Numerical investigation of the dispersive long wave equation using an adaptive moving mesh method and its stability

Journal Pre-proofs Numerical investigation of the Dispersive Long Wave Equation using an adaptive moving mesh method and its stability Abdulghani Alharbi, M.B. Almatrafi PII: DOI: Reference:

S2211-3797(19)33347-9 https://doi.org/10.1016/j.rinp.2019.102870 RINP 102870

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Results in Physics

Received Date: Accepted Date:

12 November 2019 8 December 2019

Please cite this article as: Alharbi, A., Almatrafi, M.B., Numerical investigation of the Dispersive Long Wave Equation using an adaptive moving mesh method and its stability, Results in Physics (2019), doi: https://doi.org/ 10.1016/j.rinp.2019.102870

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Numerical investigation of the Dispersive Long Wave Equation using an adaptive moving mesh method and its stability Abdulghani Alharbi and M. B. Almatrafi Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

E-mails: [email protected], [email protected]

Abstract This paper investigates the exact and numerical solutions of the dispersive long wave equations (DLWE) using the extended Jacobian elliptic function expansion method (EJEFET) and the adaptive moving mesh method (AMMT). The uniform mesh method (UMM) is also applied to obtain some numerical results. The analysis of the stability of these solutions is also presented. The achieved solutions are stable. We compare both numerical and exact solutions and obtained that the numerical results are accurate. Some 3D figures are plotted to reinforce and ensure the integrity of the qualitative and quantitative results obtained in this work. Finally, L2 and L∞ errors are analyzed to discover the proper performance of the considered techniques in solving DLWE.

Keywords: Dispersive long wave equation, extended Jacobian elliptic function expansion method, adaptive moving mesh method, MMPDEs. AMS subject classifications: 35A20, 35A99, 83C15, 65L12, 65L50, 65M50, 65N06, 65N50, 35Q51, 65Z05.

1

Introduction

Nonlinear partial differential equations (NLPDEs) are often used to model a massive number of natural phenomena, such as physical and biological phenomena, see [1, 2, 3, 4, 6, 7, 8, 9, 10]. Some empirical parameters may appear in some models of physical phenomena. Hence, it is important sometimes to perform several experiments to find appropriate values for these parameters. The exact solutions of these problems play a significant role in predicting the future behaviour of these models and demonstrating their stability [5, 11]. Constructing the exact solutions of NLPDEs has been extensively considered in recent years. Some researchers devolved and invented some powerful and effective methods, such as Darboux transformation, Cole-Hopf transformation, tanh-sech method [12, 13], Jacobi elliptic function method [14, 15], exp-function method [16, 17], homogeneous balance method [18, 19], variable separation approach, the Riccati-Bernoulli sub-ODE method [20, 21], sine-cosine method [22, 0

23], F-expansion method [24, 25], extended tanh-method [26, 27], Painleve method and ( GG )− expansion method [28, 29], for discovering the exact solutions for some PDEs. 1

The exact solution of the DLWE is investigated in this paper by using the EJEFET whose algorithm is described later. The (1 + 1)-dimensional DLWE [30] is given by the following form: ut + uux + gx = 0, 1 gt + (gu)x + uxxx = 0, 3

(1.1)

where u represents the surface velocity of water along x-direction and g represents the amplitude of surface waves. The AMMT [31, 32, 33], which is executed with a monitor function and moving mesh PDEs (MMPDEs), is applied here to show the numerical results of Eq. (1.1). This method is valid to use in a general spatial dimension, but we apply it for 1D in this paper. Sometimes, employing a large spatial domain, the boundary effects will be reduced. However, large spatial domains require an enormous number of grid elements to reach an appropriate level of computational accuracy. It can simply achieve by using the AMMT, which dynamically adapts and redistributes the points to the locations with high errors. The so-called MMPDEs contributes to moving the grid continuously in time and orderly in space by employing a PDE described later. When a large curvature or a dramatic variation occurs in the solution, the monitor function changes to reduce the error of this dramatic change. It is important to mention that this technique mostly gives successful results when a suitable monitor function is selected. Finally, this work also presents the discretization of the space-time mixed derivatives which is accomplished employing the finite differences. The most significant purpose of this work is to utilize the EJEFET and the AMMT in determining the exact solution and the numerical solution of Eq. (1.1), respectively. Through this article, we depict some figures to illustrate the behavior of the exact and numerical results of Eq. (1.1) and to demonstrate that they almost behave the same. We also present L2 and L∞ errors to predict the reliable numerical technique in solving the DLWE. In terms of accuracy, we present a comparison between the performance of the uniform mesh and adaptive moving mesh methods. This article also demonstrates that the proposed numerical methods can also be applied for various NLPDEs that have large variations on their solutions. The outline of this article is given as follows. Section 2 is devoted to providing a brief description of the EJEFET. The exact solution is analyzed in section 4, while the numerical solution obtained by using UMM is explained in subsection 5.1. Subsection 5.2 presents the AMMT and how we apply it to find the numerical solution of DLWE. Finally, Section 6 is added to summarize the significant results that appeared in this article.

2

The extended Jacobian elliptic function expansion method

This section is devoted to briefly describe the EJEFET (as illustrated in [34]). Let H(ψ, ψt , ψx , ψtt , ψxx , ....) = 0,

(2.1)

be a given PDE, where H is a general nonlinear evolution equation with some physical fields ψi (x, t) in two variables x, t.

2

Step 1. Applying the new wave transformation ψ = ψ(η), η = k(x − ct),

(2.2)

where k and c are the wave number and wave speed, respectively, Eq. (2.1) reduces to the following ordinary differential equation (ODE): G(ψ, ψ 0 , ψ 00 , ψ 000 , .....) = 0.

(2.3)

Here, the primes present the derivatives with respect to η. Step2. The Jacobian elliptic method gives the exact solution of Eq. (2.3) on the form: ψ(η) = a0 +

N X

fij−1 (η) [aj fi (η) + bj gi (η)] , i = 1, 2, 3, ...,

(2.4)

j=1

with f1 (η) = snη,

g1 (η) = cnη,

f2 (η) = snη,

g2 (η) = dnη,

f3 (η) = nsη,

g3 (η) = csη,

f4 (η) = nsη,

g4 (η) = dsη,

f5 (η) = scη,

g5 (η) = ncη,

f6 (η) = sdη,

g6 (η) = ndη,

(2.5)

where snη, cnη, dnη are the Jacobian elliptic sine function, the jacobian elliptic cosine function and the EJEFET of the third kind. Other functions, presented by Glaishers symbols, can be determined from the previous functions as follows: nsη =

1 1 1 cnη , ncη = , ndη = , scη = , snη cnη dnη snη snη dnη snη csη = , dsη = , sdη = . cnη snη dnη

(2.6)

Eqs. (2.6) can be also expressed on the forms: sn2 η + cn2 η = 1, dn2 η + m2 sn2 η = 1, ns2 η = 1 + cs2 η, ns2 η = m2 + ds2 η, sc2 η + 1 = nc2 η, m2 sd2 + 1 = nd2 η,

(2.7)

with 0 < m < 1. The derivatives of the above-mentioned functions are given by sn0 η = cnηdnη, cn0 η = −sn ηdn η, dn0 η = −m2 sn ηcn η.

(2.8)

ns0 η = −ds ηcs η, ds0 η = −cs ηns η, cs0 η = −ns ηds η.

(2.9)

sc0 η = nc ηdc η, nc0 η = sc ηdc η, cd0 η = cd ηnd η, nd0 η = m2 sd ηcd η.

(2.10)

3

The parameter N can be evaluated by balancing the highest order linear term(s) with the nonlinear one(s). Next, substituting Eq. (2.4) into Eq. (2.3) and comparing the coefficients of each power in both sides lead to a system of algebraic equations that can be straightforwardly solved by Mathematica (or another software). The periodic solutions with EJEFET is obtained when m → 1. On other words, as m → 1, snη, cnη, and dnη tends to tanhη, sechη, sechη, respectively. That is ψ(η) = a0 +

N X

tanhj−1 (η) [aj tanh(η) + bj sech(η)] ,

(2.11)

j=1

ψ(η) = a0 +

N X

cothj−1 (η) [aj coth(η) + bj coth(η)] ,

(2.12)

j=1

ψ(η) = a0 +

N X

tanj−1 (η) [aj tan(η) + bj sec(η)] ,

(2.13)

j=1

ψ(η) = a0 +

N X

cotj−1 (η) [aj cot(η) + bj csc(η)] .

(2.14)

j=1

It should be remarked that the above-mentioned technique is more powerful than the ones proposed in [35, 36, 37]. More information about the extended EJEFET method can be found in [34].

3

Stability analysis

The Hamiltonian system which shows the evolution of a given physical system is presented in this section. The most popular form of the Hamiltonian system is given by   Z 0 Z r1 1 2 2 P (c) = lim ψ(η) dη + lim ψ(η) dη . (3.1) r1 →∞ 0 2 r0 →−∞ r0 Where P (c) refers to as the momentum and ψ(η) is a solution to the differential equation Eq. (2.4). The sufficient condition of the stability is expressed as follows: ∂P (c) > 0. ∂c

(3.2)

Eq. (3.1) and Eq. (3.2) are employed to show specific intervals in which the travelling wave solutions of the DLWE is stable.

4

The exact solution of Eq. (1.1)

Through this subsection, we analyze the exact solution of Eq. (1.1) using the EJEFET. We first begin with introducing a new variable on the form: u(x, t) = ρ(η),

g(x, t) = v(η), 4

η = k(x + αt),

(4.1)

where α is a constant. Eqs. (1.1) now reduce into the ODEs k kαρη + (ρ2 )η + kvη = 0, 2

(4.2)

k3 ρηηη = 0, (4.3) 3 Taking the first integral to Eq. (4.2) and Eq. (4.3) concerning η and let the integral constant be equal to zero, yield v = −ρ(α + 0.5ρ) (4.4) kαvη + k(vρ)η +

αv + vρ +

k2 ρηη = 0. 3

(4.5)

Inserting Eq. (4.4) into Eq. (4.5), we obtain − ρ(α + ρ)(α + 0.5ρ) +

k2 ρηη = 0. 3

(4.6)

Balancing the order of ρηη with the exponent of ρ3 , gives N = 1. Hence, the extended EJEFET method expresses the solution as follows: ρ = a0 + a1 sn(η) + b1 cn(η),

(4.7)

where a0 , a1 and b1 are constants determined later. From Eq. (4.7) we have ρη = a1 cn(η) dn(η) − b1 sn(η) dn(η), ρηη = −m2 sn(η) a1 + 2 a1 sn(η)3 m2 + 2 m2 sn(η)2 cn(η) b1 − a1 sn(η) − b1 cn(η).

(4.8) (4.9)

Substituting equations (4.7) and (4.9) into equation (4.6) and equating the whole coefficients of sn3 , sn2 cn, sn2 , sncn, sn, cn, sn0 to zero, we respectively obtain a31 = 0, 2 2 2 2 −3/2 a0 a1 + 3/2 a0 b1 − 3/2 a1 α + 3/2b21 α = 0, −(3/2)a21 b1 + 1/2 b31 + 2/3 b1 k 2 m2 = 0, −(3/2) a20 a1 − 3/2 a1 b21 − 1/3 a1 k 2 − 1/3a1 k 2 m2 − 3 a0 a1 α − a1 α2 = 0, −3 a0 a1 b1 − 3 a1 b1 α = 0, −(3/2) a20 b1 − (b31 )/2 − 1/3 b1 k 2 − 3 a0 b1 α − b1 α2 = 0, −(a30 /2) − (3 a0 b21 )/2 − (3 a20 α)/2 − (3 b21 α)/2 − a0 α2 = 0. 2/3 a1 k 2 m2 + 3/2a1 b21 −

(4.10)

Solving the above system of equations yields different cases described on the next paragraph. r r 2p 2 2 2km 2p 2 2 2 a0 → ± k m + k , a1 → ± √ , b1 → 0, α → ± k (m + 1). 3 3 3 Then, the first family of equations is given by u1 (x, t) = a0 + a1 sn(k(x + αt)). As long as m → 1, Eq. (4.11) is written by 5

(4.11)

2k 2k 2k u1 (x, t) = ± √ ± √ tanh (k(x − x0 ∓ √ t)), 3 3 3 2k 1 2 g1 (x, t) = ± √ u1 (x, t) − u1 (x, t) . 2 3

(4.12)

Figure 1: Figures (a) and (b) illustrate the time evolution for u(x, t) and g(x, t) presented in Eq. (4.12), respectively. The considered constants are taken as k = 2.5 and x0 = 50. The obtained travelling wave solutions (Eqs. (4.12)) are shown in figures 1, with k = 2.5 and x0 = 50 in the interval [0, 60]. According to the conditions of stability, the travelling wave solutions (Eqs. (4.12)) are stable in the interval [0, 60].

5

Numerical results of Eq. (1.1)

To explore the numerical results of the proposed equation, the spatial derivatives are semidiscretized by using finite differences while the temporal derivative is kept continuous. It converts

6

the proposed equation to a system of PDEs given by  2 u ut + + gx = 0, 2 x 1 gt + uxxx + (g u)x = 0, 3 (x, t) ∈ [0, b] × [0, Te ].

(5.1)

Here, b and Te represent the physical domain’s length and a particular time, respectively. Since the AMMT and the UMM are applied to determine the numerical results, initial and boundary conditions are required for these methods. The initial condition is given by 2k 2k u1 (x, 0) = − √ − √ tanh (k (x − x0 )), 3 3 2k g1 (x, 0) = −u1 ( √ + 0.5u1 ), 3

(5.2)

where k, x0 are constants. The boundary conditions are deduced from Figures (a) and (b) in 1. As can be seen, the analytic solution vanishes at the end points of the physical domain. This implies that gx = ux = 0 and uxxx = 0 as x → ±∞. Hence, the boundary conditions are written as gx (0, t) = 0, gx (b, t) = 0, ∀t ∈ [0, Te ], (5.3) ux (0, t) = uxxx (0, t) = 0, ux (b, t) = uxxx (b, t) = 0.

5.1

Applying a fixed mesh method on Eqs. (5.1)

This method works on an equal subintervals of the domain. On other words, the spatial domain is split into Nx + 1 nodes where the distances between these nodes are fixed. That is xi = (i − 1)b/Nx = (i − 1)∆x, where 1 ≤ i ≤ Nx + 1. Next, the centred finite differences is employed here to discretise Eq.(5.1). Hence, 1 1 (u2 − u2i−1/2 ) + (g − gi−1/2 ) = 0, 2∆x i+1/2 ∆x i+1/2 1 (uxx )j+1/2 − (uxx )j−1/2 1 (5.4) + ((u g)i+1/2 − (u g)i−1/2 ) = 0, gt,i + 3 ∆x ∆x 1 uxx,i = (ui+1 − 2ui + ui−1 ), ∆x2 where i = 2, 3, ..., Nx . We evaluate terms indicated by uxx,i+1/2 applying spatial averages: uxx,i+1/2 = (uxx,i+1 + uxx,i )/2. Similar evaluations are taken to terms indicated by u2i+1/2 , ui+1/2 and gi+1/2 . Also, we use the same manner to approximate the terms indicated by uxx,i−1/2 , u2i−1/2 , ui−1/2 and gi−1/2 . We also use ut,1 = gt,1 = 0 and ut,N +1 = gt,N +1 = 0, to replace the boundary conditions given in Eq.(5.3), and the initial condition is chosen to be Eq. (5.2). ut,i +

5.2

Applying an adaptive mesh on Eqs. (5.1)

Another method called adaptive moving mesh is utilized in this subsection to solve Eq. (5.1) numerically with the boundary conditions given by Eq. (5.3) and the initial condition illustrated in Eqs. (5.2). We now turn to a new transform which is x = x(η, t) : [0, 1] → [0, b], t > 0, to

7

execute this method. Here, the spatial and computational coordinates are denoted by x and η, respectively. Accordingly, it can be noticed that u(x, t) = u(x(η, t), t),

g(x, t) = g(x(η, t), t).

(5.5)

The moving mesh which is corresponding to the solution of Eq. (5.5) can be now presented on the form xj (η) = x(ηj , t), j = 1, 2, · · · , Nx + 1, with x1 = 0, xNx +1 = b. Furthermore, the uniform grids is illustrated by ηj = (j − 1) Nbx , j = 1, 2, · · · , Nx + 1. When a chain rule is applied to Eq.(5.1), Eq.(5.1) is converted into  ut −  gt −

uη xη



gη xη



u2η xη

1 xt + 2 xt +

!

 gη + = 0, xη   !   (u g)η 1 uη + = 0. xη xη η xη

1 3xη



(5.6) (5.7)

η

A new mesh is obtained by considering the convergence and the error of the proposed numerical technique for some MMPDEs and monitor functions. The considered MMPDEs [31, 38, 39, 40] give accurate and reliable results. We select the most common one, which is MMPDE7. A list of MMPDEs are given by MMPDE2: σ (φxt,η )η = − (φxη )η .

MMPDE7: σxt =

(5.8)

1 (φxη )η . φ

(5.9)

MMPDE5: σ (1 − µ ∂ηη ) xt =

1 (φ xη )η . φ

(5.10)

Where σ, µ are constants, φ(x, t) is called a monitor function. As mentioned previously, different monitor functions can be used. However, we employ the most reliable ones which can be written as s The arc-length: φ(x, t) =

1 + a1

s The curvature: φ(x, t) =

gη2 u2η + a . 2 x2η x2η

1 1 + a1 2 xη



gη xη

2

1 + a1 2 xη η

(5.11)



uη xη

2 .

(5.12)

η

Here, a1 , and a2 are constants. The monitor function always provides more points to the regions that the solution has rapid changes and fewer points elsewhere. In all the numerical results shown in this paper, the parameter values of the PDEs and MMPDEs are fixed by a1 = 10, a2 = 10, k = 3.5, and x0 = 5. The reader can refer to ( [38, 39, 40, 41, 42]). Here, φ(x, t) denotes a monitor function, φj+1/2 = (φj+1 +φj )/2, φj−1/2 = (φj +φj−1 )/2 and σ ∈ (0, 1). 8

The full discretizations of MMPDE7 (Eqs. (5.9)), we use here, and the system Eqs. (5.7, 5.6) are given by  m+1  h

i x − xm 1 m+1 m+1 m+1 m+1 m m (5.13) = φ x − x − φ x − x , i+1/2 i−1/2 i+1 i i i−1 2 ∆tm σφm i ∆η i !  m+1   m −g m (ˆ um um gˆi+1 ˆi−1 gˆ − gˆm xm+1 − xm xx )i+1 − (ˆ xx )i = − m+1 m+1 m+1 m+1 ∆tm ∆tm xi+1 − xi−1 xi+2 − xi−1 i i ! m m m m (ˆ u gˆ )i+1/2 − (ˆ u gˆ )i−1/2 −2 , (5.14) m+1 xi+1 − xm+1 i−1 ! u ˆm ˆm u ˆm ˆm 2 i+1 − u i i −u i−1 (ˆ um ) = − , xx i m+1 m+1 m+1 m+1 m+1 xm+1 − x x − x x − x i+1 i−1 i+1 i i i−1 ! !  m+1   m 2 )m − (ˆ 2 )m m+1 − xm u ˆm − u ˆ (ˆ u u u ˆ −u ˆm x 1 i+1 i−1 i+1 i−1 = − m+1 m+1 m+1 m+1 ∆tm ∆t 2 x − x x − x m i i i+1 i−1 i+1 i−1 ! (5.15) m −g m gˆi+1 ˆi−1 − , i = 2, 3, . . . , Nx , m+1 xm+1 i+1 − xi−1 where ∆tm is the temporal step size and ∆η = b/Nx is the computational step size. The boundary conditions of the system Eq. (5.13) are x(0, t) = 0 and x(b, t) = b and the initial condition is taken by xj = (j − 1)

b , Nx

j = 1, 2, . . . , Nx + 1

(5.16)

The boundary conditions of the systems Eq. (5.14) and Eq. (5.15) are given by g0 = g2 , u0 = u2 , uxx,0 = uxx,2 ,

gNx = gNx +1 , ∀t ∈ [0, Te ], uNx = uNx +2 , uxx,Nx = uxx,Nx +2 .

(5.17)

The procedure of using adaptive moving mesh method for investigating the numerical results of Eqs. (5.1) is given as follows: • The monitor function φ(x, t) is computed using the current u(x, t), g(x, t) and x at t = tm . • Fixing the monitor function and solving MMPDE7 (Eq. (5.13)) yields a new mesh given by xm+1 at the time t = tm + ∆tm . m m m m • Using linear interpolation [31] to interpolate the current solutions  − − u(x  , t ), g(x , t ) on + m+1 m m x to have interpolated u ˆ , gˆ . For example, if xi ∈ xn , xn+1

gi+ =

+ − x− x+ n+1 − xi − − i − xn g + − n+1 − gn , x− x− n+1 − xn n+1 − xn

(5.18)

and u+ i is interpolated in the same manner. • Solve Eqs. (5.14) and (5.15) by starting from u ˆm , gˆm and xm+1 to investigate new solum tions for u(x, t), g(x, t) at the time t = t + ∆tm .

9

• Repeat the steps from the first step. As can be observed in Figure 2, the numerical solutions of u(x, t) and g(x, t) agree with the obtained exact solution. This ensures that the obtained results are highly accurate. In Figure 3, we compare the implement of the used numerical techniques with the analytical solution. For instance, Figure 3 (a) shows that the results of AMMT are accurate. However, the results obtained by the UMM are not completely accurate, especially, in regions with huge curvatures. Therefore, it can be noted that the AMMT is more reliable and successful to be applied in solving such an equation. As time increases from 0 to 10 the AMMT redistributes the mesh over the regions with a dramatic change to reduce the resultant error. The distribution or the movement of the nodes is presented in Figure 4. For example, the fixed mesh is impossible to be increased or decreased and thus the distribution of the points depends on the initial location of such nodes. Since there is no change in the mesh, the mesh appears as a straight line as shown in Figure 4 (c). Moreover, in Figure 5, we introduce 3D figures to indicate the similarity of the exact and numerical solutions. To be specific, Figures 5 (a) and (b) ensure that the exact and numerical solutions of u(x, t) are mostly the same.where ∆tm is the temporal step size and ∆η = b/Nx is the computational step size. The boundary conditions of the system Eq. (5.13) are x(0, t) = 0 and x(b, t) = b and the initial condition is taken by

Figure 2: A comparison between analytical and numerical solutions of u(x, t = 10), and g(x, t = 10) with ∆x = 0.01 and k = 3.5.

10

Figure 3: (a) u(x, t) and (b) g(x, t). These figures illustrate the performance of the AMMT and UMM computed with the analytical solutions.

11

Figure 4: Figures (a) and (b) show the time evolution of u(x, t) and g(x, t), respectively. Figure (c) illustrate the associated time evolution of x(η, t). The time is taken from 0 to 10. 12

Figure 5: Illustrating the comparison of the analytical and numerical solutions of u(x, t).

Figure 6: Illustrating the comparison of the analytical and numerical solutions of g(x, t).

13

∆x 0.1 0.05 0.02 0.01 0.005

L2 5.1E − 6 3.5E − 7 1.47E − 8 6.05E − 9 5.5E − 9

AMMT L∞ 2E − 3 5.54E − 4 9.4E − 5 7.4E − 5 7.4E − 5

L2 4E − 2 3.5E − 3 9.2E − 5 5.7E − 6 3.6E − 7

UMM L∞ 3.2E − 1 1E − 1 1.7E − 2 4.3E − 3 1.1E − 3

Table 1: L2 and L∞ norms present the comparison of the performance of the numerical results computed by the considered techniques.

Figure 7: It is showing the summary of the L2 norms columns taken from table (1). Table 1 reveals the L2 , L∞ norms for both the UMM and AMMT using the monitor functions Eq. (5.11) and the MMPDE7. It is observed that the results obtained using AMMT is more accurate compared to that using the UMM. All of the numerical results utilised in recording the error is taken at fixing time (t=10). Figure 7 displays the summary of the L2 norm columns that appear in Table 1.

6

Conclusions

To sum up, this article has investigated the exact and numerical solutions of the dispersive long wave equation. We have solved the proposed equation and presented four different cases for the hyperbolic solutions. The achieved travelling wave solutions (Eqs. (4.12)) are shown stable in the interval [0, 60]. The adaptive moving mesh and uniform mesh methods have been applied to obtain the numerical results of DLWE. From figures 2, 5 and 6, we observe that the obtained numerical results are almost identical to the analytical results. The solution of DLWE has rapid variations. Therefore, the uniform mesh methods do not give suitable results due to the lack of a massive number of mesh points in the regions in which the error is high. This technique can be often employed when a huge number of points is used. The L2 and L∞ are computed to test the accuracy of the considered numerical methods. Figure 3 and the given table show that

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the adaptive moving mesh method gives more appropriate performance. Finally, the adaptive moving mesh method can be strongly considered as a powerful tool for solving PDEs numerically.

References [1] M. A. E. Abdelrahman, On the shallow water equations, Z Naturforsch, 72(9)a (2017), 873-879. [2] M. A. E. Abdelrahman and M. A. Sohaly, Solitary waves for the nonlinear Schr¨odinger problem with theprobability distribution function in the stochastic input case, Eur. Phys. J. Plus., 132 (2017), 339. [3] M. Younis, S. Ali and S. A. Mahmood, Solitons for compound KdV Burgers equation with variable coefficients and power law nonlinearity, Nonlinear Dyn., 81 (2015), 1191-1196. [4] A. H. Bhrawy, An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system, Appl. Math. Comput., 247 (2014), 30-46. [5] A. R. Seadawy, Ion acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev–Petviashvili–Burgers equation in quantum plasma, Math. Meth. in App. Sci., 40 (5) (2016), 1598 - 1607. [6] Z. Korpinar, M. Inc and M. Bayram, Theory and application for the system of fractional Burger equations with Mittag leffler kernel, Applied Mathematics and Computation, 367 (2020), Article 124781. [7] E. C. Aslan and M. Inc, Optical soliton solutions of the NLSE with quadratic-cubicHamiltonian perturbations and modulation instability analysis, Optik, 196 (2019), Article 162661. [8] Z. Korpinar and M. Inc, Numerical simulations for fractional variation of (1+1)-dimensional Biswas-Milovic equation, Optik, 166 (2018), 77-85. [9] M. Inc, A. I. Aliyu, A. Yusuf and D. Baleanu, Optical solitons to the resonance nonlinear Schr¨ odinger equation by Sine-Gordon equation method, Superlattices and Microstructures, 113 (2018), 541-549. [10] M. Inc, A. I. Aliyu, A. Yusuf and D. Baleanu, Novel optical solitary waves and modulation instability analysis for the coupled nonlinear Schr¨odinger equation in monomode step-index optical fibers, Superlattices and Microstructures, 113 (2018), 745-753. [11] A. R. Seadawy, et al., Travelling wave solutions of the generalized nonlinear fifth-order KdV water wave equations and its stability, J. Taibah Univ. Sci., (2016) [12] W. Malflieta and W. Hereman, The tanh method: Exact solutions of nonlinear evolution and wave equations, Phys. Scr., 54 (1996), 563-568. [13] A. M. Wazwaz, The tanh method for travelling wave solutions of nonlinear equations, Appl. Math. Comput., 154 (2004), 714-723.

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[14] C. Q. Dai and J. F. Zhang, Jacobian elliptic function method for nonlinear differential difference equations. Chaos Solutions Fractals, 27 (2006), 1042-1049. [15] S. Liu, Z. Fu, S. Liu and Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A., 289 (2001), 69-74. [16] J. H. He and X. H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fractals, 30 (2006), 700-708. [17] H. Aminikhad, H. Moosaei and M. Hajipour, Exact solutions for nonlinear partial differential equations via Exp-function method, Numer. Methods Partial Differ. Equations, 26 (2009) 1427-1433. [18] E. Fan and H. Zhang, A note on the homogeneous balance method, Phys. Lett. A., 246 (1998), 403-406. [19] M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A., 213 (1996) 279-287. [20] A. R. Alharbi and M. B. Almatrafi, Riccati-Bernoulli Sub-ODE approach on the partial differential equations and applications, International Journal of Mathematics and Computer Science, 15 (1)(2020), 367–388. [21] X. F. Yang, Z. C. Deng and Y. Wei, A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Adv. Diff. Equa. 1 (2015), 117-133. [22] A. M. Wazwaz, Exact solutions to the double sinh-Gordon equation by the tanh method and a variable separated ODE method, Comput. Math. Appl., 50 (2005) 1685-1696. [23] A. M. Wazwaz, A sine-cosine method for handling nonlinear wave equations, Math. Comput. Modelling, 40 (2004), 499-508. [24] Y. J. Ren and H. Q. Zhang, A generalized F-expansion method to find abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional Nizhnik-Novikov-Veselov equation, Chaos Solitons Fractals, 27 (2006), 959-979. [25] J. L. Zhang, M. L. Wang, Y. M. Wang and Z. D. Fang, The improved F-expansion method and its applications, Phys. Lett. A., 350 (2006), 103-109. [26] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A., 277 (2000), 212-218. [27] A. M. Wazwaz, The extended tanh method for abundant solitary wave solutions of nonlinear wave equations, Appl. Math. Comput., 187 (2007), 1131-1142. 0

[28] M. L. Wang, J. L. Zhang and X. Z. Li, The ( GG )- expansion method and travelling wave solutions of nonlinear evolutions equations in mathematical physics, Phys. Lett. A., 372 (2008), 417-423. 0

[29] S. Zhang, J. L. Tong and W. Wang, A generalized ( GG )- expansion method for the mKdv equation with variable coefficients, Phys. Lett. A., 372 (2008), 2254-2257. 16

[30] L. J. F. Broer, Approximate equations for long water wave, Appl. Sci. Res. 31 (1975), 377-395. [31] W. Huang and R. D. Russell, The Adaptive Moving Mesh Methods, Springer, (2011). [32] C. J. Budd, W. Huang and R. D. Russell, Adaptivity with moving grid, Acta Numerica., 18 (2009), 111-241. [33] V. D. Liseikin, Grid Generation Methods, Springer, New York, NY, (2009), USA. [34] C. Liang, X. Yong, L. Zhongfei and H. Jiahua, The extended jacobian elliptic function expansion method and its application to nonlinear wave equation, Fizika A., 12(4) (2003), 161-170. [35] Z. Fu, S. Liu and Q. Zhao, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Physics Letters A., 290 (2001), 72-76. [36] E. Fan and J. Zhang, Applications of the Jacobi elliptic function method to special-type nonlinear equations, Physics Letters A., 305 (2002), 383-392. [37] Z. Yan, Abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional integrable Davey–Stewartson-type equation via a new method, Chaos Solitons Fractals, 18 (2003), 299-309. [38] A. R. Alharbi and S. Naire, An adaptive moving mesh method for thin film flow equations with surface tension, J. Comput. Appl. Math., 319(4) (2017), 365-384. [39] A. R. Alharbi and N. Shailesh, An adaptive moving mesh method for two-dimensional thin film flow equations with surface tension, J. Comput. Appl. Math., 356 (2019), 219-230. [40] A. R. Alharbi, Numerical solution of thin-film flow equations using adaptive moving mesh methods, Keele University, (2016), Ph.D. thesis. [41] E. Walsh, Moving mesh methods for problems in meteorology, University of Bath, (2010), PhD thesis. [42] R. D. Russell, W. Huang and Y. Ren, Moving Mesh Partial Differential Equations (MMPDES) Based on the Equidistribution Principle, Siam. J. Numer. Anal., 31 (1994), 709-730. [43] A. R. Alharbi, M. I. Faisal, F. A. Shah, M. Waseem, R. Ullah and S. Sherbaz, Higher order numerical approaches for nonlinear equations by decomposition technique, IEEE Access, 7 (2019), 44329-44337.

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• Nonlinear analysis approaches are used to solve the physical models. • Applying the interesting analytical method of to the nonlinear NPFDEs. • Obtaining some new solutions of the NPDEs. • Highly Performance and efficient of the proposed numerical method.

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