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International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 407-415

Published online December 25, 2024

https://doi.org/10.5391/IJFIS.2024.24.4.407

© The Korean Institute of Intelligent Systems

A Solution of Fuzzy Time Fractional Wave Equation via Two Modified Implicit Finite Difference Schemes

Maryam Almutairi and Norazrizal Aswad bin Abdul Rahman

School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia

Correspondence to :
Abdul Rahman (aswad.rahman@usm.my)

Received: October 20, 2023; Revised: August 15, 2024; Accepted: September 30, 2024

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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Fuzzy fractional partial differential equations have become a powerful approach for handling uncertainty or imprecision in real-world modeling problems. In this study, two finite difference schemes, the Crank-Nicolson and centered-time centered-space methods, were developed and used to obtain a numerical solution for double-parametric fuzzy time fractional wave equations. Fuzzy set theory principles were employed to perform fuzzy analysis and formulate the proposed numerical schemes. The Caputo formula was used to define the time-fractional derivative. To illustrate the practicality of the numerical method, a specific numerical instance was analyzed. The results are presented in tables and figures, revealing the efficacy of the schemes in terms of accuracy and ability to reduce computational expenses. A novel fuzzy computational approach known as the double-parametric form enabled these achievements.

Keywords: Fuzzy Caputo formula, Fuzzy time fractional wave equation, Finite difference method, Double parametric form

1. Introduction

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Partial differential equations of fractional order are frequently employed in various fields, including physics, engineering, finance, and medical sciences because of their ability to offer a more accurate and detailed representation of models that cannot be captured by traditional integer-order differential equations [1–4]. Recently, the fractional wave equation has been the focus of numerous studies in areas such as acoustics, electromagnetism, and seismic analysis [5–7]. This equation can also describe the movement of objects such as strings, wires, and fluid surfaces [8, 9]. The behavior of any wave or motion can be represented as a combination of sinusoidal waves. In 1747, d’Alembert introduced the notion of a traveling-wave solution to the wave equation. Analytical methods for solving fractional wave equations do not generate precise solutions, leading researchers to rely on numerical or approximation methods. The Adomian decomposition method (ADM) was applied by Jafari and Daftardar-Gejji (2006) [10] to obtain approximate solutions to both nonlinear and linear fractional wave equations. Jafari and Momani [11] solved linear and nonlinear fractional wave equations by applying a homotopy perturbation method. Odibat and Momani (2006) [12] implemented the Adomian decomposition technique to handle the time-fractional wave equation (TFWE) under boundary conditions. Using the Caputo formula to describe the fractional derivative, they found that the ADM is effective and convenient for solving TFWE.

A commonly used numerical approach is the finite-difference scheme, which has been extensively discussed by various researchers [13–17]. This method is significant in solving the fractional wave equation because it allows discretizing the equation and solving it numerically while accurately handling the nonlocal and memory effects of the system. Ghode et al. [16] developed an explicit finite-difference method (FDM) to solve TFWE.

Liu et al. [17] introduced a methodology for addressing the initial boundary value problem (IBVP) of a variable-order TFWE by combining the central difference in space using the quadratic Charles Hermite and Newton (H2N2) estimation in time. An energy analysis method was used to evaluate the convergence of the proposed method. A numerical example is presented to demonstrate the effectiveness of the numerical results. The conventional approach to modeling processes uses fractional wave equations and assumes that the variables and parameters are precisely defined. However, these parameters can often be imprecise and uncertain because of errors in experiments and measurements, leading to the use of fuzzy fractional wave equations. In the past few years, the focus on studying fuzzy fractional wave equations has increased, with various contributions documented in prior research studies [18–25].

The analytical solution of fuzzy fractional wave equations is often impractical because of the complexity of the equations. Therefore, the interest in obtaining approximate solutions via numerical methods is increasing. The understanding and analysis of these problems can be enhanced using solutions obtained by numerical methods, among which, the FDM is one of the most frequently used for its simplicity and universal applicability. To the best of our knowledge, based on literature reviews, no research studies have solved the fuzzy time fractional wave equation (FTFWE) using the FDM. The aim of this study was to obtain a numerical solution for FTFWE, whereby two different finite element methods were developed and applied to solve the FTFWE in double parametric form.

The structure of this paper is as follows: Section 2 provides an overview of the time fractional wave equation in a fuzzy environment, and alternative representations of fuzzy numbers are discussed for two parameters. The centered-time centered-space (CTCS) method is reformulated and applied in Section 3 to solve FTFWE. In Section 4, the Crank-Nicholson (C-N) method is developed and implemented to solve FTFWE. Section 5 presents an example to validate the proficiency of both proposed schemes. Finally, a brief conclusion is provided.

2. Wave Equation with Time Fractional Derivative in Fuzzy Form

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In this section, the overall structure of the FTFWE is presented based on the Hukuhara derivative using a fuzzy technique called the double parametric form.

Considering the FTFWE representation, incorporating given boundary and initial conditions [19], we have

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

In accordance with the singular parametric form of the Hukuhara derivatives, Eq. (1) is expressed as

[∂αu_(x,t,α;r)∂αt,∂αu¯(x,t,α;r)∂αt]=[k_(x,t;r),k¯(x,t;r)] [∂2u_(x,t;r)∂x2, ∂2u¯(x,t;r)∂x2]+[b_(x,t;r), b¯(x,t,;r)].

Including the effects of uncertain boundaries and initial conditions:

[u_(x,0;r), u¯(x,0;r)]=[∅1(x,0;r), ∅1(x,0;r)],[∂u_∂t(x,0;r),∂u¯∂t(x,0;r)]=[∅2(x,0;r), ∅2(x,0;r)],[u_(0,t;r), u¯(0,t;r)]=[v_(0,t;r), v¯(0,t;r)],[u_(l,t;r), u¯(l,t;r)]=[y_(l,t;r), y¯(l,t;r)].

Using the double-parametric form in [19], Eq. (2) can be rewritten as follows:

[β (∂αu¯(x,t,α;r)∂αt-∂αu_(x,t,α;r)∂αt)+∂αu_(x,t,α;r)∂αt]=[β [k¯(x,t;r)-k_(x,t,r)]+k_(x,t,r)×[β (∂2u¯(x,t;r)∂x2-∂2u_(x,t;r)∂x2)+∂αu_(x,t;r)∂x2]+[β (b¯(x,t;r)-b_(x,t;r))+b_(x,t;r)].

Including imprecise boundary and initial conditions, we obtain

(β(u¯(x,0;r)-u_(x,0;r)+u_(x,0;r))=(β(∅¯1(x;r)-∅¯1(x;r))+∅¯1(x;r)),(β (∂u¯∂t(x,0;r)-∂u_∂t(x,0;r))+∂u_∂t(x,0;r))=(β(∅¯2(x;r)-∅_2(x;r))+∅_2(x;r)),(β(u¯(0,t;r)-u_(0,t;r))+u_(0,t;r))=(β(v¯(x;r)-v_(x;r))+v_(x;r)),(β(u¯(l,t;r)-u_(l,t;r))+u_(l,t;r))=(β(y¯(x;r)-y_(x;r))+y_(x;r)),

where β ∈ [0, 1].

The fuzzy functions are converted to crisp functions as follows:

∂αu˜(x,t;r,β)∂αt=β (∂αu¯(x,t,α;r)∂αt-∂αu_(x,t,α;r)∂αt)+∂αu_(x,t,α;r)∂αt,∂2u˜(x,t,r,β)∂x2=β (∂2u¯(x,t;r)∂x2-∂2u_(x,t;r)∂x2)+∂2u_(x,t;r)∂x2,k˜(x,t;r,β)=(k¯(x,t,r)-k_(x,t,r))+k_(x,t,r),b˜(x,t;r,β)=β(b¯(x,t;r)-b_(x,t;r))+b_(x,t;r),u˜(x,0,r,β)=β(u¯(x,0;r)-u_(x,0;r))+u_(x,0;r),∂u˜∂t(x,0;r,β)=β (∂u¯∂t(x,0;r)-∂u_∂t(x,0;r))+∂u_∂t(x,0;r),∅˜1(x,r,β)=β (∅¯1(x;r)-∅_1(x;r))+∅_1(x;r),∅˜2(x,r,β)=β (∅¯2(x;r)-∅_2(x;r))+∅_2(x;r),u˜(0,t,r,β)=β (u_(0,t;r)-u¯(0,t;r))+u_(0,t;r),v˜(x,r,β)=β (v¯(x;r)-v_(x;r))+v_(x;r),u˜(l,t,r,β)=β (u_(l,t;r)-u¯(l,t;r))+u_(l,t;r),y˜(x,r,β)=β (y¯(x;r)-y_(x;r))+y_(x;r).

By substituting these values into Eq. (3), we have

∂αu˜(x,t,α;r,β)∂αt=k˜(x,t;r,β)∂2u˜(x,t;r,β)∂x2+b˜(x,t;r,β),0≤r≤1, 0≤β≤1,u˜(x,0;r,β)=∅˜1(x,r,β),∂u˜∂t(x,0;r,β)=∅˜2(x,r,β),u˜(0,t,β)=v˜(x,r,β), u˜(l,t,β)=y˜(x,r,β).

This single parametric form allows us to determine the upper and lower bounds of the solutions by assuming β = 1 and β = 0, respectively, which is expressed as follows:

u˜(x,t;r,1)=u¯(x,t;r) and u˜(x,t;r,0)=u_(x,t;r).

3. Centered-Time Centered-Space in Double Parametric Form

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In this section, the CTCS method is reformulated and implemented in double-parametric form by replacing the fractional time derivative in the governing equation with the Caputo derivative and substituting the second-order space derivative with a central difference approximation to solve the FTFWE. Based on the definition of the Caputo derivative [26], the discretization of the fractional time derivative in Eq. (1) can be implemented as follows:

∂αu˜(x,t,α;r,β)∂αt= Δt-αΓ(3-α)[u˜in+1(x,t,α;r,β) -2u˜in(x,t,α;r,β)+u˜in-1(x,t,α;r,β) +∑j=1nbj(u˜in-j+1(x,t,α;r,β) -2u˜in-j(x,t,α;r,β) +u˜in-j-1(x,t,α;r,β))]+O(k),

where bj = (j + 1)2–α – (j)2–α, j = 1, 2, 3, ...

Furthermore, the central difference in space is used to discretize the second-order space derivative.

∂2u˜(x,t,α;r,β)∂x2=u˜i+1n(x,t;r,β)-2u˜in(x,t;r,β)+u˜i-1n(x,t;r,β)h2.

Now, substituting Eqs. (5) and (6) into Eq. (4) we have

Δt-αΓ(3-α)[u˜in+1(x,t,α;r,β)-2u˜in(x,t,α;r,β)         +u˜in-1(x,t,α;r,β)+∑j=1nbj(u˜in-j+1(x,t,α;r,β)         -2u˜in-j(x,t,α;r,β)+u˜in-j-1(x,t,α;r,β))]=k˜(x,t;r,β)u˜i+1n(x,t;r,β)-2u˜in(x,t;r,β)+u˜i-1n(x,t;r,β)h2         +b˜(x,t;r,β).

Assuming s˜(x,t;r)=k¯(x,t;r,β)ΔtαΓ(3-α)h2, from Eq. (7), we obtain for all r, β ∈ [0, 1],

u˜in-j(x,t;r,β)=s˜(u˜(i+1)n (x,t;r,β)+u˜(i-1)n(x,t;r,β))         +(2-2s˜)u˜in(x,t;r,β)-u˜in-1(x,t;r,β)         -∑j=1nbj(u˜in-j+1(x,t,α;r,β)-2u˜in-j(x,t,α;r,β)         +u˜in-j-1(x,t,α;r,β))+kαΓ(3-α)b˜(x,t;r,β).

4. Crank-Nicholson in Double Parametric Fuzzy Form

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The C-N method was reformulated, analyzed, and implemented in double-parametric fuzzy form by replacing the fractional time derivative in the governing equation with the Caputo derivative and replacing the second-order space derivative with a central difference approximation at the time level (n+12) to solve the FTFWE.

Based on the definition of the Caputo derivative derived in [26], the fractional time derivative in Eq. (1) can be discretized as follows:

∂αu˜(x,t,α;r,β)∂αt= Δt-αΓ(3-α)[u˜in+1(x,t,α;r,β) -2u˜in(x,t,α;r,β)+u˜in-1(x,t,α;r,β) +∑j=1nbj(u˜in-j+1(x,t,α;r,β) -2u˜in-j(x,t,α;r,β) +u˜in-j-1(x,t,α;r,β))]+O(k),

where bj = (j + 1)2–α – (j)2–α, j = 1, 2, 3, ...

In addition, the definition of the central difference at time level (n+12) in space is employed to discretize the second-order space derivative

∂2u˜(x,t,α;r,β)∂x2=u˜i+1n+12(x,t;r,β)-2u˜in+12(x,t;r,β)+u˜i-1n+12(x,t;r,β)h2.

Now, substituting Eqs. (9) and (10) into Eq. (4), we obtain

Δt-αΓ(3-α)[u˜in+1-2u˜in+u˜in-1+∑j=1nbj(u˜in-j+1-2u˜in-j+u˜in-j-1)]=k˜(x,t;r,β)×u˜i+1n+12(x,t;r,β)-2u˜in+12(x,t;r,β)+u˜i-1n-12(x,t;r,β)h2+b˜(x,t;r,β).

Simplifying Eq. (11) results in the following:

Δt-αΓ(3-α)[u˜in+1-2u˜in+u˜in-1+∑j=1nbj(u˜in-j+1-2u˜in-j+u˜in-j-1)]=k˜(x,t;r,β)2×[u˜i+1n+1(x,t;r,β)-2u˜in+1(x,t;r,β)+u˜i+1n-1(x,t;r,β)h2+u˜i+1n(x,t;r,β)-2u˜in(x,t;r,β)+u˜i-1n(x,t;r,β)h2]+b˜(x,t;r,β).

Assuming s˜(x,t;r)=k˜ΔtαΓ(3-α)2h2 for all r, β ∈ [0, 1] and from Eq. (12), we have

(1+2s)u˜in+1(x,t;r,β)         -s(u˜i+1n+1(x,t;r,β)+u˜i-1n+1(x,t;r,β))=s(u˜i+1n(x,t;r,β)+u˜i-1n(x,t;r,β))         +(2-2s)u˜in(x,t;r,β)-u˜in-1(x,t;r,β)         -∑j=1nbj(u˜i+1n-j+1-2u˜in-j+u˜i-1n-j-1)         +kαΓ(3-α)b˜(x,t;r,β).

5. Numerical Example

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Consider FTFWE [27] expressed as

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

Based on the given boundary conditions ũ(0, t) = ũ(L, t) = 0 and initial condition

u˜(x,0)=μ˜[sin(5πx)+2sin(7πx)].

the fuzzy number remains identical when expressed in single parametric form as follows:

μ˜(r)=[μ_(r),μ¯(r)]=[r-1,1-r]   for all r∈[0,1].

The analytical solution to Eq. (15) is provided in [23]:

{u_(x,t,α;r)=μ_[sin(5πx)cos(5πt)+2sin(7πx)cos(7πt)],u¯(x,t,α;r)=μ¯[sin(5πx)cos(5πt)+2sin(7πx)cos(7πt)].

The error for the fuzzy solution of Eq. (15) is defined as follows:

[E˜]r=∣U˜(t,x;r)-u˜(t,x;r)∣={[E_]r=∣U_(t,x;r)-u_(t,x;r)∣,[E¯]r=∣U¯(t,x;r)-u¯(t,x;r)∣.

At Δx = h = 0.2 and Δtα = (0.001)1.5, the fuzzy number, when expressed in double parametric form, remains unchanged and can be represented as μ̃(r) = [β(2 – 2r) + (r – 1)].

Tables 1–2 and Figures 1–3 demonstrate a strong correlation between the proposed methods and the exact solution, indicating a high level of agreement for all r, β ∈ [0, 1] at t = 0.005, and α = 1.5. Furthermore, they fulfill the characteristics of the double-parametric representation of fuzzy numbers, resulting in the adoption of a triangular shape. Additionally, the CTCS scheme yields marginally more precise outcomes than the C-N method. In Fig. 4, the proposed methods yield more precise outcomes at locations near the inflection point (β = 0.5) in terms of numerical results. The utilization of the double parametric form was found to be a comprehensive and efficient method for transitioning the governing equation from an uncertain to a precise state. This approach not only decreases the computational burden but also yields more precise outcomes in comparison to the single parametric form.

6. Conclusion

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In this study, two FDM schemes were reformulated and implemented to obtain a numerical solution for FTFWE in double parametric form. The Caputo definition was used for the Hukuhara time fractional derivative. The CTCS and C-N schemes yielded outcomes that conformed to the characteristics of fuzzy numbers by adopting the form of triangular fuzzy numbers. The CTCS scheme was found to produce more precise solutions than the C-N method, as demonstrated by a comparative analysis of the exact and numerical solutions. These schemes can be extended to solve nonlinear fuzzy fractional diffusion equations and other types of fuzzy fractional partial differential equations, which will be investigated in detail at a later stage.

Figures

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Fig. 1.

Analytical lower solution of Eq. (15) at t = 0.005, x = 1.8, and r = 0.


Fig. 2.

Analytical upper solution of Eq. (15) at t = 0.005, x = 1.8, and r = 0.


Fig. 3.

Analytical and FDM solutions to Eq. (15) at α = 1.5, x = 1.8, t = 0.005 for all r ∈ [0, 1].


Fig. 4.

Fuzzy analytical and fuzzy numerical solutions of Eq. (15) by (a) CTCS and (b) C-N at t = 0.005 and x = 1.8 for all r, β ∈ [0, 1].


Tables

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Table. 1.

Table 1. Numerical results of Eq. (15) by CTCS and C-N at t = 0.005 and x = 0.9 for all r, β ∈ [0, 1].

βrCTCSC-N
ũ(1.8, 0.05; r, β)Ẽ(1.8, 0.05; r, β)ũ(1.8, 0.05; r, β)Ẽ (1.8, 0.05; r, β)
β = 0 Lower solution0–1.8920941.46803 × 10−3–1.8770981.35277 × 10−2
0.1–1.7028851.32123 × 10−3–1.6893891.21749 × 10−2
0.3–1.3244661.02762 × 10−3–1.3139689.46938 × 10−3
0.5–0.9460477.34015 × 10−4–0.9385496.76384 × 10−3
0.7–0.5676284.40409 × 10−4–0.5631294.0583 × 10−3
0.9–0.18920941.46803 × 10−4–0.1877091.35277 × 10−3
10000
β = 1 Upper solution01.8920941.46803 × 10−31.8770981.35277 × 10−2
0.11.7028851.32123 × 10−31.6893891.21749 × 10−2
0.31.3244661.02762 × 10−31.3139689.46938 × 10−3
0.50.9460477.34015 × 10−40.9385496.76384 × 10−3
0.70.5676284.40409 × 10−40.5631294.0583 × 10−3
0.90.18920941.46803 × 10−40.1877091.35277 × 10−3
10000

Table. 2.

Table 2. Numerical solutions to Eq. (15) by CTCS and C-N at t = 0.005 and x = 1.8 for all r, β ∈ [0, 1].

βrCTCSC-N
ũ(1.8, 0.05; r, β)Ẽ(1.8, 0.05; r, β)ũ(1.8, 0.05; r, β)Ẽ (1.8, 0.05; r, β)
β = 0.40–0.37841892.93606 × 10−4–0.3754192.70554 × 10−3
0.1–0.3405772.64245 × 10−4–0.3378782.43498 × 10−3
0.3–0.2648932.05524 × 10−4–0.2627941.89388 × 10−3
0.5–0.1892091.46803 × 10−4–0.1877101.35277 × 10−3
0.7–0.1135268.80818 × 10−5–0.1111408.11661 × 10−4
0.9–0.0378422.93606 × 10−5–0.0375422.70554 × 10−4
10000
β = 0.600.37841892.93606 × 10−40.3754192.70554 × 10−3
0.10.3405772.64245 × 10−40.3378782.43498 × 10−3
0.30.2648932.05524 × 10−40.2627941.89388 × 10−3
0.50.1892091.46803 × 10−40.1877101.35277 × 10−3
0.70.1135268.80818 × 10−50.1111408.11661 × 10−4
0.90.0378422.93606 × 10−50.0375422.70554 × 10−4
10000

References

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  1. Samko, SG, Kilbas, AA, and Marichev, O (1993). Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach Science Publishers
  2. Podlubny, I (1999). Fractional Differential Equations. San Diego, CA: Academic Press
  3. Oldham, K, and Spanier, J (1974). The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. New York, NY: Academic Press
  4. Miller, KS, and Ross, B (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. New York, NY: John Willey & Sons
  5. Carcione, JM (2007). Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media. Oxford, UK: Elsevier
  6. Chen, W, Sun, H, and Li, X (2022). Fractional Derivative Modeling in Mechanics and Engineering. Singapore: Springer https://doi.org/10.1007/978-981-16-8802-7
    CrossRef
  7. 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
    CrossRef
  8. Sullivan, J, Iadecola, T, and Williamson, DJ (). Planar p-string condensation: chiral fracton phases from fractional quantum Hall layers and beyond. Physical Review B. 103, 2021. article no 205301
  9. Jian, CM, and Qi, XL (2014). Layer construction of 3D topological states and string braiding statistics. Physical Review X. 4. article no 041043
    CrossRef
  10. Zureigat, H, Ismail, AI, and Sathasivam, S (2021). Numerical solutions of fuzzy time fractional advection-diffusion equations in double parametric form of fuzzy number. Mathematical Methods in the Applied Sciences. 44, 7956-7968. https://doi.org/10.1002/mma.5573
    CrossRef
  11. 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
    CrossRef
  12. Zureigat, H, Ismail, AI, and Sathasivam, S (2020). A compact Crank–Nicholson scheme for the numerical solution of fuzzy time fractional diffusion equations. Neural Computing and Applications. 32, 6405-6412. https://doi.org/10.1007/s00521-019-04148-2
    CrossRef
  13. Sweilam, NH, Khader, MM, and Adel, M (2012). On the stability analysis of weighted average finite difference methods for fractional wave equation. Fractional Differential Calculus. 2, 17-29. https://doi.org/10.7153/fdc-02-02
    CrossRef
  14. Lyu, P, Liang, Y, and Wang, Z (2020). A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation. Applied Numerical Mathematics. 151, 448-471. https://doi.org/10.1016/j.apnum.2019.11.012
    CrossRef
  15. Wang, YM (2015). A compact finite difference method for a class of time fractional convection-diffusion-wave equations with variable coefficients. Numerical Algorithms. 70, 625-651. https://doi.org/10.1007/s11075-015-9965-x
    CrossRef
  16. Ghode, K, Takale, K, and Gaikwad, S. (2021) . New technique for solving time fractional wave equation: Python. Journal of Mathematical and Computational Science. 11, 5327-5343. https://www.scik.org/index.php/jmcs/article/view/6048
  17. Liu, X, Bo, Y, and Jin, Y (2022). A numerical method for the variable-order time-fractional wave equations based on the H2N2 approximation. Journal of Function Spaces, 2022. article no 3438289
  18. Arfan, M, Shah, K, Ullah, A, Salahshour, S, Ahmadian, A, and Ferrara, M (2022). A novel semi-analytical method for solutions of two dimensional fuzzy fractional wave equation using natural transform. Discrete and Continuous Dynamical Systems – Series S. 15, 315-338. https://doi.org/10.3934/dcdss.2021011
    CrossRef
  19. Behera, D, and Chakraverty, S (2015). New approach to solve fully fuzzy system of linear equations using single and double parametric form of fuzzy numbers. Sadhana. 40, 35-49. https://doi.org/10.1007/s12046-014-0295-9
    CrossRef
  20. Osman, M, Gong, ZT, and Mohammed, A (2021). Differential transform method for solving fuzzy fractional wave equation. Journal of Computational Analysis and Application. 29, 431-453.
  21. Ghaffari, M, Allahviranloo, T, Abbasbandy, S, and Azhini, M (2022). Generalized Hukuhara conformable fractional derivative and its application to fuzzy fractional partial differential equations. Soft Computing. 26, 2135-2146. https://doi.org/10.1007/s00500-021-06637-w
    CrossRef
  22. Qayyum, M, Ahmad, E, Akgul, A, and El Din, SM (). Fuzzy-fractional modeling of Korteweg-de Vries equations in Gaussian-Caputo sense: new solutions via extended He-Mahgoub algorithm. Ain Shams Engineering Journal. 15, 2024. article no 102623
  23. Rashid, S, Ashraf, R, and Hammouch, Z (2023). New generalized fuzzy transform computations for solving fractional partial differential equations arising in oceanography. Journal of Ocean Engineering and Science. 8, 55-78. https://doi.org/10.1016/j.joes.2021.11.004
    CrossRef
  24. Sartanpara, PP, and Meher, R (2023). A robust fuzzy-fractional approach for the atmospheric internal wave model. Journal of Ocean Engineering and Science. 8, 308-322. https://doi.org/10.1016/j.joes.2022.02.001
    CrossRef
  25. Zureigat, HH (2023). Modified natural Daftardar-Jafari Method for solving fuzzy time-fractional wave equation. International Journal of Fuzzy Logic and Intelligent Systems. 23, 409-417. https://doi.org/10.5391/IJFIS.2023.23.4.409
    CrossRef
  26. Li, L, and Liu, JG (2018). A generalized definition of Caputo derivatives and its application to fractional ODEs. SIAM Journal on Mathematical Analysis. 50, 2867-2900. https://doi.org/10.1137/17M1160318
    CrossRef
  27. Salah, A, Khan, M, and Gondal, MA (2013). A novel solution procedure for fuzzy fractional heat equations by homotopy analysis transform method. Neural Computing and Applications. 23, 269-271. https://doi.org/10.1007/s00521-012-0855-z
    CrossRef

Biographies

Go to

Maryam Almutairi is a Ph.D. student at the University of Science, Malaysia (USM). She received her M.A. from the University of Science, Malaysia (USM). Her research interests include numerical solutions and computational optimization fields in fuzzy fractional partial differential equations.

E-mail: maryam.almutairi@student.usm.my

Norazrizal Aswad bin Abdul Rahman is a senior lecturer in the School of Mathematical Sciences at the University of Science Malaysia (USM). His research interests include fuzzy set theory and fuzzy logic, statistical analysis (SPSS, Minitab), graphics technology, operational research, and mathematics education.

E-mail: aswad.rahman@usm.my

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 407-415

Published online December 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.4.407

Copyright © The Korean Institute of Intelligent Systems.

A Solution of Fuzzy Time Fractional Wave Equation via Two Modified Implicit Finite Difference Schemes

Maryam Almutairi and Norazrizal Aswad bin Abdul Rahman

School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia

Correspondence to:Abdul Rahman (aswad.rahman@usm.my)

Received: October 20, 2023; Revised: August 15, 2024; Accepted: September 30, 2024

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

Fuzzy fractional partial differential equations have become a powerful approach for handling uncertainty or imprecision in real-world modeling problems. In this study, two finite difference schemes, the Crank-Nicolson and centered-time centered-space methods, were developed and used to obtain a numerical solution for double-parametric fuzzy time fractional wave equations. Fuzzy set theory principles were employed to perform fuzzy analysis and formulate the proposed numerical schemes. The Caputo formula was used to define the time-fractional derivative. To illustrate the practicality of the numerical method, a specific numerical instance was analyzed. The results are presented in tables and figures, revealing the efficacy of the schemes in terms of accuracy and ability to reduce computational expenses. A novel fuzzy computational approach known as the double-parametric form enabled these achievements.

Keywords: Fuzzy Caputo formula, Fuzzy time fractional wave equation, Finite difference method, Double parametric form

1. Introduction

Partial differential equations of fractional order are frequently employed in various fields, including physics, engineering, finance, and medical sciences because of their ability to offer a more accurate and detailed representation of models that cannot be captured by traditional integer-order differential equations [1–4]. Recently, the fractional wave equation has been the focus of numerous studies in areas such as acoustics, electromagnetism, and seismic analysis [5–7]. This equation can also describe the movement of objects such as strings, wires, and fluid surfaces [8, 9]. The behavior of any wave or motion can be represented as a combination of sinusoidal waves. In 1747, d’Alembert introduced the notion of a traveling-wave solution to the wave equation. Analytical methods for solving fractional wave equations do not generate precise solutions, leading researchers to rely on numerical or approximation methods. The Adomian decomposition method (ADM) was applied by Jafari and Daftardar-Gejji (2006) [10] to obtain approximate solutions to both nonlinear and linear fractional wave equations. Jafari and Momani [11] solved linear and nonlinear fractional wave equations by applying a homotopy perturbation method. Odibat and Momani (2006) [12] implemented the Adomian decomposition technique to handle the time-fractional wave equation (TFWE) under boundary conditions. Using the Caputo formula to describe the fractional derivative, they found that the ADM is effective and convenient for solving TFWE.

A commonly used numerical approach is the finite-difference scheme, which has been extensively discussed by various researchers [13–17]. This method is significant in solving the fractional wave equation because it allows discretizing the equation and solving it numerically while accurately handling the nonlocal and memory effects of the system. Ghode et al. [16] developed an explicit finite-difference method (FDM) to solve TFWE.

Liu et al. [17] introduced a methodology for addressing the initial boundary value problem (IBVP) of a variable-order TFWE by combining the central difference in space using the quadratic Charles Hermite and Newton (H2N2) estimation in time. An energy analysis method was used to evaluate the convergence of the proposed method. A numerical example is presented to demonstrate the effectiveness of the numerical results. The conventional approach to modeling processes uses fractional wave equations and assumes that the variables and parameters are precisely defined. However, these parameters can often be imprecise and uncertain because of errors in experiments and measurements, leading to the use of fuzzy fractional wave equations. In the past few years, the focus on studying fuzzy fractional wave equations has increased, with various contributions documented in prior research studies [18–25].

The analytical solution of fuzzy fractional wave equations is often impractical because of the complexity of the equations. Therefore, the interest in obtaining approximate solutions via numerical methods is increasing. The understanding and analysis of these problems can be enhanced using solutions obtained by numerical methods, among which, the FDM is one of the most frequently used for its simplicity and universal applicability. To the best of our knowledge, based on literature reviews, no research studies have solved the fuzzy time fractional wave equation (FTFWE) using the FDM. The aim of this study was to obtain a numerical solution for FTFWE, whereby two different finite element methods were developed and applied to solve the FTFWE in double parametric form.

The structure of this paper is as follows: Section 2 provides an overview of the time fractional wave equation in a fuzzy environment, and alternative representations of fuzzy numbers are discussed for two parameters. The centered-time centered-space (CTCS) method is reformulated and applied in Section 3 to solve FTFWE. In Section 4, the Crank-Nicholson (C-N) method is developed and implemented to solve FTFWE. Section 5 presents an example to validate the proficiency of both proposed schemes. Finally, a brief conclusion is provided.

2. Wave Equation with Time Fractional Derivative in Fuzzy Form

In this section, the overall structure of the FTFWE is presented based on the Hukuhara derivative using a fuzzy technique called the double parametric form.

Considering the FTFWE representation, incorporating given boundary and initial conditions [19], we have

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

In accordance with the singular parametric form of the Hukuhara derivatives, Eq. (1) is expressed as

[∂αu_(x,t,α;r)∂αt,∂αu¯(x,t,α;r)∂αt]=[k_(x,t;r),k¯(x,t;r)] [∂2u_(x,t;r)∂x2, ∂2u¯(x,t;r)∂x2]+[b_(x,t;r), b¯(x,t,;r)].

Including the effects of uncertain boundaries and initial conditions:

[u_(x,0;r), u¯(x,0;r)]=[∅1(x,0;r), ∅1(x,0;r)],[∂u_∂t(x,0;r),∂u¯∂t(x,0;r)]=[∅2(x,0;r), ∅2(x,0;r)],[u_(0,t;r), u¯(0,t;r)]=[v_(0,t;r), v¯(0,t;r)],[u_(l,t;r), u¯(l,t;r)]=[y_(l,t;r), y¯(l,t;r)].

Using the double-parametric form in [19], Eq. (2) can be rewritten as follows:

[β (∂αu¯(x,t,α;r)∂αt-∂αu_(x,t,α;r)∂αt)+∂αu_(x,t,α;r)∂αt]=[β [k¯(x,t;r)-k_(x,t,r)]+k_(x,t,r)×[β (∂2u¯(x,t;r)∂x2-∂2u_(x,t;r)∂x2)+∂αu_(x,t;r)∂x2]+[β (b¯(x,t;r)-b_(x,t;r))+b_(x,t;r)].

Including imprecise boundary and initial conditions, we obtain

(β(u¯(x,0;r)-u_(x,0;r)+u_(x,0;r))=(β(∅¯1(x;r)-∅¯1(x;r))+∅¯1(x;r)),(β (∂u¯∂t(x,0;r)-∂u_∂t(x,0;r))+∂u_∂t(x,0;r))=(β(∅¯2(x;r)-∅_2(x;r))+∅_2(x;r)),(β(u¯(0,t;r)-u_(0,t;r))+u_(0,t;r))=(β(v¯(x;r)-v_(x;r))+v_(x;r)),(β(u¯(l,t;r)-u_(l,t;r))+u_(l,t;r))=(β(y¯(x;r)-y_(x;r))+y_(x;r)),

where β ∈ [0, 1].

The fuzzy functions are converted to crisp functions as follows:

∂αu˜(x,t;r,β)∂αt=β (∂αu¯(x,t,α;r)∂αt-∂αu_(x,t,α;r)∂αt)+∂αu_(x,t,α;r)∂αt,∂2u˜(x,t,r,β)∂x2=β (∂2u¯(x,t;r)∂x2-∂2u_(x,t;r)∂x2)+∂2u_(x,t;r)∂x2,k˜(x,t;r,β)=(k¯(x,t,r)-k_(x,t,r))+k_(x,t,r),b˜(x,t;r,β)=β(b¯(x,t;r)-b_(x,t;r))+b_(x,t;r),u˜(x,0,r,β)=β(u¯(x,0;r)-u_(x,0;r))+u_(x,0;r),∂u˜∂t(x,0;r,β)=β (∂u¯∂t(x,0;r)-∂u_∂t(x,0;r))+∂u_∂t(x,0;r),∅˜1(x,r,β)=β (∅¯1(x;r)-∅_1(x;r))+∅_1(x;r),∅˜2(x,r,β)=β (∅¯2(x;r)-∅_2(x;r))+∅_2(x;r),u˜(0,t,r,β)=β (u_(0,t;r)-u¯(0,t;r))+u_(0,t;r),v˜(x,r,β)=β (v¯(x;r)-v_(x;r))+v_(x;r),u˜(l,t,r,β)=β (u_(l,t;r)-u¯(l,t;r))+u_(l,t;r),y˜(x,r,β)=β (y¯(x;r)-y_(x;r))+y_(x;r).

By substituting these values into Eq. (3), we have

∂αu˜(x,t,α;r,β)∂αt=k˜(x,t;r,β)∂2u˜(x,t;r,β)∂x2+b˜(x,t;r,β),0≤r≤1, 0≤β≤1,u˜(x,0;r,β)=∅˜1(x,r,β),∂u˜∂t(x,0;r,β)=∅˜2(x,r,β),u˜(0,t,β)=v˜(x,r,β), u˜(l,t,β)=y˜(x,r,β).

This single parametric form allows us to determine the upper and lower bounds of the solutions by assuming β = 1 and β = 0, respectively, which is expressed as follows:

u˜(x,t;r,1)=u¯(x,t;r) and u˜(x,t;r,0)=u_(x,t;r).

3. Centered-Time Centered-Space in Double Parametric Form

In this section, the CTCS method is reformulated and implemented in double-parametric form by replacing the fractional time derivative in the governing equation with the Caputo derivative and substituting the second-order space derivative with a central difference approximation to solve the FTFWE. Based on the definition of the Caputo derivative [26], the discretization of the fractional time derivative in Eq. (1) can be implemented as follows:

∂αu˜(x,t,α;r,β)∂αt= Δt-αΓ(3-α)[u˜in+1(x,t,α;r,β) -2u˜in(x,t,α;r,β)+u˜in-1(x,t,α;r,β) +∑j=1nbj(u˜in-j+1(x,t,α;r,β) -2u˜in-j(x,t,α;r,β) +u˜in-j-1(x,t,α;r,β))]+O(k),

where bj = (j + 1)2–α – (j)2–α, j = 1, 2, 3, ...

Furthermore, the central difference in space is used to discretize the second-order space derivative.

∂2u˜(x,t,α;r,β)∂x2=u˜i+1n(x,t;r,β)-2u˜in(x,t;r,β)+u˜i-1n(x,t;r,β)h2.

Now, substituting Eqs. (5) and (6) into Eq. (4) we have

Δt-αΓ(3-α)[u˜in+1(x,t,α;r,β)-2u˜in(x,t,α;r,β)         +u˜in-1(x,t,α;r,β)+∑j=1nbj(u˜in-j+1(x,t,α;r,β)         -2u˜in-j(x,t,α;r,β)+u˜in-j-1(x,t,α;r,β))]=k˜(x,t;r,β)u˜i+1n(x,t;r,β)-2u˜in(x,t;r,β)+u˜i-1n(x,t;r,β)h2         +b˜(x,t;r,β).

Assuming s˜(x,t;r)=k¯(x,t;r,β)ΔtαΓ(3-α)h2, from Eq. (7), we obtain for all r, β ∈ [0, 1],

u˜in-j(x,t;r,β)=s˜(u˜(i+1)n (x,t;r,β)+u˜(i-1)n(x,t;r,β))         +(2-2s˜)u˜in(x,t;r,β)-u˜in-1(x,t;r,β)         -∑j=1nbj(u˜in-j+1(x,t,α;r,β)-2u˜in-j(x,t,α;r,β)         +u˜in-j-1(x,t,α;r,β))+kαΓ(3-α)b˜(x,t;r,β).

4. Crank-Nicholson in Double Parametric Fuzzy Form

The C-N method was reformulated, analyzed, and implemented in double-parametric fuzzy form by replacing the fractional time derivative in the governing equation with the Caputo derivative and replacing the second-order space derivative with a central difference approximation at the time level (n+12) to solve the FTFWE.

Based on the definition of the Caputo derivative derived in [26], the fractional time derivative in Eq. (1) can be discretized as follows:

∂αu˜(x,t,α;r,β)∂αt= Δt-αΓ(3-α)[u˜in+1(x,t,α;r,β) -2u˜in(x,t,α;r,β)+u˜in-1(x,t,α;r,β) +∑j=1nbj(u˜in-j+1(x,t,α;r,β) -2u˜in-j(x,t,α;r,β) +u˜in-j-1(x,t,α;r,β))]+O(k),

where bj = (j + 1)2–α – (j)2–α, j = 1, 2, 3, ...

In addition, the definition of the central difference at time level (n+12) in space is employed to discretize the second-order space derivative

∂2u˜(x,t,α;r,β)∂x2=u˜i+1n+12(x,t;r,β)-2u˜in+12(x,t;r,β)+u˜i-1n+12(x,t;r,β)h2.

Now, substituting Eqs. (9) and (10) into Eq. (4), we obtain

Δt-αΓ(3-α)[u˜in+1-2u˜in+u˜in-1+∑j=1nbj(u˜in-j+1-2u˜in-j+u˜in-j-1)]=k˜(x,t;r,β)×u˜i+1n+12(x,t;r,β)-2u˜in+12(x,t;r,β)+u˜i-1n-12(x,t;r,β)h2+b˜(x,t;r,β).

Simplifying Eq. (11) results in the following:

Δt-αΓ(3-α)[u˜in+1-2u˜in+u˜in-1+∑j=1nbj(u˜in-j+1-2u˜in-j+u˜in-j-1)]=k˜(x,t;r,β)2×[u˜i+1n+1(x,t;r,β)-2u˜in+1(x,t;r,β)+u˜i+1n-1(x,t;r,β)h2+u˜i+1n(x,t;r,β)-2u˜in(x,t;r,β)+u˜i-1n(x,t;r,β)h2]+b˜(x,t;r,β).

Assuming s˜(x,t;r)=k˜ΔtαΓ(3-α)2h2 for all r, β ∈ [0, 1] and from Eq. (12), we have

(1+2s)u˜in+1(x,t;r,β)         -s(u˜i+1n+1(x,t;r,β)+u˜i-1n+1(x,t;r,β))=s(u˜i+1n(x,t;r,β)+u˜i-1n(x,t;r,β))         +(2-2s)u˜in(x,t;r,β)-u˜in-1(x,t;r,β)         -∑j=1nbj(u˜i+1n-j+1-2u˜in-j+u˜i-1n-j-1)         +kαΓ(3-α)b˜(x,t;r,β).

5. Numerical Example

Consider FTFWE [27] expressed as

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

Based on the given boundary conditions ũ(0, t) = ũ(L, t) = 0 and initial condition

u˜(x,0)=μ˜[sin(5πx)+2sin(7πx)].

the fuzzy number remains identical when expressed in single parametric form as follows:

μ˜(r)=[μ_(r),μ¯(r)]=[r-1,1-r]   for all r∈[0,1].

The analytical solution to Eq. (15) is provided in [23]:

{u_(x,t,α;r)=μ_[sin(5πx)cos(5πt)+2sin(7πx)cos(7πt)],u¯(x,t,α;r)=μ¯[sin(5πx)cos(5πt)+2sin(7πx)cos(7πt)].

The error for the fuzzy solution of Eq. (15) is defined as follows:

[E˜]r=∣U˜(t,x;r)-u˜(t,x;r)∣={[E_]r=∣U_(t,x;r)-u_(t,x;r)∣,[E¯]r=∣U¯(t,x;r)-u¯(t,x;r)∣.

At Δx = h = 0.2 and Δtα = (0.001)1.5, the fuzzy number, when expressed in double parametric form, remains unchanged and can be represented as μ̃(r) = [β(2 – 2r) + (r – 1)].

Tables 1–2 and Figures 1–3 demonstrate a strong correlation between the proposed methods and the exact solution, indicating a high level of agreement for all r, β ∈ [0, 1] at t = 0.005, and α = 1.5. Furthermore, they fulfill the characteristics of the double-parametric representation of fuzzy numbers, resulting in the adoption of a triangular shape. Additionally, the CTCS scheme yields marginally more precise outcomes than the C-N method. In Fig. 4, the proposed methods yield more precise outcomes at locations near the inflection point (β = 0.5) in terms of numerical results. The utilization of the double parametric form was found to be a comprehensive and efficient method for transitioning the governing equation from an uncertain to a precise state. This approach not only decreases the computational burden but also yields more precise outcomes in comparison to the single parametric form.

6. Conclusion

In this study, two FDM schemes were reformulated and implemented to obtain a numerical solution for FTFWE in double parametric form. The Caputo definition was used for the Hukuhara time fractional derivative. The CTCS and C-N schemes yielded outcomes that conformed to the characteristics of fuzzy numbers by adopting the form of triangular fuzzy numbers. The CTCS scheme was found to produce more precise solutions than the C-N method, as demonstrated by a comparative analysis of the exact and numerical solutions. These schemes can be extended to solve nonlinear fuzzy fractional diffusion equations and other types of fuzzy fractional partial differential equations, which will be investigated in detail at a later stage.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.

Fig 1.

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Figure 1.

Analytical lower solution of Eq. (15) at t = 0.005, x = 1.8, and r = 0.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 407-415https://doi.org/10.5391/IJFIS.2024.24.4.407

Fig 2.

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Figure 2.

Analytical upper solution of Eq. (15) at t = 0.005, x = 1.8, and r = 0.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 407-415https://doi.org/10.5391/IJFIS.2024.24.4.407

Fig 3.

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Figure 3.

Analytical and FDM solutions to Eq. (15) at α = 1.5, x = 1.8, t = 0.005 for all r ∈ [0, 1].

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 407-415https://doi.org/10.5391/IJFIS.2024.24.4.407

Fig 4.

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Figure 4.

Fuzzy analytical and fuzzy numerical solutions of Eq. (15) by (a) CTCS and (b) C-N at t = 0.005 and x = 1.8 for all r, β ∈ [0, 1].

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 407-415https://doi.org/10.5391/IJFIS.2024.24.4.407

Table 1 . Numerical results of Eq. (15) by CTCS and C-N at t = 0.005 and x = 0.9 for all r, β ∈ [0, 1].

βrCTCSC-N
ũ(1.8, 0.05; r, β)Ẽ(1.8, 0.05; r, β)ũ(1.8, 0.05; r, β)Ẽ (1.8, 0.05; r, β)
β = 0 Lower solution0–1.8920941.46803 × 10−3–1.8770981.35277 × 10−2
0.1–1.7028851.32123 × 10−3–1.6893891.21749 × 10−2
0.3–1.3244661.02762 × 10−3–1.3139689.46938 × 10−3
0.5–0.9460477.34015 × 10−4–0.9385496.76384 × 10−3
0.7–0.5676284.40409 × 10−4–0.5631294.0583 × 10−3
0.9–0.18920941.46803 × 10−4–0.1877091.35277 × 10−3
10000
β = 1 Upper solution01.8920941.46803 × 10−31.8770981.35277 × 10−2
0.11.7028851.32123 × 10−31.6893891.21749 × 10−2
0.31.3244661.02762 × 10−31.3139689.46938 × 10−3
0.50.9460477.34015 × 10−40.9385496.76384 × 10−3
0.70.5676284.40409 × 10−40.5631294.0583 × 10−3
0.90.18920941.46803 × 10−40.1877091.35277 × 10−3
10000

Table 2 . Numerical solutions to Eq. (15) by CTCS and C-N at t = 0.005 and x = 1.8 for all r, β ∈ [0, 1].

βrCTCSC-N
ũ(1.8, 0.05; r, β)Ẽ(1.8, 0.05; r, β)ũ(1.8, 0.05; r, β)Ẽ (1.8, 0.05; r, β)
β = 0.40–0.37841892.93606 × 10−4–0.3754192.70554 × 10−3
0.1–0.3405772.64245 × 10−4–0.3378782.43498 × 10−3
0.3–0.2648932.05524 × 10−4–0.2627941.89388 × 10−3
0.5–0.1892091.46803 × 10−4–0.1877101.35277 × 10−3
0.7–0.1135268.80818 × 10−5–0.1111408.11661 × 10−4
0.9–0.0378422.93606 × 10−5–0.0375422.70554 × 10−4
10000
β = 0.600.37841892.93606 × 10−40.3754192.70554 × 10−3
0.10.3405772.64245 × 10−40.3378782.43498 × 10−3
0.30.2648932.05524 × 10−40.2627941.89388 × 10−3
0.50.1892091.46803 × 10−40.1877101.35277 × 10−3
0.70.1135268.80818 × 10−50.1111408.11661 × 10−4
0.90.0378422.93606 × 10−50.0375422.70554 × 10−4
10000

References

  1. Samko, SG, Kilbas, AA, and Marichev, O (1993). Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach Science Publishers
  2. Podlubny, I (1999). Fractional Differential Equations. San Diego, CA: Academic Press
  3. Oldham, K, and Spanier, J (1974). The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. New York, NY: Academic Press
  4. Miller, KS, and Ross, B (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. New York, NY: John Willey & Sons
  5. Carcione, JM (2007). Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media. Oxford, UK: Elsevier
  6. Chen, W, Sun, H, and Li, X (2022). Fractional Derivative Modeling in Mechanics and Engineering. Singapore: Springer https://doi.org/10.1007/978-981-16-8802-7
    CrossRef
  7. 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
    CrossRef
  8. Sullivan, J, Iadecola, T, and Williamson, DJ (). Planar p-string condensation: chiral fracton phases from fractional quantum Hall layers and beyond. Physical Review B. 103, 2021. article no 205301
  9. Jian, CM, and Qi, XL (2014). Layer construction of 3D topological states and string braiding statistics. Physical Review X. 4. article no 041043
    CrossRef
  10. Zureigat, H, Ismail, AI, and Sathasivam, S (2021). Numerical solutions of fuzzy time fractional advection-diffusion equations in double parametric form of fuzzy number. Mathematical Methods in the Applied Sciences. 44, 7956-7968. https://doi.org/10.1002/mma.5573
    CrossRef
  11. 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
    CrossRef
  12. Zureigat, H, Ismail, AI, and Sathasivam, S (2020). A compact Crank–Nicholson scheme for the numerical solution of fuzzy time fractional diffusion equations. Neural Computing and Applications. 32, 6405-6412. https://doi.org/10.1007/s00521-019-04148-2
    CrossRef
  13. Sweilam, NH, Khader, MM, and Adel, M (2012). On the stability analysis of weighted average finite difference methods for fractional wave equation. Fractional Differential Calculus. 2, 17-29. https://doi.org/10.7153/fdc-02-02
    CrossRef
  14. Lyu, P, Liang, Y, and Wang, Z (2020). A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation. Applied Numerical Mathematics. 151, 448-471. https://doi.org/10.1016/j.apnum.2019.11.012
    CrossRef
  15. Wang, YM (2015). A compact finite difference method for a class of time fractional convection-diffusion-wave equations with variable coefficients. Numerical Algorithms. 70, 625-651. https://doi.org/10.1007/s11075-015-9965-x
    CrossRef
  16. Ghode, K, Takale, K, and Gaikwad, S. (2021) . New technique for solving time fractional wave equation: Python. Journal of Mathematical and Computational Science. 11, 5327-5343. https://www.scik.org/index.php/jmcs/article/view/6048
  17. Liu, X, Bo, Y, and Jin, Y (2022). A numerical method for the variable-order time-fractional wave equations based on the H2N2 approximation. Journal of Function Spaces, 2022. article no 3438289
  18. Arfan, M, Shah, K, Ullah, A, Salahshour, S, Ahmadian, A, and Ferrara, M (2022). A novel semi-analytical method for solutions of two dimensional fuzzy fractional wave equation using natural transform. Discrete and Continuous Dynamical Systems – Series S. 15, 315-338. https://doi.org/10.3934/dcdss.2021011
    CrossRef
  19. Behera, D, and Chakraverty, S (2015). New approach to solve fully fuzzy system of linear equations using single and double parametric form of fuzzy numbers. Sadhana. 40, 35-49. https://doi.org/10.1007/s12046-014-0295-9
    CrossRef
  20. Osman, M, Gong, ZT, and Mohammed, A (2021). Differential transform method for solving fuzzy fractional wave equation. Journal of Computational Analysis and Application. 29, 431-453.
  21. Ghaffari, M, Allahviranloo, T, Abbasbandy, S, and Azhini, M (2022). Generalized Hukuhara conformable fractional derivative and its application to fuzzy fractional partial differential equations. Soft Computing. 26, 2135-2146. https://doi.org/10.1007/s00500-021-06637-w
    CrossRef
  22. Qayyum, M, Ahmad, E, Akgul, A, and El Din, SM (). Fuzzy-fractional modeling of Korteweg-de Vries equations in Gaussian-Caputo sense: new solutions via extended He-Mahgoub algorithm. Ain Shams Engineering Journal. 15, 2024. article no 102623
  23. Rashid, S, Ashraf, R, and Hammouch, Z (2023). New generalized fuzzy transform computations for solving fractional partial differential equations arising in oceanography. Journal of Ocean Engineering and Science. 8, 55-78. https://doi.org/10.1016/j.joes.2021.11.004
    CrossRef
  24. Sartanpara, PP, and Meher, R (2023). A robust fuzzy-fractional approach for the atmospheric internal wave model. Journal of Ocean Engineering and Science. 8, 308-322. https://doi.org/10.1016/j.joes.2022.02.001
    CrossRef
  25. Zureigat, HH (2023). Modified natural Daftardar-Jafari Method for solving fuzzy time-fractional wave equation. International Journal of Fuzzy Logic and Intelligent Systems. 23, 409-417. https://doi.org/10.5391/IJFIS.2023.23.4.409
    CrossRef
  26. Li, L, and Liu, JG (2018). A generalized definition of Caputo derivatives and its application to fractional ODEs. SIAM Journal on Mathematical Analysis. 50, 2867-2900. https://doi.org/10.1137/17M1160318
    CrossRef
  27. Salah, A, Khan, M, and Gondal, MA (2013). A novel solution procedure for fuzzy fractional heat equations by homotopy analysis transform method. Neural Computing and Applications. 23, 269-271. https://doi.org/10.1007/s00521-012-0855-z
    CrossRef

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