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A comparative study of Bagley–Torvik equation under nonsingular kernel derivatives using Weeks method

  • Kamran EMAIL logo , Muhammad Asif , Aiman Mukheimer , Kamal Shah , Thabet Abdeljawad EMAIL logo and Fahad M. Alotaibi
Published/Copyright: January 2, 2024
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Open Physics
From the journal Open Physics Volume 22 Issue 1

Abstract

Modeling several physical events leads to the Bagley–Torvik equation (BTE). In this study, we have taken into account the BTE, including the Caputo–Fabrizio and Atangana–Baleanu derivatives. It becomes challenging to find the analytical solution to these kinds of problems using standard methods in many circumstances. Therefore, to arrive at the required outcome, numerical techniques are used. The Laplace transform is a promising method that has been utilized in the literature to address a variety of issues that come up when modeling real-world data. For complicated functions, the Laplace transform approach can make the analytical inversion of the Laplace transform excessively laborious. As a result, numerical techniques are utilized to invert the Laplace transform. The numerical inverse Laplace transform is generally an ill-posed problem. Numerous numerical techniques for inverting the Laplace transform have been developed as a result of this challenge. In this article, we use the Weeks method, which is one of the most efficient numerical methods for inverting the Laplace transform. In our proposed methodology, first the BTE is transformed into an algebraic equation using Laplace transform. Then the reduced equation solved the Laplace domain. Finally, the Weeks method is used to convert the obtained solution from the Laplace domain into the real domain. Three test problems with Caputo–Fabrizio and Atangana–Baleanu derivatives are considered to demonstrate the accuracy, effectiveness, and feasibility of the proposed numerical method.

Keywords: Laplace transform; Bagley–Torvik equation; Caputo–Fabrizio derivative; Atangana–Baleanu derivative; numerical inversion; Weeks method; Laguerre polynomials

1 Introduction

A realistic modeling of a physical phenomenon, such as viscoelasticity, heat conduction, electrode–electrolyte polarization, electromagnetic waves, diffusion, and control theory, can be successfully accomplished by employing fractional calculus, which has caught the attention of a lot of investigators across many disciplines of applied science and engineering [1–4]. In this article, we consider the fractional Bagley–Torvik equation (BTE), first appeared in an innovative work [5]. Their work was about modeling the viscoelastic behavior of geological strata, metals, and glasses using fractional differential equations, demonstrating that this approach is successful in describing structures with both elastic and viscoelastic components. BTE is an extremely important equation used to solve many applied scientific and engineering problems. More specifically, BTE can be used to represent any linearly damped fractional oscillator with damping term having fractional derivative of order 3 2 . Particularly, the models of materials whose damping varies on frequency can be predicted by an equation with a 1 2 -order or 3 2 -order derivative. It may also model the motion of a rigid plate submerged in a viscous fluid and a gas in a fluid, describing the motion of actual physical systems [2]. Generalized form of the BTE is written as follows:

(1) a 1 D ξ β W ( ξ ) + a 2 D ξ γ + 1 W ( ξ ) + a 3 W ( ξ ) = f ( ξ ) , 1 < β ≤ 2 , 0 < γ ≤ 1 ,

with initial data

W ( 0 ) = W 0 , W ′ ( 0 ) = W 1 , W 0 , W 1 ∈ R .

We will consider Eq. (1) for β = 2 and γ = 1 2 . In literature, D ξ β and D ξ γ + 1 fractional derivatives are used in Riemann–Liouville and Liouville–Caputo sense due to their convenient status. For example, Ji et al. [6] considered the BTE equation in Liouville–Caputo sense and studied its numerical solution using shifted Chebyshev operational matrix. Ray and Bera [7] obtained the solution of BTE using Adomian decomposition method. Jena and Chakraverty [8] obtained the analytic solution of BTE using Sumudu transformation. Çenesiz et al. [9] obtained the numerical solution of BTE using the generalized Taylor collocation method. Mashayekhi and Razzaghi [10] studied the numerical solution of BTE using the hybrid functions approximation. They also derived the error bounds for the presented method. Gülsu et al. [11] utilized the Taylor matrix method for the approximation of the solution of BTE. In the study by Yüzbaşı [12], the numerical solution of BTE was obtained via the Bessel collocation method. In the study by Pinar [13], the authors obtained the analytic solution of BTE with conformable fractional derivative using the sine-Gordon expansion method and the Bernouli equation method. Raja et al. [14] studied the solution of BTE system arising in fluid dynamic model via feed-forward fractional artificial neural networks and sequential quadratic programming algorithm. However, these derivatives contain singular kernels, and they face problems when trying to model nonlocal phenomenon.

Caputo and Fabrizio in 2015 introduced a new fractional differential operator based on the exponential kernel function known as Caputo–Fabrizio derivative (CFD) to overcome the problem of the singular kernel function involved in the Riemann–Liouville and Liouville–Caputo fractional differential operators [16]. They demonstrated that CFD was suitable for modeling some physical problems. Atangana and Alqahtani [17] used the CFD for modeling the ground water pollution. Hasan et al. [18] studied the numerical solution of BTE under the CFD using a modified reproducing kernel Hilbert space method. Al-Smadi et al. [19] studied the solution of a nonlinear differential equation with CFD using a reproducing kernel algorithm. Moore et al. [20] developed a CFD model for HIV/AIDS epidemic. They obtained the numerical solution of the proposed model using a three step Adams–Bashforth predictor method. Joshi et al. [21] framed a fractional order mathematical model in the sense of CFD, to investigate the role of buffer and calcium concentration on fibroblast cells. For more information on CFD the readers can refer to the previous studies [22–24]. However, some issues were also pointed out against the considered derivatives, as the kernel in integral was nonsingular but was not nonlocal.

To overcome these issues, Atangana and Baleanu [25] proposed a new fractional operator based upon the Mittag–Leffler function known as Atangana–Baleanu derivative (ABD). Their operator includes a nonlocal and nonsingular kernel with all the benefits of Riemann–Liouville, Liouville–Caputo, and Caputo–Fabrizio operators. In addition to these features, the derivative was found very useful in thermal science material. Due to these powerful features, researchers have applied it to many phenomena [26,27]. Atangana [28] applied the ABD to the nonlinear Fisher’s reaction–diffusion equation and obtained the solution of the modified equation using an iterative scheme. Gómez-Aguilar et al. [29] applied the ABD to electromagnetic waves in dielectric media. Ghanbari et al. [30] applied the ABD to three species predator-prey model. They obtained the desired solution using the product integration rule. Khan et al. [31] studied some necessary and sufficient conditions for the existence of the solutions of differential equations with modified ABD. Joshi et al. [32] proposed a nonsingular SIR model with the Mittag–Leffler law. The author used the nonlinear Beddington–DeAngelis infection rate and Holling type II treatment rate. The qualitative properties of the SIR model and the local and global stability of the model were also discussed. More information on the applications of ABD can be found in previous studies [33–38].

The goal of this study is to use both the Caputo–Fabrizio and ABDs with fractional order to extend the BTE to the realm of fractional calculus. In this work, we have used the Laplace transform method for this purpose. However, using the Laplace transform some times makes the analytic inversion very hard to compute for complicated functions. The literature on the numerical inversion of Laplace transform is extensive. Readers looking for a survey on the comparison of numerical inverse Laplace transform methods should begin with [39] and then proceed to the more recent work [40]. The numerical comparisons reported in these articles reveal supremacy of three numerical methods: the Talbot’s method, the enhanced trapezoidal rule, and the Weeks method. The latter method is the subject of the current study, specifically the issue of choosing the two free parameters that determine method’s accuracy. The Weeks method holds one major advantage over the Talbot’s method and the enhanced trapezoidal rule: In particular, it assumes that a smooth function can be well approximated by an expansion in terms of orthonormal Laguerre functions [41]. Laguerre functions are used because the quadrature formulas involving them are similar to the Laplace transform operator. In this method, the unknown coefficients are evaluated once for all for any given transformed function. It is highly efficient for multiple evaluations in the time domain. Computing the function at a new time with these other methods requires essentially restarting the numerical inversion procedure. Furthermore, it is equally applicable to real and complex time-domain functions [42].

2 Basic definitions

Here, we present some important definitions related to our work.

Definition 2.1

The Caputo derivative of a function W ( ξ ) with fractional order γ ∈ ( l − 1 , l ] , l ∈ Z + is defined as follows [1]:

(2) D ξ γ η C W ( ξ ) = 1 Γ ( l − γ ) ∫ 0 ξ d l W ( s ) d s l ( ξ − s ) γ − l + 1 d s .

Definition 2.2

Let W ∈ ℋ l ( η , δ ) , δ > η , γ ∈ ( l − 1 , l ] , l ∈ Z + and not necessarily differentiable, then the CFD with base point η at point ξ ∈ ( η , δ ) is defined as follows [25]:

(3) D ξ γ η CFD W ( ξ ) = G ( γ ) 1 − γ ∫ η ξ d l W ( s ) d s l e − γ 1 − γ ( ξ − s ) d s ,

where G ( γ ) = 1 − γ + γ Γ ( γ ) .

Definition 2.3

Let W ∈ ℋ l ( η , δ ) , δ > η , γ ∈ ( l − 1 , l ] , l ∈ Z + , then the ABD with base point η at point ξ ( η , δ ) is defined as follows [25]:

(4) D ξ γ η A B D W ( ξ ) = G ( γ ) 1 − γ ∫ η ξ d l W ( s ) d s l E γ − γ 1 − γ ( ξ − s ) γ d s ,

where E γ ( ξ ) = ∑ k = 0 ∞ ξ k Γ ( γ k + 1 ) is Mittag–Leffler function and ℋ l is the lth-order Sobolev space on a domain Ω ⊂ R defined as ℋ l ( Ω ) = { W ∈ L 2 ( Ω ) : W m ∈ L 2 ( Ω ) , ∀ m ≤ l } .

Definition 2.4

The Laplace transform of a piecewise continuous function W ( ξ ) for ξ > 0 is defined as follows [2]:

(5) L { W ( ξ ) } = W ^ ( z ) = ∫ 0 ∞ e ( − z ξ ) W ( ξ ) d ξ ,

where z is the Laplace parameter.

Definition 2.5

If l ∈ N , γ ∈ [ 0 , 1 ] , then the Laplace transform of the CFD of a function W ( ξ ) is defined as follows [25]:

(6) L { D ξ γ + l 0 CFD W ( ξ ) } = 1 z + γ ( 1 − z ) [ z l + 1 W ^ ( z ) − z l W ( 0 ) − z l − 1 W ′ ( 0 ) − ⋯ − W l ( 0 ) ] ,

if l = 0 , then

(7) L { D ξ γ 0 CFD W ( ξ ) } = 1 z + γ ( 1 − z ) [ z W ^ ( z ) − W ( 0 ) ] ,

and if l = 1 , then we have

(8) L { D ξ γ + 1 0 CFD W ( ξ ) } = 1 z + γ ( 1 − z ) [ z 2 W ^ ( z ) − z W ( 0 ) − W ′ ( 0 ) ] .

Definition 2.6

The Laplace transform of ABD of a function W ( ξ ) is defined as follows [25]:

(9) L { D ξ γ 0 ABD W ( ξ ) } = Q ( γ ) z γ ( 1 − γ ) + γ [ z γ W ^ ( z ) − z γ − 1 W ( 0 ) ] .

3 Proposed scheme

This section covers the proposed numerical scheme for modeling BTE with CFD and ABD. The method has three major steps: (i) first, a BTE with CFD/ABD is considered and transformed to an algebraic equation via the Laplace transform; (ii) second, the reduced equation is solved in Laplace transform domain; (iii) finally, the desired solution is retrieved using numerical inverse Laplace transform method.

3.1 BTE with CFD

We consider BTE with CFD as follows:

(10) a 1 D ξ β W ( ξ ) + a 2 D ξ γ + 1 0 CFD W ( ξ ) + a 3 W ( ξ ) = f ( ξ ) , l − 1 < γ ≤ l ,

with initial conditions

(11) W ( j ) ( 0 ) = W 0 ( j ) , j = 0 , 1 , … , l − 1 .

By taking the Laplace transform of Eq. (10), we obtain

(12) L { a 1 D ξ β W ( ξ ) + a 2 D ξ γ + 1 0 CFD W ( ξ ) + a 3 W ( ξ ) } = L { f ( ξ ) } ,

which implies

(13) a 1 [ z β W ^ ( z ) − z β − 1 W ( 0 ) − z β − 2 W ′ ( 0 ) ] + a 2 1 z + γ ( 1 − z ) [ z 2 W ^ ( z ) − z W ( 0 ) − W ′ ( 0 ) ] + a 3 W ^ ( z ) = F ^ ( z ) .

By solving for W ^ ( z ) , we obtain

(14) W ^ ( z ) = a 1 z β − 1 W ( 0 ) + a 1 β − 2 W ′ ( 0 ) + a 2 z W ( 0 ) z + γ ( 1 − z ) + a 2 W ′ ( 0 ) z + γ ( 1 − z ) + F ^ ( z ) a 1 z β + a 2 z 2 z + γ ( 1 − z ) + a 3 .

3.2 BTE with ABD

We consider BTE with ABD as follows:

(15) a 1 D ξ β W ( ξ ) + a 2 D ξ γ + 1 0 A B D W ( ξ ) + a 3 W ( ξ ) = f ( ξ ) . l − 1 < γ ≤ l ,

with initial conditions

(16) W ( j ) ( 0 ) = W 0 ( j ) , j = 0 , 1 , … , p − 1 .

By taking the Laplace transform of Eq. (15), we obtain

(17) L { a 1 D ξ β W ( ξ ) + a 2 D ξ γ + 1 0 A B D W ( ξ ) + a 3 W ( ξ ) } = L { f ( ξ ) } ,

which implies

(18) a 1 [ z β W ^ ( z ) − z β − 1 W ( 0 ) − z β − 2 W ′ ( 0 ) ] + a 2 G ( γ + 1 ) ( 1 − z γ + 1 ) γ + 1 [ z γ + 1 W ^ ( z ) − z γ W ( 0 ) − z γ − 1 W ′ ( 0 ) ] + a 3 W ^ ( z ) = F ^ ( z ) .

Solving for W ^ ( z ) , we obtain

(19) W ^ ( z ) = a 1 z β − 1 W ( 0 ) + a 1 z β − 2 W ′ ( 0 ) + a 2 G ( γ + 1 ) z γ W ( 0 ) ( 1 − z γ + 1 ) γ + 1 + a 2 G ( γ + 1 ) z γ − 1 W ′ ( 0 ) ( 1 − z γ + 1 ) γ + 1 + F ^ ( z ) a 1 z β + a 2 G ( γ + 1 ) z γ + 1 ( 1 − z γ + 1 ) γ + 1 + a 3 .

3.3 Inverse Laplace transform

By taking the inverse Laplace transform of (14) or (19), we have

(20) W ( ξ ) = 1 2 π i ∫ σ − i ∞ σ + i ∞ W ^ ( z ) e z ξ d z = 1 2 π i ∫ Θ W ^ ( z ) e z ξ d z , σ > σ 0 .

The integral in Eq. (20) is known as Bromwich integral, σ 0 is the abscissa of convergence, and Θ is an appropriately selected line joining σ − i ∞ and σ + i ∞ to restrict all the singularities of the transform function W ^ ( z ) to the left of Θ . The analytic computation of the integral in Eq. (20) can be hard for complicated functions. Therefore, the research community have developed various approaches for the numerical approximation of the integral in Eq. (20). Each individual approach has its own application and is suitable for a specific problem. All the approaches are based on approximations used to evaluate the integral given in Eq. (20). In the current study, we use the Weeks method for the numerical approximation of the integral in Eq. (20).

3.4 Weeks method

The Weeks method is one of the most effective numerical strategies for inverting the Laplace transform, as long as the two free parameters in the Laguerre expansion on which it is based are chosen well. The Weeks technique has one significant benefit over the enhanced trapezoidal rule and Talbot’s method: it gives a function expansion, notably the Laguerre series expansion. This indicates that for each given W ^ ( z ) , the unknown coefficients in the Laguerre series expansion may be determined once and for all. In Weeks method, the Bromwich line is parameterized as z = σ + i y , y ∈ R to obtain the Fourier integral

(21) W ( ξ ) = e σ ξ 2 π ∫ − ∞ ∞ e i ξ y W ^ ( σ + i y ) d y .

The function W ^ ( σ + i y ) is expanded as follows:

(22) W ^ ( σ + i y ) = ∑ k = − ∞ ∞ a k ( − ς + i y ) k ( ς + i y ) k + 1 , ς > 0 , y ∈ R .

By using Eq. (22) in Eq. (21), we obtain

(23) W ( ξ ) = e σ ξ 2 π ∑ k = − ∞ ∞ a k ψ k ( ξ ; ς ) ,

where

(24) ψ k ( ξ ; ς ) = ∫ − ∞ ∞ e i ξ y ( − ς + i y ) k ( ς + i y ) k + 1 d y .

We may use residues to evaluate the Fourier integral, and for ξ > 0 , one obtains

(25) ψ k ( ξ ; ς ) = 2 π e − ς ξ L k ( 2 ς ξ ) , k ≥ 0 , 0 , k < 0 ,

where L k ( ξ ) is Laguerre polynomial of degree k , σ > σ 0 , σ 0 is the abscissa of convergence, and σ , ς ∈ R are positive parameters. The polynomials L k ( ξ ) are defined as follows:

(26) L k ( ξ ) = e ξ k ! d k d ξ k ( e − ξ ξ k ) ,

where a k denotes the coefficients in the Taylor series. See (Figure 1):

(27) Q ( ω ) = 2 ς 1 − ω W ^ σ + 2 ς 1 − ω − ς = ∑ k = 0 ∞ a k ω k , ∣ ω ∣ < R ,

where R denotes the radius of convergence of the Maclaurin series (27). The coefficients a k are computed as follows:

(28) a k = 1 2 π i ∫ ∣ ω ∣ = 1 Q ( ω ) ω k + 1 d ω = 1 2 π ∫ − π π Q ( e i ϑ ) e − i k ϑ d ϑ .

Figure 1 Laguerre polynomials of orders 0 through k k .
Figure 1

Laguerre polynomials of orders 0 through k .

The integral in Eq. (28) is the well-known Cauchy’s formula, which can be approximated as follows:

(29) a ˜ k = e − i k h ∕ 2 2 M ∑ j = − M M − 1 Q ( e i ϑ j + 1 ∕ 2 ) e − i k ϑ j , k = 0 , 1 , 2 , … , M − 1 ,

where ϑ j = j h , h = π M .

3.4.1 Error analysis of the method

This section is devoted to the error analysis. Weideman [43] analyzed the error of the Weeks method. The following observations were made during their study for the following expansion:

(30) W ( ξ ) = exp ( σ ξ ) ∑ k = 0 ∞ a k exp ( − ς ξ ) L k ( 2 ς ξ ) .

Three main factors that contribute to error were identified:

  • The first factor is truncating the series at M terms.

  • The numerical computation of the coefficients is the second factor.

  • Third is the inversion of Laplace transform numerically. Any inaccuracy in the evaluated coefficients increases with rising ξ when σ > 0 , which is how the error in (30) may be seen.

To model these three factors of error, the real expansion is

(31) W ˜ ( ξ ) = exp ( σ ξ ) ∑ k = 0 N − 1 a ˜ k ( 1 + ϖ k ) exp ( − ς ξ ) L k ( 2 ς ξ ) ,

where ϖ k denotes the relative error of the coefficients in the floating-point representation, i.e., f l ( a ˜ k ) = a ˜ k ( 1 + ϖ k ) . From Eqs (31) and (30), we have

∣ W ( ξ ) − W ˜ ( ξ ) ∣ ≤ exp ( σ ξ ) ( T ( error ) + D ( error ) + C ( error ) ) ,

with assumption ∑ k = 0 ∞ ∣ a k ∣ < ∞ , where T ( error ) = ∑ k = M ∞ ∣ a k ∣ is the truncation error bound, D ( error ) = ∑ k = 0 M − 1 ∣ a k − a ˜ k ∣ is the discretization error bound , C ( error ) = ϖ ∑ k = 0 M − 1 ∣ a ˜ k ∣ is the conditioning error bound, and ϖ k denotes the roundoff unit of machine satisfying the condition max 0 ≤ k ≤ M − 1 ∣ ϖ k ∣ ≤ ϖ with the fact that ∣ exp ( − ς ξ ) L k ( 2 ς ξ ) ∣ ≤ 1 . We can neglect the D ( error ) in comparison with T ( error ) and C ( error ) [43]. Therefore, we refer to T ( error ) and C ( error ) . For T ( error ) and C ( error ) , the upper bound were reported as follows [43]:

T ( error ) ≤ k ( χ ) χ M ( χ − 1 ) , C ( error ) ≤ ϖ χ k ( χ ) χ − 1 ,

which holds for χ ∈ ( 1 , R ) . Therefore, we have the following error bound:

(32) error est ≤ k ( χ ) χ M ( χ − 1 ) + ϖ χ k ( χ ) χ − 1 .

To have an optimal value of error est , Weideman [43] proposed two algorithms for obtaining the optimal values of σ and ς . We have used Algorithm 1:

Algorithm 1: Algorithm for optimal values of ( σ , ς )
The algorithm requires W ( z ) , ξ , and M , and [ σ 0 , σ max ] × [ 0 , ς max ] which are expected to have the best values of σ and ς . The algorithm then operates as follows:
σ = { σ ∈ [ σ 0 , σ max ] ∣ error est ( σ , ς ( σ ) ) = minimum } ,
where
ς ( σ ) = { ς ∈ [ 0 , ς max ] ∣ T ( error ) ( σ , ς ) = minimum } .

4 Applications

This section illustrates the effectiveness of the Weeks method discussed earlier for the solution of BTE with CFD and ABD. Three numerical examples are used to support the proposed scheme. For all the numerical experiments, the fractional derivative D ξ γ + 1 in (1) is considered in CFD and ABD sense with 0 < γ ≤ 1 , and the fractional derivative D ξ β is considered in Caputo’s sense with β = 2 . The computational results demonstrate the accuracy and convergence of the proposed method. The absolute error ( AbS ( error ) ) and the relative error ( RlE ( error ) ) are used to measure the numerical error. The two error norms are defined as follows:

AbS ( error ) = ∣ W ( ξ ) − W Aprrox ( ξ ) ∣ ,

and

RlE ( error ) = W ( ξ ) − W Aprrox ( ξ ) W ( ξ ) .

The forcing term f ( ξ ) and the initial boundary data are calculated using the exact solution for each example.

Problem 1

The first problem is solved using the Weeks method in ABD and CFD sense with exact solution W ( ξ ) = γ ξ 2 . In Table 1 the AbS ( error ) , the RlE ( error ) , and the error est for different values of ξ with M = 65 corresponding to Problem 1 are shown. Table 2 shows the AbS ( error ) , the RlE ( error ) , and the error est for different values of M at ξ = 1 corresponding to Problem 1. Figure 2 shows the plots of exact solution and Weeks solution. A comparison between the AbS ( error ) , the RlE ( error ) , and the error est of the proposed numerical scheme for problem 1 with ABD and CFD for different values of ξ with M = 65 is shown in Figure 3(a) and (b), respectively. Similarly, the comparison between the AbS ( error ) , the RlE ( error ) , and the error est of the proposed numerical scheme for problem 1 with ABD and CFD for different values of M at ξ = 1 using the proposed method is shown in Figure 4(a) and (b), respectively. The results demonstrates the efficiency of the method for BTE with two different fractional derivatives.

Table 1

The AbS ( error ) , RlE ( error ) , and error est corresponding to Problem 1

M σ ς AbS ( error ) RlE ( error ) error est
ABD CFD ABD CFD ABD CFD ABD CFD ABD CFD
0.1 2.6146 3.4347 1.8936 1.2117 1.7347 × 1 0 ‒ 18 1.7347 × 1 0 ‒ 18 1.1565 × 1 0 ‒ 16 1.1565 × 1 0 ‒ 16 4.8094 × 1 0 ‒ 16 1.5691 × 1 0 ‒ 16
0.2 3.3654 3.4708 1.1703 1.1481 6.9389 × 1 0 ‒ 18 6.9389 × 1 0 ‒ 18 1.1565 × 1 0 ‒ 16 1.1565 × 1 0 ‒ 16 2.2407 × 1 0 ‒ 16 2.1615 × 1 0 ‒ 16
0.3 2.6146 3.1652 1.8936 1.1481 2.7756 × 1 0 ‒ 17 2.7756 × 1 0 ‒ 17 2.0560 × 1 0 ‒ 16 2.0560 × 1 0 ‒ 16 8.1133 × 1 0 ‒ 16 3.2886 × 1 0 ‒ 16
0.4 2.6146 3.1652 1.8936 1.1481 5.5511 × 1 0 ‒ 17 2.7756 × 1 0 ‒ 17 2.3130 × 1 0 ‒ 16 1.1565 × 1 0 ‒ 16 1.0538 × 1 0 ‒ 15 4.5131 × 1 0 ‒ 16
0.5 2.7000 3.4708 3.3401 1.4116 1.1102 × 1 0 ‒ 16 5.5511 × 1 0 ‒ 17 2.9606 × 1 0 ‒ 16 1.4803 × 1 0 ‒ 16 1.5692 × 1 0 ‒ 15 6.2770 × 1 0 ‒ 16
0.6 2.7000 3.4511 3.3401 1.3399 1.1102 × 1 0 ‒ 16 1.1102 × 1 0 ‒ 16 2.0560 × 1 0 ‒ 16 2.0560 × 1 0 ‒ 16 2.0556 × 1 0 ‒ 15 8.1381 × 1 0 ‒ 16
0.7 2.7000 3.1472 3.3401 1.1703 1.1102 × 1 0 ‒ 16 1.1102 × 1 0 ‒ 16 1.5105 × 1 0 ‒ 16 1.5105 × 1 0 ‒ 16 2.6928 × 1 0 ‒ 15 1.1399 × 1 0 ‒ 15
0.8 2.7000 3.1472 3.3401 1.1703 3.3307 × 1 0 ‒ 16 3.3307 × 1 0 ‒ 16 3.4694 × 1 0 ‒ 16 3.4694 × 1 0 ‒ 16 3.5275 × 1 0 ‒ 15 1.5615 × 1 0 ‒ 15
0.9 2.7000 3.1472 3.3401 1.1703 2.2204 × 1 0 ‒ 16 2.2204 × 1 0 ‒ 16 1.8275 × 1 0 ‒ 16 1.8275 × 1 0 ‒ 16 4.6209 × 1 0 ‒ 15 2.1391 × 1 0 ‒ 15
1 3.2934 2.5881 1.1703 0.8872 2.2204 × 1 0 ‒ 16 2.2204 × 1 0 ‒ 16 1.4803 × 1 0 ‒ 16 1.4803 × 1 0 ‒ 16 3.2704 × 1 0 ‒ 15 2.2446 × 1 0 ‒ 15
Table 2

The AbS ( error ) , RlE ( error ) , and error est corresponding to Problem 1

M σ ς AbS ( error ) RlE ( error ) error est
ABD CFD ABD CFD ABD CFD ABD CFD ABD CFD
40 4.9699 4.2745 3.1153 2.4922 4.4409 × 1 0 ‒ 16 2.2204 × 1 0 ‒ 16 2.9606 × 1 0 ‒ 16 1.4803 × 1 0 ‒ 16 6.2431 × 1 0 ‒ 15 5.7162 × 1 0 ‒ 15
45 4.3000 2.5069 2.0592 1.8034 2.2204 × 1 0 ‒ 16 0 1.4803 × 1 0 ‒ 16 0 5.0408 × 1 0 ‒ 15 2.4480 × 1 0 ‒ 15
50 3.8254 2.4621 1.7308 1.1146 4.4409 × 1 0 ‒ 16 2.2204 × 1 0 ‒ 16 2.9606 × 1 0 ‒ 16 1.4803 × 1 0 ‒ 16 3.5918 × 1 0 ‒ 15 2.1400 × 1 0 ‒ 15
55 3.1754 2.5209 1.6716 1.1930 2.2204 × 1 0 ‒ 16 0 1.4803 × 1 0 ‒ 16 0 3.1507 × 1 0 ‒ 15 2.5050 × 1 0 ‒ 15
60 3.2776 3.3708 1.4373 1.2837 2.2204 × 1 0 ‒ 16 2.2204 × 1 0 ‒ 16 1.4803 × 1 0 ‒ 16 2.9606 × 1 0 ‒ 16 3.2848 × 1 0 ‒ 15 3.1068 × 1 0 ‒ 15
65 3.1652 2.4074 1.0677 0.8610 2.2204 × 1 0 ‒ 16 2.2204 × 1 0 ‒ 16 1.4803 × 1 0 ‒ 16 1.4803 × 1 0 ‒ 16 2.9024 × 1 0 ‒ 15 2.3612 × 1 0 ‒ 15
70 3.1950 3.1931 1.0031 1.3932 2.2204 × 1 0 ‒ 16 0 1.4803 × 1 0 ‒ 16 0 2.9266 × 1 0 ‒ 15 3.1576 × 1 0 ‒ 15
75 2.8348 2.6966 1.0677 0.8610 4.4409 × 1 0 ‒ 16 2.2204 × 1 0 ‒ 16 2.9606 × 1 0 ‒ 16 1.4803 × 1 0 ‒ 16 3.7614 × 1 0 ‒ 15 2.5670 × 1 0 ‒ 15
80 3.5043 2.3172 0.9581 0.8166 2.2204 × 1 0 ‒ 16 2.2204 × 1 0 ‒ 16 1.4803 × 1 0 ‒ 16 1.4803 × 1 0 ‒ 16 3.3670 × 1 0 ‒ 15 2.3426 × 1 0 ‒ 15
85 3.7103 2.7967 1.0677 0.7618 2.2204 × 1 0 ‒ 16 0 1.4803 × 1 0 ‒ 16 0 3.4057 × 1 0 ‒ 15 2.5169 × 1 0 ‒ 15
Figure 2 Exact solution vs Weeks solution of problem 1.
Figure 2

Exact solution vs Weeks solution of problem 1.

Figure 3 (a) The AbS ( error ) {\rm{AbS}}\left({\rm{error}}) , the RlE ( error ) {\rm{RlE}}\left({\rm{error}}) , and the error est {{\rm{error}}}_{{\rm{est}}} versus ξ \xi using M = 65 M=65 with ABD {\rm{ABD}} corresponding to problem 1. (b) The AbS ( error ) {\rm{AbS}}\left({\rm{error}}) , the RlE ( error ) {\rm{RlE}}\left({\rm{error}}) , and the error est {{\rm{error}}}_{{\rm{est}}} versus ξ \xi using M = 65 M=65 with CFD {\rm{CFD}} corresponding to problem 1.
Figure 3

(a) The AbS ( error ) , the RlE ( error ) , and the error est versus ξ using M = 65 with ABD corresponding to problem 1. (b) The AbS ( error ) , the RlE ( error ) , and the error est versus ξ using M = 65 with CFD corresponding to problem 1.

Figure 4 (a) The AbS ( error ) {\rm{AbS}}\left({\rm{error}}) , the RlE ( error ) {\rm{RlE}}\left({\rm{error}}) , and the error est {{\rm{error}}}_{{\rm{est}}} versus M M at ξ = 1 \xi =1 with ABD {\rm{ABD}} corresponding to problem 1. (b) The AbS ( error ) {\rm{AbS}}\left({\rm{error}}) , the RlE ( error ) {\rm{RlE}}\left({\rm{error}}) , and the error est {{\rm{error}}}_{{\rm{est}}} versus M M at ξ = 1 \xi =1 with CFD {\rm{CFD}} corresponding to problem 1.
Figure 4

(a) The AbS ( error ) , the RlE ( error ) , and the error est versus M at ξ = 1 with ABD corresponding to problem 1. (b) The AbS ( error ) , the RlE ( error ) , and the error est versus M at ξ = 1 with CFD corresponding to problem 1.

Problem 2

The second problem is solved using the Weeks method with ABD and CFD and exact solution W ( ξ ) = ( γ 3 + γ − 1 ) ξ + 1 . In Table 3, the AbS ( error ) , the RlE ( error ) , and the error est of the proposed method for different values of ξ with M = 65 corresponding to Problem 2 are presented. Table 4 shows the the AbS ( error ) , the RlE ( error ) , and the error est obtained using the proposed method for different values of M at ξ = 1 corresponding to Problem 2. Figure 5(a) shows the plots of exact solution and Weeks solution. A comparison between the AbS ( error ) , the RlE ( error ) , and the error est of the proposed numerical scheme for problem 1 with ABD and CFD for different values of ξ with M = 65 is shown in Figure 6(a) and (b), respectively. Similarly, the comparison between the AbS ( error ) , the RlE ( error ) , and the error est of the proposed method for problem 1 with ABD and CFD for different values of M at ξ = 1 using the proposed numerical scheme is shown in Figure 7(a) and (b), respectively. An excellent agreement between the theoretical and computed error is observed. In this case also, we see that the proposed numerical scheme has efficiently approximated the solution of BTE in both cases.

Table 3

The AbS ( error ) , RlE ( error ) , and error est corresponding to Problem 2

ξ σ ς AbS ( error ) RlE ( error ) error est
ABD CFD ABD CFD ABD CFD ABD CFD ABD CFD
0.1 3.3575 2.8414 1.0950 0.9623 8.8818 × 1 0 ‒ 16 8.8818 × 1 0 ‒ 16 2.0837 × 1 0 ‒ 16 2.0837 × 1 0 ‒ 16 3.3116 × 1 0 ‒ 15 3.6615 × 1 0 ‒ 15
0.2 3.2764 2.7190 1.4466 1.2817 8.8818 × 1 0 ‒ 16 8.8818 × 1 0 ‒ 16 1.9101 × 1 0 ‒ 16 1.9101 × 1 0 ‒ 16 4.6647 × 1 0 ‒ 15 4.7458 × 1 0 ‒ 15
0.3 3.1784 2.6885 1.0017 1.2230 8.8818 × 1 0 ‒ 16 8.8818 × 1 0 ‒ 16 1.7631 × 1 0 ‒ 16 1.7631 × 1 0 ‒ 16 6.1218 × 1 0 ‒ 15 6.0351 × 1 0 ‒ 15
0.4 3.2631 3.0666 1.0950 0.9623 1.7764 × 1 0 ‒ 15 2.6645 × 1 0 ‒ 15 3.2744 × 1 0 ‒ 16 4.9116 × 1 0 ‒ 16 8.3950 × 1 0 ‒ 15 8.0995 × 1 0 ‒ 15
0.5 3.2631 2.9472 1.0950 1.0843 8.8818 × 1 0 ‒ 16 2.6645 × 1 0 ‒ 15 1.5280 × 1 0 ‒ 16 4.5841 × 1 0 ‒ 16 1.1634 × 1 0 ‒ 14 1.0483 × 1 0 ‒ 14
0.6 3.2631 2.9472 1.0950 1.0843 1.7764 × 1 0 ‒ 15 2.6645 × 1 0 ‒ 15 2.8651 × 1 0 ‒ 16 4.2976 × 1 0 ‒ 16 1.6123 × 1 0 ‒ 14 1.4076 × 1 0 ‒ 14
0.7 3.3116 2.9472 1.5047 1.0843 8.8818 × 1 0 ‒ 16 1.7764 × 1 0 ‒ 15 1.3483 × 1 0 ‒ 16 2.6966 × 1 0 ‒ 16 2.4405 × 1 0 ‒ 14 1.8901 × 1 0 ‒ 14
0.8 3.3116 2.9472 1.5047 1.0843 2.6645 × 1 0 ‒ 15 3.5527 × 1 0 ‒ 15 3.8201 × 1 0 ‒ 16 5.0935 × 1 0 ‒ 16 3.3985 × 1 0 ‒ 14 2.5379 × 1 0 ‒ 14
0.9 3.2631 2.9472 1.0950 1.0843 1.7764 × 1 0 ‒ 15 5.3291 × 1 0 ‒ 15 2.4127 × 1 0 ‒ 16 7.2381 × 1 0 ‒ 16 4.2914 × 1 0 ‒ 14 3.4078 × 1 0 ‒ 14
1 3.2631 2.9472 1.0950 1.0843 2.6645 × 1 0 ‒ 15 6.2172 × 1 0 ‒ 15 3.4381 × 1 0 ‒ 16 8.0223 × 1 0 ‒ 16 5.9472 × 1 0 ‒ 14 4.5758 × 1 0 ‒ 14
Table 4

The AbS ( error ) , RlE ( error ) , and error est corresponding to Problem 2

M σ ς AbS ( error ) RlE ( error ) error est
ABD CFD ABD CFD ABD CFD ABD CFD ABD CFD
40 3.3361 2.1992 1.8698 1.4711 8.8818 × 1 0 ‒ 16 8.8818 × 1 0 ‒ 16 1.1460 × 1 0 ‒ 16 1.1460 × 1 0 ‒ 16 6.5568 × 1 0 ‒ 14 2.2671 × 1 0 ‒ 14
45 3.2776 2.2218 1.4159 0.9623 2.6645 × 1 0 ‒ 15 2.6645 × 1 0 ‒ 15 3.4381 × 1 0 ‒ 16 3.4381 × 1 0 ‒ 16 5.7867 × 1 0 ‒ 14 2.2004 × 1 0 ‒ 14
50 3.5237 2.2910 1.3375 1.2817 2.6645 × 1 0 ‒ 15 1.7764 × 1 0 ‒ 15 3.4381 × 1 0 ‒ 16 2.2921 × 1 0 ‒ 16 7.6392 × 1 0 ‒ 14 2.3616 × 1 0 ‒ 14
55 3.7631 2.7798 1.6532 1.5258 7.9936 × 1 0 ‒ 15 8.8818 × 1 0 ‒ 16 1.0314 × 1 0 ‒ 15 1.1460 × 1 0 ‒ 16 8.8526 × 1 0 ‒ 14 3.5755 × 1 0 ‒ 14
60 3.8559 2.7416 1.3935 1.1284 3.5527 × 1 0 ‒ 15 3.5527 × 1 0 ‒ 15 4.5841 × 1 0 ‒ 16 4.5841 × 1 0 ‒ 16 1.0943 × 1 0 ‒ 13 3.9962 × 1 0 ‒ 14
65 3.7631 2.8375 1.1383 0.8898 3.5527 × 1 0 ‒ 15 8.8818 × 1 0 ‒ 16 4.5841 × 1 0 ‒ 16 1.1460 × 1 0 ‒ 16 8.8963 × 1 0 ‒ 14 4.1297 × 1 0 ‒ 14
70 3.7116 2.8292 1.2984 0.7921 8.8818 × 1 0 ‒ 16 2.6645 × 1 0 ‒ 15 1.1460 × 1 0 ‒ 16 3.4381 × 1 0 ‒ 16 9.1073 × 1 0 ‒ 14 4.3553 × 1 0 ‒ 14
75 3.7854 2.8163 1.1004 0.9332 5.3291 × 1 0 ‒ 15 8.8818 × 1 0 ‒ 16 6.8762 × 1 0 ‒ 16 1.1460 × 1 0 ‒ 16 9.6980 × 1 0 ‒ 14 4.2486 × 1 0 ‒ 14
80 3.7244 2.7791 1.7039 0.7101 1.7764 × 1 0 ‒ 15 8.8818 × 1 0 ‒ 16 2.2921 × 1 0 ‒ 16 1.1460 × 1 0 ‒ 16 1.0826 × 1 0 ‒ 13 3.8936 × 1 0 ‒ 14
85 3.7146 2.6875 1.2202 0.7411 5.3291 × 1 0 ‒ 15 2.6645 × 1 0 ‒ 15 6.8762 × 1 0 ‒ 16 3.4381 × 1 0 ‒ 16 9.4354 × 1 0 ‒ 14 3.9239 × 1 0 ‒ 14
Figure 5 Exact solution vs Weeks solution of problem 2.
Figure 5

Exact solution vs Weeks solution of problem 2.

Figure 6 (a) The AbS ( error ) {\rm{AbS}}\left({\rm{error}}) , the RlE ( error ) {\rm{RlE}}\left({\rm{error}}) , and the error est {{\rm{error}}}_{{\rm{est}}} versus ξ \xi using M = 65 M=65 with ABD {\rm{ABD}} corresponding to problem 2. (b) The AbS ( error ) {\rm{AbS}}\left({\rm{error}}) , the RlE ( error ) {\rm{RlE}}\left({\rm{error}}) , and the error est {{\rm{error}}}_{{\rm{est}}} versus ξ \xi using M = 65 M=65 with CFD {\rm{CFD}} corresponding to problem 2.
Figure 6

(a) The AbS ( error ) , the RlE ( error ) , and the error est versus ξ using M = 65 with ABD corresponding to problem 2. (b) The AbS ( error ) , the RlE ( error ) , and the error est versus ξ using M = 65 with CFD corresponding to problem 2.

Figure 7 (a) The AbS ( error ) {\rm{AbS}}\left({\rm{error}}) , the RlE ( error ) {\rm{RlE}}\left({\rm{error}}) , and the error est {{\rm{error}}}_{{\rm{est}}} versus M M at ξ = 1 \xi =1 with ABD {\rm{ABD}} corresponding to problem 2. (b) The AbS ( error ) {\rm{AbS}}\left({\rm{error}}) , the RlE ( error ) {\rm{RlE}}\left({\rm{error}}) , and the error est {{\rm{error}}}_{{\rm{est}}} versus M M at ξ = 1 \xi =1 with CFD {\rm{CFD}} corresponding to problem 2.
Figure 7

(a) The AbS ( error ) , the RlE ( error ) , and the error est versus M at ξ = 1 with ABD corresponding to problem 2. (b) The AbS ( error ) , the RlE ( error ) , and the error est versus M at ξ = 1 with CFD corresponding to problem 2.

Problem 3

The third problem is solved using the Weeks method with ABD and CFD and exact solution W ( ξ ) = γ 2 ξ . In Table 5, the AbS ( error ) , the RlE ( error ) , and the error est of the proposed method for different values of ξ with M = 65 corresponding to Problem 2 are presented. Table 6 shows the the AbS ( error ) , the RlE ( error ) , and the error est obtained using the proposed method for different values of M at ξ = 1 corresponding to Problem 2. Figure 8(a) shows the plots of exact solution and Weeks solution. A comparison between the AbS ( error ) , the RlE ( error ) , and the error est of the proposed numerical scheme for problem 1 with ABD and CFD for different values of ξ with M = 65 is shown in Figure 9(a) and (b), respectively. Similarly, the comparison between the AbS ( error ) , the RlE ( error ) , and the error est of the proposed numerical scheme for problem 1 with ABD and CFD for different values of M at ξ = 1 using the proposed numerical scheme is shown in Figure 10(a) and (b), respectively. We can see that for this problem also, the method has produced very accurate and stable results.

Table 5

The AbS ( error ) , RlE ( error ) , and error est corresponding to Problem 3

ξ σ ς AbS ( error ) RlE ( error ) error est
ABD CFD ABD CFD ABD CFD ABD CFD ABD CFD
0.1 3.4836 3.2814 1.1134 1.2312 2.7756 × 1 0 ‒ 17 5.5511 × 1 0 ‒ 17 1.2336 × 1 0 ‒ 16 2.4672 × 1 0 ‒ 16 3.5605 × 1 0 ‒ 16 4.1059 × 1 0 ‒ 16
0.2 3.5572 3.3504 1.2628 1.3932 5.5511 × 1 0 ‒ 17 1.1102 × 1 0 ‒ 16 1.2336 × 1 0 ‒ 16 2.4672 × 1 0 ‒ 16 5.5464 × 1 0 ‒ 16 5.9594 × 1 0 ‒ 16
0.3 2.8325 3.1001 0.9141 1.2111 2.2204 × 1 0 ‒ 16 1.1102 × 1 0 ‒ 16 3.2895 × 1 0 ‒ 16 1.6448 × 1 0 ‒ 16 8.6165 × 1 0 ‒ 16 8.2063 × 1 0 ‒ 16
0.4 2.8325 3.3397 0.9141 1.1870 1.1102 × 1 0 ‒ 16 2.2204 × 1 0 ‒ 16 1.2336 × 1 0 ‒ 16 2.4672 × 1 0 ‒ 16 1.1438 × 1 0 ‒ 15 1.0626 × 1 0 ‒ 15
0.5 2.8180 2.3605 0.9564 0.7803 2.2204 × 1 0 ‒ 16 2.2204 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 1.6122 × 1 0 ‒ 15 1.2562 × 1 0 ‒ 15
0.6 2.4168 2.8056 0.9299 1.2600 4.4409 × 1 0 ‒ 16 2.2204 × 1 0 ‒ 16 3.2895 × 1 0 ‒ 16 1.6448 × 1 0 ‒ 16 1.9765 × 1 0 ‒ 15 1.9023 × 1 0 ‒ 15
0.7 2.8325 3.2416 0.9141 1.0171 2.2204 × 1 0 ‒ 16 2.2204 × 1 0 ‒ 16 1.4098 × 1 0 ‒ 16 1.4098 × 1 0 ‒ 16 2.6754 × 1 0 ‒ 15 2.8818 × 1 0 ‒ 15
0.8 2.8180 3.2416 0.9564 1.0171 4.4409 × 1 0 ‒ 16 2.2204 × 1 0 ‒ 16 2.4672 × 1 0 ‒ 16 1.2336 × 1 0 ‒ 16 3.7548 × 1 0 ‒ 15 3.9851 × 1 0 ‒ 15
0.9 2.8180 2.8348 0.9564 1.0144 4.4409 × 1 0 ‒ 16 4.4409 × 1 0 ‒ 16 2.1930 × 1 0 ‒ 16 2.1930 × 1 0 ‒ 16 4.9771 × 1 0 ‒ 15 4.5504 × 1 0 ‒ 15
1 2.4229 2.8348 0.7804 1.0144 4.4409 × 1 0 ‒ 16 4.4409 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 4.7645 × 1 0 ‒ 15 6.0418 × 1 0 ‒ 15
Table 6

The AbS ( error ) , RlE ( error ) , and error est corresponding to Problem 3

M σ ς AbS ( error ) RlE ( error ) error est
ABD CFD ABD CFD ABD CFD ABD CFD ABD CFD
40 2.7218 2.4075 1.8961 1.1591 4.4409 × 1 0 ‒ 16 4.4409 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 6.7274 × 1 0 ‒ 15 4.4139 × 1 0 ‒ 15
45 2.8798 2.7267 1.4047 1.3932 8.8818 × 1 0 ‒ 16 4.4409 × 1 0 ‒ 16 3.9475 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 6.2411 × 1 0 ‒ 15 5.3304 × 1 0 ‒ 15
50 2.9416 2.6065 1.5047 1.0828 4.4409 × 1 0 ‒ 16 4.4409 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 6.4160 × 1 0 ‒ 15 4.7877 × 1 0 ‒ 15
55 2.7754 3.0227 1.5047 1.1244 4.4409 × 1 0 ‒ 16 4.4409 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 7.1677 × 1 0 ‒ 15 6.1922 × 1 0 ‒ 15
60 2.9242 3.4810 1.0137 1.3214 4.4409 × 1 0 ‒ 16 4.4409 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 6.7855 × 1 0 ‒ 15 8.4881 × 1 0 ‒ 15
65 3.1167 3.0459 1.2628 1.3932 4.4409 × 1 0 ‒ 16 4.4409 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 7.5523 × 1 0 ‒ 15 6.8758 × 1 0 ‒ 15
70 3.1341 3.1361 0.9299 1.0091 8.8818 × 1 0 ‒ 16 8.8818 × 1 0 ‒ 16 3.9475 × 1 0 ‒ 16 3.9475 × 1 0 ‒ 16 6.9303 × 1 0 ‒ 15 6.5436 × 1 0 ‒ 15
75 3.1167 2.8493 0.9299 0.8020 4.4409 × 1 0 ‒ 16 4.4409 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 1.9737 × 1 0 ‒ 16 6.4809 × 1 0 ‒ 15 5.2802 × 1 0 ‒ 15
80 3.2249 3.2820 0.9299 0.8604 8.8818 × 1 0 ‒ 16 1.3323 × 1 0 ‒ 15 3.9475 × 1 0 ‒ 16 5.9212 × 1 0 ‒ 16 7.5662 × 1 0 ‒ 15 8.3731 × 1 0 ‒ 15
85 2.9590 3.1725 0.9299 0.8851 4.4409 × 1 0 ‒ 16 1.7764 × 1 0 ‒ 15 1.9737 × 1 0 ‒ 16 7.8949 × 1 0 ‒ 16 6.7840 × 1 0 ‒ 15 7.2423 × 1 0 ‒ 15
Figure 8 Exact solution vs Weeks solution of problem 3.
Figure 8

Exact solution vs Weeks solution of problem 3.

Figure 9 (a) The AbS ( error ) {\rm{AbS}}\left({\rm{error}}) , the RlE ( error ) {\rm{RlE}}\left({\rm{error}}) , and the error est {{\rm{error}}}_{{\rm{est}}} versus ξ \xi using M = 65 M=65 with ABD {\rm{ABD}} corresponding to problem 3. (b) The AbS ( error ) {\rm{AbS}}\left({\rm{error}}) , the RlE ( error ) {\rm{RlE}}\left({\rm{error}}) , and the error est {{\rm{error}}}_{{\rm{est}}} versus ξ \xi using M = 65 M=65 with CFD {\rm{CFD}} corresponding to problem 3.
Figure 9

(a) The AbS ( error ) , the RlE ( error ) , and the error est versus ξ using M = 65 with ABD corresponding to problem 3. (b) The AbS ( error ) , the RlE ( error ) , and the error est versus ξ using M = 65 with CFD corresponding to problem 3.

Figure 10 (a) The AbS ( error ) {\rm{AbS}}\left({\rm{error}}) , the RlE ( error ) {\rm{RlE}}\left({\rm{error}}) , and the error est {{\rm{error}}}_{{\rm{est}}} versus M M at ξ = 1 \xi =1 with ABD {\rm{ABD}} corresponding to problem 3. (b) The AbS ( error ) {\rm{AbS}}\left({\rm{error}}) , the RlE ( error ) {\rm{RlE}}\left({\rm{error}}) , and the error est {{\rm{error}}}_{{\rm{est}}} versus M M at ξ = 1 \xi =1 with CFD {\rm{CFD}} corresponding to problem 3.
Figure 10

(a) The AbS ( error ) , the RlE ( error ) , and the error est versus M at ξ = 1 with ABD corresponding to problem 3. (b) The AbS ( error ) , the RlE ( error ) , and the error est versus M at ξ = 1 with CFD corresponding to problem 3.

5 Conclusion

The considered scheme has been applied for nonlocal and nonsingular fractional-order problems involving CFD and ABD. The proposed scheme has the ability to avoid discretization and complex calculation to approximate various problems of fractional orders. The computational cost is low as compared to other numerical or analytic methods for the considered problems. Our conclusion from this study is that: (i) This method provides a stable and accurate approach to the numerical calculation of the solutions to the considered BTEs, (ii) the method is highly sensitive to a proper choice for the two free parameters in the Laguerre expansion, and (iii) The proposed scheme is very easy to implement. In the future work, our aim is to use the proposed scheme coupled with some spatial discretization methods for numerical modeling of time-fractional PDEs.

Acknowledgments

Aiman Mukheimer, Kamal Shah, and Thabet Abdeljawad are thankful to Prince Sultan University for paying the APC and support through the TAS research lab.

  1. Funding information: Aiman Mukheimer, Kamal Shah, and Thabet Abdeljawad are thankful to Prince Sultan University for paying the APC and support through the TAS research lab.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2023-08-17
Revised: 2023-10-19
Accepted: 2023-11-12
Published Online: 2024-01-02

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  1. Regular Articles
  2. Numerical study of flow and heat transfer in the channel of panel-type radiator with semi-detached inclined trapezoidal wing vortex generators
  3. Homogeneous–heterogeneous reactions in the colloidal investigation of Casson fluid
  4. High-speed mid-infrared Mach–Zehnder electro-optical modulators in lithium niobate thin film on sapphire
  5. Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative
  6. Mononuclear nanofluids undergoing convective heating across a stretching sheet and undergoing MHD flow in three dimensions: Potential industrial applications
  7. Heat transfer characteristics of cobalt ferrite nanoparticles scattered in sodium alginate-based non-Newtonian nanofluid over a stretching/shrinking horizontal plane surface
  8. The electrically conducting water-based nanofluid flow containing titanium and aluminum alloys over a rotating disk surface with nonlinear thermal radiation: A numerical analysis
  9. Growth, characterization, and anti-bacterial activity of l-methionine supplemented with sulphamic acid single crystals
  10. A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium
  11. Optoelectronic–thermomagnetic effect of a microelongated non-local rotating semiconductor heated by pulsed laser with varying thermal conductivity
  12. Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder
  13. Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field
  14. Numerical analysis of thermophoretic particle deposition on 3D Casson nanofluid: Artificial neural networks-based Levenberg–Marquardt algorithm
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  19. Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations
  20. Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model
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https://doi.org/10.1515/phys-2023-0161
Keywords for this article
Laplace transform; Bagley–Torvik equation; Caputo–Fabrizio derivative; Atangana–Baleanu derivative; numerical inversion; Weeks method; Laguerre polynomials
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Numerical study of flow and heat transfer in the channel of panel-type radiator with semi-detached inclined trapezoidal wing vortex generators
  3. Homogeneous–heterogeneous reactions in the colloidal investigation of Casson fluid
  4. High-speed mid-infrared Mach–Zehnder electro-optical modulators in lithium niobate thin film on sapphire
  5. Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative
  6. Mononuclear nanofluids undergoing convective heating across a stretching sheet and undergoing MHD flow in three dimensions: Potential industrial applications
  7. Heat transfer characteristics of cobalt ferrite nanoparticles scattered in sodium alginate-based non-Newtonian nanofluid over a stretching/shrinking horizontal plane surface
  8. The electrically conducting water-based nanofluid flow containing titanium and aluminum alloys over a rotating disk surface with nonlinear thermal radiation: A numerical analysis
  9. Growth, characterization, and anti-bacterial activity of l-methionine supplemented with sulphamic acid single crystals
  10. A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium
  11. Optoelectronic–thermomagnetic effect of a microelongated non-local rotating semiconductor heated by pulsed laser with varying thermal conductivity
  12. Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder
  13. Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field
  14. Numerical analysis of thermophoretic particle deposition on 3D Casson nanofluid: Artificial neural networks-based Levenberg–Marquardt algorithm
  15. Analyzing fuzzy fractional Degasperis–Procesi and Camassa–Holm equations with the Atangana–Baleanu operator
  16. Bayesian estimation of equipment reliability with normal-type life distribution based on multiple batch tests
  17. Chaotic control problem of BEC system based on Hartree–Fock mean field theory
  18. Optimized framework numerical solution for swirling hybrid nanofluid flow with silver/gold nanoparticles on a stretching cylinder with heat source/sink and reactive agents
  19. Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations
  20. Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model
  21. Analysis of the heat transfer enhancement in water-based micropolar hybrid nanofluid flow over a vertical flat surface
  22. Microscopic seepage simulation of gas and water in shale pores and slits based on VOF
  23. Model of conversion of flow from confined to unconfined aquifers with stochastic approach
  24. Study of fractional variable-order lymphatic filariasis infection model
  25. Soliton, quasi-soliton, and their interaction solutions of a nonlinear (2 + 1)-dimensional ZK–mZK–BBM equation for gravity waves
  26. Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering
  27. Nonlinear fractional-order differential equations: New closed-form traveling-wave solutions
  28. Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo–Christov model
  29. Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function
  30. Magnetohydrodynamic water-based hybrid nanofluid flow comprising diamond and copper nanoparticles on a stretching sheet with slips constraints
  31. Super-resolution reconstruction method of the optical synthetic aperture image using generative adversarial network
  32. A two-stage framework for predicting the remaining useful life of bearings
  33. Influence of variable fluid properties on mixed convective Darcy–Forchheimer flow relation over a surface with Soret and Dufour spectacle
  34. Inclined surface mixed convection flow of viscous fluid with porous medium and Soret effects
  35. Exact solutions to vorticity of the fractional nonuniform Poiseuille flows
  36. In silico modified UV spectrophotometric approaches to resolve overlapped spectra for quality control of rosuvastatin and teneligliptin formulation
  37. Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems
  38. Substituent effect on the electronic and optical properties of newly designed pyrrole derivatives using density functional theory
  39. A comparative analysis of shielding effectiveness in glass and concrete containers
  40. Numerical analysis of the MHD Williamson nanofluid flow over a nonlinear stretching sheet through a Darcy porous medium: Modeling and simulation
  41. Analytical and numerical investigation for viscoelastic fluid with heat transfer analysis during rollover-web coating phenomena
  42. Influence of variable viscosity on existing sheet thickness in the calendering of non-isothermal viscoelastic materials
  43. Analysis of nonlinear fractional-order Fisher equation using two reliable techniques
  44. Comparison of plan quality and robustness using VMAT and IMRT for breast cancer
  45. Radiative nanofluid flow over a slender stretching Riga plate under the impact of exponential heat source/sink
  46. Numerical investigation of acoustic streaming vortices in cylindrical tube arrays
  47. Numerical study of blood-based MHD tangent hyperbolic hybrid nanofluid flow over a permeable stretching sheet with variable thermal conductivity and cross-diffusion
  48. Fractional view analytical analysis of generalized regularized long wave equation
  49. Dynamic simulation of non-Newtonian boundary layer flow: An enhanced exponential time integrator approach with spatially and temporally variable heat sources
  50. Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source
  51. Estimation of monotone α-quantile of past lifetime function with application
  52. Numerical simulation for the slip impacts on the radiative nanofluid flow over a stretched surface with nonuniform heat generation and viscous dissipation
  53. Study of fractional telegraph equation via Shehu homotopy perturbation method
  54. An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion
  55. Establishing breather and N-soliton solutions for conformable Klein–Gordon equation
  56. An electro-optic half subtractor from a silicon-based hybrid surface plasmon polariton waveguide
  57. CFD analysis of particle shape and Reynolds number on heat transfer characteristics of nanofluid in heated tube
  58. Abundant exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation
  59. A short report on a probability-based interpretation of quantum mechanics
  60. Study on cavitation and pulsation characteristics of a novel rotor-radial groove hydrodynamic cavitation reactor
  61. Optimizing heat transport in a permeable cavity with an isothermal solid block: Influence of nanoparticles volume fraction and wall velocity ratio
  62. Linear instability of the vertical throughflow in a porous layer saturated by a power-law fluid with variable gravity effect
  63. Thermal analysis of generalized Cattaneo–Christov theories in Burgers nanofluid in the presence of thermo-diffusion effects and variable thermal conductivity
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  67. Thermal analysis of extended surfaces using deep neural networks
  68. Steady-state thermodynamic process in multilayered heterogeneous cylinder
  69. Multiresponse optimisation and process capability analysis of chemical vapour jet machining for the acrylonitrile butadiene styrene polymer: Unveiling the morphology
  70. Modeling monkeypox virus transmission: Stability analysis and comparison of analytical techniques
  71. Fourier spectral method for the fractional-in-space coupled Whitham–Broer–Kaup equations on unbounded domain
  72. The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model
  73. Research on optimization of combustor liner structure based on arc-shaped slot hole
  74. Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method
  75. Effectiveness of microwave ablation using two simultaneous antennas for liver malignancy treatment
  76. Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives
  77. Reliability of two-dimensional steady magnetized Jeffery fluid over shrinking sheet with chemical effect
  78. Generalized model of thermoelasticity associated with fractional time-derivative operators and its applications to non-simple elastic materials
  79. Migration of two rigid spheres translating within an infinite couple stress fluid under the impact of magnetic field
  80. A comparative investigation of neutron and gamma radiation interaction properties of zircaloy-2 and zircaloy-4 with consideration of mechanical properties
  81. New optical stochastic solutions for the Schrödinger equation with multiplicative Wiener process/random variable coefficients using two different methods
  82. Physical aspects of quantile residual lifetime sequence
  83. Synthesis, structure, I–V characteristics, and optical properties of chromium oxide thin films for optoelectronic applications
  84. Smart mathematically filtered UV spectroscopic methods for quality assurance of rosuvastatin and valsartan from formulation
  85. A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform
  86. Homotopic dynamic solution of hydrodynamic nonlinear natural convection containing superhydrophobicity and isothermally heated parallel plate with hybrid nanoparticles
  87. A novel tetra hybrid bio-nanofluid model with stenosed artery
  88. Propagation of traveling wave solution of the strain wave equation in microcrystalline materials
  89. Innovative analysis to the time-fractional q-deformed tanh-Gordon equation via modified double Laplace transform method
  90. A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
  91. New soliton solutions of the conformable time fractional Drinfel'd–Sokolov–Wilson equation based on the complete discriminant system method
  92. Irradiation of hydrophilic acrylic intraocular lenses by a 365 nm UV lamp
  93. Inflation and the principle of equivalence
  94. The use of a supercontinuum light source for the characterization of passive fiber optic components
  95. Optical solitons to the fractional Kundu–Mukherjee–Naskar equation with time-dependent coefficients
  96. A promising photocathode for green hydrogen generation from sanitation water without external sacrificing agent: silver-silver oxide/poly(1H-pyrrole) dendritic nanocomposite seeded on poly-1H pyrrole film
  97. Photon balance in the fiber laser model
  98. Propagation of optical spatial solitons in nematic liquid crystals with quadruple power law of nonlinearity appears in fluid mechanics
  99. Theoretical investigation and sensitivity analysis of non-Newtonian fluid during roll coating process by response surface methodology
  100. Utilizing slip conditions on transport phenomena of heat energy with dust and tiny nanoparticles over a wedge
  101. Bismuthyl chloride/poly(m-toluidine) nanocomposite seeded on poly-1H pyrrole: Photocathode for green hydrogen generation
  102. Infrared thermography based fault diagnosis of diesel engines using convolutional neural network and image enhancement
  103. On some solitary wave solutions of the Estevez--Mansfield--Clarkson equation with conformable fractional derivatives in time
  104. Impact of permeability and fluid parameters in couple stress media on rotating eccentric spheres
  105. Review Article
  106. Transformer-based intelligent fault diagnosis methods of mechanical equipment: A survey
  107. Special Issue on Predicting pattern alterations in nature - Part II
  108. A comparative study of Bagley–Torvik equation under nonsingular kernel derivatives using Weeks method
  109. On the existence and numerical simulation of Cholera epidemic model
  110. Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions
  111. Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks
  112. Prediction of COVID-19 spread with models in different patterns: A case study of Russia
  113. Study of chronic myeloid leukemia with T-cell under fractal-fractional order model
  114. Accumulation process in the environment for a generalized mass transport system
  115. Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications
  116. Special Issue on Nanomaterial utilization and structural optimization - Part II
  117. Numerical study on flow and heat transfer performance of a spiral-wound heat exchanger for natural gas
  118. Study of ultrasonic influence on heat transfer and resistance performance of round tube with twisted belt
  119. Numerical study on bionic airfoil fins used in printed circuit plate heat exchanger
  120. Improving heat transfer efficiency via optimization and sensitivity assessment in hybrid nanofluid flow with variable magnetism using the Yamada–Ota model
  121. Special Issue on Nanofluids: Synthesis, Characterization, and Applications
  122. Exact solutions of a class of generalized nanofluidic models
  123. Stability enhancement of Al2O3, ZnO, and TiO2 binary nanofluids for heat transfer applications
  124. Thermal transport energy performance on tangent hyperbolic hybrid nanofluids and their implementation in concentrated solar aircraft wings
  125. Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method
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