α[N[R(u_(x,t))]+N[F(u_(x,t))]],ψ¯(x,s,u)   =(ms)α∑i=0p-11s(sm)α-iy(i)(0)+(ms)αN [q¯(x,t)]         -(ms)α[N[R(u¯(x,t))]+N[F(u¯(x,t))]].

In the second step, we consider the inverse natural transform on both sides of Eq. (20) to obtain

{u_(x,t)   =N-1 [(ms)α∑i=0p-11s(sm)α-iy(i)(0)+(ms)αN [q_(x,t)]]         -N-1[(ms)μ[N[R(u_(x,t))]+N[F(u_(x,t))]]],u¯(x,t)   =N-1 [(ms)α∑i=0p-11s(sm)α-iy(i)(0)+(ms)αN [q¯(x,t)]]         -N-1[(ms)μ[N[R(u¯(x,t))]+N[F(u¯(x,t))]]].

Then, Eq. (21) can be rewritten as follows:

{u_(x,t)   =Q_(x,t)-N-1[(ms)α[N[R(u_(x,t))]+N[F(u_(x,t))]]],u¯(x,t)   =Q¯(x,t)-N-1[(ms)α[N[R(u¯(x,t))]+N[F(u¯(x,t))]]],

where Q(x, t), Q̄(x, t) are the terms corresponding to the lower and upper initial conditions, respectively.

In the final step, an iterative method known as the fuzzy Daftardar-Jafari method [11] is applied, and the solution of Eq. (17) is written as an infinite series as follows:

{u_(x,t)=∑n=0∞un_(x,t),u¯(x,t)=∑n=0∞un¯(x,t).

Substituting Eq. (23) into Eq. (22), we obtain

{∑n=0∞un_(x,t)   =Q_(x,t)-N-1[(ms)α[ N [R (∑n=0∞un_)]         +N [F(∑n=0∞un_)]]],∑n=0∞un¯(x,t)   =Q¯(x,t)-N-1[(ms)α[ N [R (∑n=0∞un¯)]         +N [F(∑n=0∞un¯)]]].

The nonlinear term is decomposed as [11]

{F (∑n=0∞u_(x,t))   =F (u0_(x,t))+∑n=1∞ [F (∑k=0nuk_)-F (∑k=0n-1uk_)],F (∑n=0∞u¯(x,t))   =F (u0¯(x,t))+∑n=1∞ [F (∑k=0nuk¯)-F (∑k=0n-1uk¯)].

Substituting Eq. (25) into Eq. (24), we get

{∑n=0∞un_(x,t)=Q_(x,t)-N-1[(ms)αN [R∑n=0∞un_(x,t)]]         -N-1 [(ms)αN [F (u0_(x,t))         +∑n=1∞[F (∑k=0nuk_)-F (∑k=0n-1uk_)]]],∑n=0∞un¯(x,t)=Q¯(x,t)-N-1[(ms)αN [R∑n=0∞un¯(x,t)]]         -N-1 [(ms)αN [F (u0¯(x,t))         +∑n=1∞[F (∑k=0nuk¯)-F (∑k=0n-1uk¯)]]].

The following iteration is then deduced:

u0_(x,t)=Q_(x,t),u1_(x,t)=-N-1[(ms)αN [R (u0_(x,t))] -N-1[(ms)αN [F (u0_(x,t))]]],u2_(x,t)=-N-1[(ms)αN [R (u1_(x,t))] -N-1[(ms)αN [F (u1_+α0)-F (u0_)]]],un_(x,t)=-N-1[(ms)αN [R (un-1_)]] -N-1[(ms)αN [F (u0_+⋯+un-1_) -F (u0_+…+un-2_)]], n=3,4,…,u0¯(x,t)=Q¯(x,t),u1¯(x,t)=-N-1[(ms)αN [R (u0¯(x,t))] -N-1[(ms)αN [F (u0¯(x,t))]]],u2¯(x,t)=-N-1[(ms)αN [R (u1¯(x,t))] -N-1[(ms)αN [F (u1¯+α0)-F (u0¯)]]],un¯(x,t)=-N-1[(ms)αN [R (un-1¯)]] -N-1[(ms)αN [F (u0¯+⋯+un-1¯) -F (u0¯+…+un-2¯)]], n=3,4,…,

where Eqs. (26) and (27), respectively represent the fuzzy lower and upper approximation solutions of Eq. (17). The n + 1 term approximate solution of Eq. (17) is given by

{u_(x,t)=u_0+u_1+⋯+u_n,u¯(x,t)=u¯0+u¯1+⋯+u¯n.

4. Convergence of the Fuzzy Natural Daftardar-Jafari Method

In this section, the fuzzy convergence condition of the FNDJM is discussed based on the following two lemmas:

Lemma 1 [15]

If N is C∞ in a neighborhood of ũ0 and ||Nn(ũ0)|| ≤ L for any n and for some real L > 0 and||ũi|| < M <e−1, i = 1, 2, . . . , then the series ∑n=0∞H˜n is absolutely convergent and

‖H˜n‖≤L Mnen-1(e-1), n=1,2,….

Lemma 2 [16]

If N is C∞ and ||Nn(u0˜)|| ≤M<e-1 ∀n, then the series ∑n=0∞H˜n is absolutely convergent.

5. Example

Consider the one-dimensional of FTFWE with the initial and boundary conditions [16]

∂αu˜(x,t,α)∂αt=12x2∂2u˜(x,t)∂x2,1<α≤2, (x,t)∈Ω=[0, L]×[0, T],

subject to the fuzzy boundary conditions ũ(0, t) = 0, ũ(l, t) = 0, and fuzzy initial conditions

u˜(x,0)=k˜(r)x, ∂u˜∂t(x,0)=k˜(r)x2,

where k̃(r) = [0.75 + 0.25r, 1.25 − 0.25r] for all r ∈ [0, 1].

Now, let R˜(u(x,t))=12 x2∂2u˜(x,t)∂x2, F̃(u(x, t)) = 0 and q̃(x, t) = 0.

We consider the natural transform of both sides of Eq. (28) and use the initial conditions to yield

ψ_(x,s,u)=(0.75+0.25r) (xs+mx2s2)+(ms)α [N [12x2∂2u˜(x,t)∂x2]],ψ¯(x,s,u)=(1.25-0.25r) (xs+mx2s2)+(ms)α [N [12x2∂2u˜(x,t)∂x2]].

Now, we consider the inverse natural transform of Eq. (15) to get

u_(x,t)=(0.75+0.25r) (x+tx2)+N-1[(ms)α [N [12x2∂2u˜(x,t)∂x2]]],u¯(x,t)=(1.25-0.25r) (x+tx2)+N-1[(ms)α [N [12x2∂2u˜(x,t)∂x2]]].

The, we implement the FNDJM to obtain the following terms:

u0_(x,t)=(0.75+0.25r)(x+tx2),u1_(x,t)=N-1 [(ms)α [N [12x2∂2u0_(x,t)∂x2]]]=(r-1)x2tα+1Γ(α+2),u2_(x,t)=N-1 [(ms)α [N [12x2∂2u1_(x,t)∂x2]]]=(r-1)x2t2α+1Γ(2α+2),un_(x,t)=N-1 [(ms)α [N [12x2∂2un-1_(x,t)∂x2]]]=(r-1)x2tnα+1Γ (nα+2),u0¯(x,t)=(1.25-0.25r)(x+tx2),u1¯(x,t)=N-1 [(ms)α [N [12x2∂2u0_(x,t)∂x2]]]=(r-1)x2tα+1Γ(α+2),u2¯(x,t)=N-1 [(ms)α [N [12x2∂2u1_(x,t)∂x2]]]=(r-1)x2t2α+1Γ(2α+2),un¯(x,t)=N-1 [(ms)α [N [12x2∂2un-1_(x,t)∂x2]]]=(r-1)x2tnα+1Γ (nα+2).

By simplifying Eqs. (31) and (32), we obtain the general solution of the FTFWE

{u_(x,t)=[(0.75+0.25r)(x+x2)]∑n=0∞tnα+1Γ(nα+2),u¯(x,t)=[(1.25-0.25r)(x+x2)]∑n=0∞tnα+1Γ(nα+2).

We obtained the same solution as in Eq. (24), where the Laplace homotopy perturbation method was used to solve the problem.

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 α indicates that the FNDJM is feasible, accurate, and satisfies the fuzzy properties and theories. Furthermore, the obtained results agree with the theoretical results.

6. Conclusion

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.

Fig 1.

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Figure 1.

Fuzzy solution of Eq. (28) by FNDJM at t = 0.5, x = 0.5 and α = 1.5 for all r ∈ [0, 1].

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 409-417https://doi.org/10.5391/IJFIS.2023.23.4.409

Fig 2.

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Figure 2.

Fuzzy solution of Eq. (28) by FNDJM at t = 0.5, x = 0.5 and α = 1.2, 1.4, 1.5, 1.8 for all r ∈ [0, 1].

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 409-417https://doi.org/10.5391/IJFIS.2023.23.4.409

Table 1 . Fuzzy solutions of Eq. (28) by the FNDJM with α = 1.5, x = 0.5, r = 0 for all t ∈ [0, 1].

tFuzzy lower solution by FNDJMFuzzy upper solution by FNDJM
0.10.0567875871748062950.09464597862467715
0.20.115565540080291160.19260923346715192
0.30.177285991671075560.2954766527851259
0.40.242740430494307020.40456738415717836
0.50.31267957545949540.5211326257658256
0.60.387856467532891660.6464274458881527
0.70.46905064603211750.7817510767201958
0.80.5570850806196790.9284751343661315
0.90.65283990727897911.0880665121316317
10.75726476004145781.2621079334024297

Table 2 . Fuzzy solutions of Eq. (28) by the FNDJM with x = 0.5, t = 0.5 and α = 1.5 for all r ∈ [0, 1].

rFuzzy lower solution by FNDJMFuzzy upper solution by FNDJM
00.31267957545949540.5211326257658256
0.20.333524880490128460.5002873207351926
0.40.35437018552076140.4794420157045595
0.60.375215490551394470.4585967106739266
0.80.396060795582027450.43775140564329357
10.416906100612660540.41690610061266054

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