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Abundant accurate analytical and semi-analytical solutions of the positive Gardner–Kadomtsev–Petviashvili equation

  • Dexu Zhao , Raghda A. M. Attia , Jian Tian , Samir A. Salama , Dianchen Lu and Mostafa M. A. Khater EMAIL logo
Published/Copyright: February 4, 2022
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Open Physics
From the journal Open Physics Volume 20 Issue 1

Abstract

This research article examines the correctness of two new analytical methods for solving the internal solitary waves of shallow seas. To get the computational solutions to the positive (2 + 1)-dimensional Gardner–Kadomtsev–Petviashvili model, the extended simplest equation and modified Kudryashov methods are used. Numerous new traveling wave solutions in various forms are developed in order to assess the starting conditions required for the variational iteration technique, one of the most accurate semi-analytical methods. The semi-analytical solutions are used to demonstrate the precision of the solutions obtained and the analytical methods employed. The dynamical behavior of internal solitary waves in shallow waters is shown using many three-dimensional drawings. The performance of the used schemes demonstrates their efficacy and power, as well as their capacity to handle a large number of nonlinear evolution equations.

Keywords: positive (2 + 1)-D GKP equation; analytical and semi-analytical solutions

1 Introduction

In recent years, the shallow water wave phenomena have emerged as an attractive option for researching and analyzing the bidirectional propagating water wave surface’s dynamical and physical behavior [1]. The flow of a fluid under a pressure surface illustrates the shallow water wave attitude through a set of hyperbolic nonlinear partial differential equations that have been recently constructed [2]. Adhemar Jean Claude Barr’e de Saint–Venant has derived these unidirectional hyperbolic equations as a model of transitory open-channel flow and surface runoff [3,4, 5,6]. As a result, these equations are referred to as Saint–Venant equations, a contraction of the two-dimensional (2D) shallow water equations [7]. Additionally, the famous Navier–Stokes equations are considered a subset of shallow water wave equations because they are used when the horizontal length scale of the fluid is slightly greater than the vertical length scale, and mass conservation ensures that the fluid’s vertical velocity scale is small in comparison to its horizontal speed scale [8,9]. Additionally, the 2D Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation is included in this hyperbolic collection [10,11,12]. The shallow water wave has many applications in a variety of disciplines, including electromagnetic theory, astrophysics, electrochemistry, fluid dynamics, plasma physics, acoustics, and cosmology [13,14,15].

Three kinds of computational schemes (analytical, semi-analytical, and numerical methods) have been derived such as refs [16,17,18, 19,20]. Each of these methods has been applied to a large number of nonlinear evolution problems. Numerous computational, semi-analytical, and numerical solutions have been developed, but none of them is universally applicable to all nonlinear evolution equations; therefore, the quest for this unifying approach continues. The tremendous development in computer technology has been the most beneficial instrument in this research; nevertheless, this technology has been utilized only to find new methods, and no one has used it to verify the correctness of previously obtained systems [21,22].

This manuscript studies the ( 2 + 1 )-D Gardner–Kadomtsev–Petviashvili equation, which is given by refs [23,24,25]

(1) ( G t + p G G x ± G 2 G x + G x x x ) x + G y y = 0 ,

where p , q are arbitrary constants to be evaluated while G = G ( x , y , t ) is a function of space and time which explains the evolution of nonlinear waves where the effect of the surface tension and the viscosity is negligible. Employing the following wave transformation G ( x , y , t ) = K ( ψ ) , ψ = x + y + r t , where r is the wave velocity, and integrating the result once with zero constants of the integration convert the positive form of equation (1) into the following ordinary differential equation:

(2) ( r + 1 ) K + p K K ′ + q K 2 K ′ + K ‴ = 0 .

Using the homogeneous balance principles and the following auxiliary equations for extended simplest equation (ESE) and modified Kudryashov (MKud) methods [26,27, 28,29] for equation (2), respectively, F ′ ( J ) = l 1 + l 2 F ( J ) + l 3 F ( J ) 2 and E ′ ( J ) = ln ( k ) ( E ( J ) 2 − E ( J ) ) , where l 1 , l 2 , l 3 , k are arbitrary constants to be constructed later, given n = 1 . Thus, the general solutions of equation (2) based on the suggested analytical schemes (ESE and MKud) [30,31, 32,33] are formulated in the following forms:

(3) K ( J ) = ∑ i = − n n a i F ( J ) i = a 1 F ( J ) + a − 1 F ( J ) + a 0 , ∑ i = 0 n a i E ( J ) i = a 1 E ( J ) + a 0 ,

where a 0 , a 1 , a − 1 are arbitrary constants.

The rest of the article is organized as follows. Section 2 investigates the analytical and semi-analytical solutions of the considered model. Section 3 explains the paper’s contributions and novelty. Section 4 gives the conclusion of the article.

2 Accuracy of computational solutions

Investigating the computational solutions of the investigated model with the ESE and Mkud methods then checking the solutions’ accuracy by employing the VI semi-analytical schemes, shows the applied methods’ performance and accuracy as following:

2.1 ESE method’s solutions

Calculating the abovementioned parameters through the ESE method’s framework gives the following families:

Family I

a 1 → 0 , r → a − 1 2 ( − l 2 2 ) − 2 a − 1 2 l 1 l 3 + 6 a 0 a − 1 l 1 l 2 − 6 a 0 2 l 1 2 − a − 1 2 a − 1 2 , p → − 6 ( a − 1 l 1 l 2 − 2 a 0 l 1 2 ) a − 1 2 , q → − 6 l 1 2 a − 1 2 .

Family II

a − 1 → 0 , r → a 1 2 ( − l 2 2 ) − 2 a 1 2 l 1 l 3 + 6 a 0 a 1 l 2 l 3 − 6 a 0 2 l 3 2 − a 1 2 a 1 2 , p → − 6 ( a 1 l 2 l 3 − 2 a 0 l 3 2 ) a 1 2 , q → − 6 l 3 2 a 1 2 .

Family III

a − 1 → − i 6 l 1 q , a 1 → − i 6 l 3 q , r → i 6 a 0 l 2 q + a 0 2 q − l 2 2 + 4 l 1 l 3 − 1 , p → − 2 a 0 q − i 6 l 2 q , where ( q < 0 ) .

Thus, the computational solutions of the positive nonlinear ( 2 + 1 )-D GKP equation are constructed as follows: For l 2 = 0 , l 1 l 3 > 0 , we obtain

(4) G I , 1 ( x , y , t ) = a 0 − a − 1 l 1 l 3 l 1 × cot l 1 l 3 ( a − 1 2 ( − η + 2 l 1 l 3 t + t − x − y ) + 6 a 0 2 l 1 2 t ) a − 1 2 ,

(5) G I , 2 ( x , y , t ) = a 0 − a − 1 l 1 l 3 l 1 tan × l 1 l 3 ( a − 1 2 ( − η + 2 l 1 l 3 t + t − x − y ) + 6 a 0 2 l 1 2 t ) a − 1 2 ,

(6) G II , 1 ( x , y , t ) = a 0 − a 1 l 1 l 3 l 3 tan × l 1 l 3 ( a 1 2 ( − η + 2 l 1 l 3 t + t − x − y ) + 6 a 0 2 l 3 2 t ) a 1 2 ,

(7) G II , 2 ( x , y , t ) = a 0 − a 1 l 1 l 3 l 3 cot l 1 l 3 ( a 1 2 ( − η + 2 l 1 l 3 t + t − x − y ) + 6 a 0 2 l 3 2 t ) a 1 2 ,

(8) G III , 1 ( x , y , t ) = a 0 − 2 i 6 l 1 l 3 q csc ( 2 l 1 l 3 ( t ( a 0 2 q + 4 l 1 l 3 − 1 ) + η + x + y ) ) .

For l 2 = 0 , l 1 l 3 < 0 , we obtain

(9) G I , 3 ( x , y , t ) = a 0 − a − 1 − l 1 l 3 l 1 coth − l 1 l 3 ( a − 1 2 ( − 2 l 1 l 3 t − t + x + y ) − 6 a 0 2 l 1 2 t ) a − 1 2 ∓ log ( η ) 2 ,

(10) G I , 4 ( x , y , t ) = a 0 − a − 1 − l 1 l 3 l 1 tanh − l 1 l 3 ( a − 1 2 ( − 2 l 1 l 3 t − t + x + y ) − 6 a 0 2 l 1 2 t ) a − 1 2 ∓ log ( η ) 2 ,

(11) G II , 3 ( x , y , t ) = a 1 − l 1 l 3 l 3 tanh − l 1 l 3 ( a 1 2 ( − 2 l 1 l 3 t − t + x + y ) − 6 a 0 2 l 3 2 t ) a 1 2 ∓ log ( η ) 2 + a 0 ,

(12) G II , 4 ( x , y , t ) = a 1 − l 1 l 3 l 3 coth − l 1 l 3 ( a 1 2 ( − 2 l 1 l 3 t − t + x + y ) − 6 a 0 2 l 3 2 t ) a 1 2 ∓ log ( η ) 2 + a 0 ,

(13) G III , 2 ( x , y , t ) = a 0 + 2 i 6 − l 1 l 3 q csch 2 − l 1 l 3 ( t ( a 0 2 q + 4 l 1 l 3 − 1 ) + x + y ) ∓ log ( η ) 2 .

For l 1 = 0 , l 2 > 0 , we obtain

(14) G I , 5 ( x , y , t ) = a − 1 e l 2 ( − η + l 2 2 t + t − x − y ) l 2 − a − 1 l 3 l 2 + a 0 ,

(15) G II , 5 ( x , y , t ) = a 1 l 2 exp l 2 t − 6 a 0 l 3 l 2 a 1 + 6 a 0 2 l 3 2 a 1 2 + l 2 2 − η + t − x − y − l 3 + a 0 ,

(16) G III , 3 ( x , y , t ) = i 6 l 2 q ( − 1 + l 3 exp ( l 2 ( t ( i 6 a 0 l 2 q + a 0 2 q − l 2 2 − 1 ) + η + x + y ) ) ) + a 0 + i 6 l 2 q .

For l 1 = 0 , l 2 < 0 , we obtain

(17) G I , 6 ( x , y , t ) = − a − 1 e l 2 ( − η + l 2 2 t + t − x − y ) l 3 − a − 1 + a 0 ,

(18) G II , 6 ( x , y , t ) = a 0 − a 1 l 3 exp l 2 t − 6 a 0 l 3 l 2 a 1 + 6 a 0 2 l 3 2 a 1 2 + l 2 2 − η + t − x − y + l 3 ,

(19) G III , 4 ( x , y , t ) = − i 6 l 3 q ( 1 + l 3 exp ( l 2 ( t ( i 6 a 0 l 2 q + a 0 2 q − l 2 2 − 1 ) + η + x + y ) ) ) + a 0 + i 6 l 3 q .

For 4 l 1 l 3 > l 2 2 , we obtain

(20) G I , 7 ( x , y , t ) = a 0 − 2 a − 1 l 3 4 l 1 l 3 − l 2 2 tan 4 l 1 l 3 − l 2 2 ( a − 1 2 ( − η + l 2 2 t + 2 l 1 l 3 t + t − x − y ) − 6 a 0 a − 1 l 1 l 2 t + 6 a 0 2 l 1 2 t ) 2 a − 1 2 + l 2 ,

(21) G I , 8 ( x , y , t ) = a 0 − 2 a − 1 l 3 4 l 1 l 3 − l 2 2 cot 4 l 1 l 3 − l 2 2 ( a − 1 2 ( − η + l 2 2 t + 2 l 1 l 3 t + t − x − y ) − 6 a 0 a − 1 l 1 l 2 t + 6 a 0 2 l 1 2 t ) 2 a − 1 2 + l 2 ,

(22) G II , 7 ( x , y , t ) = − a 1 4 l 1 l 3 − l 2 2 2 l 3 tan 4 l 1 l 3 − l 2 2 2 a 1 2 ( a 1 2 ( − η + l 2 2 t + 2 l 1 l 3 t + t − x − y ) − 6 a 0 a 1 l 2 l 3 t + 6 a 0 2 l 3 2 t ) − a 1 l 2 2 l 3 + a 0 ,

(23) G II , 8 ( x , y , t ) = − a 1 4 l 1 l 3 − l 2 2 2 l 3 cot 4 l 1 l 3 − l 2 2 2 a 1 2 ( a 1 2 ( − η + l 2 2 t + 2 l 1 l 3 t + t − x − y ) − 6 a 0 a 1 l 2 l 3 t + 6 a 0 2 l 3 2 t ) − a 1 l 2 2 l 3 + a 0 ,

(24) G III , 5 ( x , y , t ) = − i 3 2 q 4 l 1 l 3 − l 2 2 tan 1 2 4 l 1 l 3 − l 2 2 ( t ( i 6 a 0 l 2 q + a 0 2 q − l 2 2 + 4 l 1 l 3 − 1 ) + η + x + y ) + ( 2 i 6 l 1 l 3 ) / q l 2 − 4 l 1 l 3 − l 2 2 tan 1 2 4 l 1 l 3 − l 2 2 ( t ( i 6 a 0 l 2 q + a 0 2 q − l 2 2 + 4 l 1 l 3 − 1 ) + η + x + y ) + a 0 + i 3 2 l 2 q ,

(25) G III , 6 ( x , y , t ) = − i 3 2 q 4 l 1 l 3 − l 2 2 cot 1 2 4 l 1 l 3 − l 2 2 ( t ( i 6 a 0 l 2 q + a 0 2 q − l 2 2 + 4 l 1 l 3 − 1 ) + η + x + y ) + ( 2 i 6 l 1 l 3 ) / q l 2 − 4 l 1 l 3 − l 2 2 cot 1 2 4 l 1 l 3 − l 2 2 ( t ( i 6 a 0 l 2 q + a 0 2 q − l 2 2 + 4 l 1 l 3 − 1 ) + η + x + y ) + a 0 + i 3 2 l 2 q ,

where η is the arbitrary constant [30,31].

2.1.1 Semi-analytical solutions

Applying the VI method [34] for equation (1) with the following initial condition G I ( x , y , 0 ) = 0.5 tanh ( 0.4 x + 0.4 y ) + 0.01 , based on the VI method, the correction function of equation (10) is given by

(26) G n + 1 ( x , y , t ) = G n ( x , y , t ) + ∫ 0 s λ ( ( G t + 0.0612245 G G x − 3.06122 G 2 G x + G x x x ) x + G y y ) d s ,

where 1 + λ ∣ s = t = 0 , λ ′ = 0 . Thus, the Lagrangian multiplier gives ( λ = − 1 ) . Consequently, the semi-analytical solution of equation (10) is given by:

(27) G I , 1 ( x , y , t ) = t sech 6 ( 0.4 x + 0.4 y ) ( 0.1 sinh ( 0.7 x + 0.7 y ) + 0.004 sinh ( 1.4 x + 1.4 y ) + 0.1 cosh ( 0.7 x + 0.7 y ) − 0.005 cosh ( 1.4 x + 1.4 y ) − 0.2 ) + 0.5 tanh ( 0.4 x + 0.4 y ) + 0.01 .

Using the same techniques, it is easy to obtain G i ( x , y , t ) , where i = 2 , 3 , 4 , …

2.2 MKud method’s solutions

Calculating the above mentioned parameters through the MKud method’s framework gives:

a 0 → − i ( 6 q log ( k ) − i p ) 2 q , a 1 → i 6 log ( k ) q , r → 2 q log 2 ( k ) + p 2 − 4 q 4 q , where q < 0 .

Thus, the computational solutions of the positive nonlinear ( 2 + 1 )-D GKP equation are constructed as follows:

(28) G ( x , y , t ) = − 1 2 q − 2 i 6 q log ( k ) 1 ± k t ( 2 q log 2 ( k ) + p 2 − 4 q ) 4 q + x + y + i 6 q log ( k ) + p .

2.2.1 Semi-analytical solutions

Applying the VI method for equation (1) with the following initial condition G ( x , y , 0 ) = 1 8 4 6 1 ± e x + y − 2 6 + 1 , based on the VI method, the correction function of equation (10) is given by

(29) G n + 1 ( x , y , t ) = G n ( x , y , t ) + ∫ 0 s λ ( ( G t + G G x − 4 G 2 G x + G x x x ) x + G y y ) d s ,

where 1 + λ ∣ s = t = 0 , λ ′ = 0 . Thus, the Lagrangian multiplier gives ( λ = − 1 ) . Consequently, the semi-analytical solution of equation (10) is given by:

(30) G 1 ( x , y , t ) = 1 16 ( e x + y + 1 ) 6 e 3 ( x + y ) − 16 ( 26 6 + 9 ) t cosh ( x + y ) + 8 ( 2 6 + 9 ) t cosh ( 2 ( x + y ) ) − 2 6 sinh ( x + y ) ( ( 45 t + 32 ) cosh ( x + y ) − 51 t + 8 cosh ( 2 ( x + y ) ) + 24 ) + 24 ( 22 6 − 9 ) t + 128 cosh 6 x + y 2 .

Using the same techniques, it is easy to obtain G i ( x , y , t ) , where i = 2 , 3 , 4 , …

3 Results’ interpretation

Here, the analytical and semi-analytical obtained results are explained and discussed to show the objectives of this manuscript. Two computational schemes have been successfully implemented to the positive nonlinear ( 2 + 1 ) D-GKP equation, and many novel solitary wave solutions in various forms such as exponential, trigonometric, hyperbolic have been constructed. The physical characterization of the evolution of nonlinear waves where the effect of the surface tension and the viscosity is negligible is explained through Figures 1 and 2 for the, respectively, values of the parameters ( a 0 = 2 , a − 1 = 3 , η = 1 , l 1 = − 2 , l 3 = 8 and k = e , p = 1 , q = − 4 ). Comparing our obtained solutions with those obtained by Kalim U Tariq et al., who have used the auxiliary equation method, finds just one match between both solutions ( equations (5) and (33), [23] when p λ 1 = 2 a 1 a 0 , 3 ( 4 ω − 5 ) l 1 l 3 = 2 a 1 a − 1 l 1 ) and all our solutions are new and different from their solutions. The evaluated analytical solutions have been used to calculate the initial condition for the studied model. Thus, the VI method has been productively handling the semi-analytical solutions that have been used to calculate the absolute error between the exact and semi-analytical solutions, which show the match between them (Tables 1, 2 and Figures 3, 4). The superiority of the MKud method has been shown over the ESE method (Figure 5).

Figure 1 Solitary wave solution equation (10) in (a) three, (b) two, and (c) contour plots.
Figure 1

Solitary wave solution equation (10) in (a) three, (b) two, and (c) contour plots.

Figure 2 Solitary wave solution equation (28) in (a) three, (b) two, and (c) contour plots.
Figure 2

Solitary wave solution equation (28) in (a) three, (b) two, and (c) contour plots.

Table 1

Absolute error between analytical and semi-analytical solutions based on ESE and VI schemes

Value of x t = 2 t = 4 t = 6 t = 8 t = 10 t = 12 t = 14 t = 16 t = 18 t = 20
0 7.5233 × 1 0 − 5 0.00015047 0.0002257 0.00030093 0.00037616 0.0004514 0.00052663 0.00060186 0.000677095 0.000752327
1 3.998 × 1 0 − 5 7.9959 × 1 0 − 5 0.00011994 0.00015992 0.0001999 0.00023988 0.00027986 0.00031984 0.000359816 0.000399796
2 2.0417 × 1 0 − 5 4.0835 × 1 0 − 5 6.1252 × 1 0 − 5 8.1669 × 1 0 − 5 0.00010209 0.0001225 0.00014292 0.00016334 0.000183756 0.000204173
3 1.0239 × 1 0 − 5 2.0478 × 1 0 − 5 3.0717 × 1 0 − 5 4.0955 × 1 0 − 5 5.1194 × 1 0 − 5 6.1433 × 1 0 − 5 7.1672 × 1 0 − 5 8.1911 × 1 0 − 5 9.21496 × 1 0 − 5 0.000102388
4 5.0902 × 1 0 − 6 1.018 × 1 0 − 5 1.5271 × 1 0 − 5 2.0361 × 1 0 − 5 2.5451 × 1 0 − 5 3.0541 × 1 0 − 5 3.5632 × 1 0 − 5 4.0722 × 1 0 − 5 4.58122 × 1 0 − 5 5.09024 × 1 0 − 5
5 2.52 × 1 0 − 6 5.04 × 1 0 − 6 7.56 × 1 0 − 6 1.008 × 1 0 − 5 1.26 × 1 0 − 5 1.512 × 1 0 − 5 1.764 × 1 0 − 5 2.016 × 1 0 − 5 2.26801 × 1 0 − 5 2.52001 × 1 0 − 5
6 1.245 × 1 0 − 6 2.49 × 1 0 − 6 3.735 × 1 0 − 6 4.98 × 1 0 − 6 6.225 × 1 0 − 6 7.4701 × 1 0 − 6 8.7151 × 1 0 − 6 9.9601 × 1 0 − 6 1.12051 × 1 0 − 5 1.24501 × 1 0 − 5
7 6.1448 × 1 0 − 7 1.229 × 1 0 − 6 1.8434 × 1 0 − 6 2.4579 × 1 0 − 6 3.0724 × 1 0 − 6 3.6869 × 1 0 − 6 4.3013 × 1 0 − 6 4.9158 × 1 0 − 6 5.53028 × 1 0 − 6 6.14476 × 1 0 − 6
8 3.0312 × 1 0 − 7 6.0625 × 1 0 − 7 9.0937 × 1 0 − 7 1.2125 × 1 0 − 6 1.5156 × 1 0 − 6 1.8187 × 1 0 − 6 2.1219 × 1 0 − 6 2.425 × 1 0 − 6 2.72812 × 1 0 − 6 3.03125 × 1 0 − 6
9 1.495 × 1 0 − 7 2.9899 × 1 0 − 7 4.4849 × 1 0 − 7 5.9799 × 1 0 − 7 7.4748 × 1 0 − 7 8.9698 × 1 0 − 7 1.0465 × 1 0 − 6 1.196 × 1 0 − 6 1.34547 × 1 0 − 6 1.49497 × 1 0 − 6
10 7.3721 × 1 0 − 8 1.4744 × 1 0 − 7 2.2116 × 1 0 − 7 2.9488 × 1 0 − 7 3.686 × 1 0 − 7 4.4233 × 1 0 − 7 5.1605 × 1 0 − 7 5.8977 × 1 0 − 7 6.63488 × 1 0 − 7 7.37209 × 1 0 − 7
11 3.6352 × 1 0 − 8 7.2703 × 1 0 − 8 1.0905 × 1 0 − 7 1.4541 × 1 0 − 7 1.8176 × 1 0 − 7 2.1811 × 1 0 − 7 2.5446 × 1 0 − 7 2.9081 × 1 0 − 7 3.27164 × 1 0 − 7 3.63516 × 1 0 − 7
12 1.7924 × 1 0 − 8 3.5849 × 1 0 − 8 5.3773 × 1 0 − 8 7.1697 × 1 0 − 8 8.9622 × 1 0 − 8 1.0755 × 1 0 − 7 1.2547 × 1 0 − 7 1.4339 × 1 0 − 7 1.61319 × 1 0 − 7 1.79243 × 1 0 − 7
13 8.8381 × 1 0 − 9 1.7676 × 1 0 − 8 2.6514 × 1 0 − 8 3.5352 × 1 0 − 8 4.419 × 1 0 − 8 5.3028 × 1 0 − 8 6.1866 × 1 0 − 8 7.0704 × 1 0 − 8 7.95425 × 1 0 − 8 8.83805 × 1 0 − 8
14 4.3578 × 1 0 − 9 8.7156 × 1 0 − 9 1.3073 × 1 0 − 8 1.7431 × 1 0 − 8 2.1789 × 1 0 − 8 2.6147 × 1 0 − 8 3.0505 × 1 0 − 8 3.4862 × 1 0 − 8 3.92202 × 1 0 − 8 4.3578 × 1 0 − 8
15 2.1487 × 1 0 − 9 4.2974 × 1 0 − 9 6.4461 × 1 0 − 9 8.5948 × 1 0 − 9 1.0744 × 1 0 − 8 1.2892 × 1 0 − 8 1.5041 × 1 0 − 8 1.719 × 1 0 − 8 1.93383 × 1 0 − 8 2.1487 × 1 0 − 8
16 1.0595 × 1 0 − 9 2.1189 × 1 0 − 9 3.1784 × 1 0 − 9 4.2378 × 1 0 − 9 5.2973 × 1 0 − 9 6.3568 × 1 0 − 9 7.4162 × 1 0 − 9 8.4757 × 1 0 − 9 9.53513 × 1 0 − 9 1.05946 × 1 0 − 8
17 5.2239 × 1 0 − 10 1.0448 × 1 0 − 9 1.5672 × 1 0 − 9 2.0895 × 1 0 − 9 2.6119 × 1 0 − 9 3.1343 × 1 0 − 9 3.6567 × 1 0 − 9 4.1791 × 1 0 − 9 4.70148 × 1 0 − 9 5.22386 × 1 0 − 9
18 2.5757 × 1 0 − 10 5.1514 × 1 0 − 10 7.7272 × 1 0 − 10 1.0303 × 1 0 − 9 1.2879 × 1 0 − 9 1.5454 × 1 0 − 9 1.803 × 1 0 − 9 2.0606 × 1 0 − 9 2.31815 × 1 0 − 9 2.57572 × 1 0 − 9
19 1.27 × 1 0 − 10 2.54 × 1 0 − 10 3.81 × 1 0 − 10 5.08 × 1 0 − 10 6.35 × 1 0 − 10 7.6201 × 1 0 − 10 8.8901 × 1 0 − 10 1.016 × 1 0 − 9 1.14301 × 1 0 − 9 1.27001 × 1 0 − 9
20 6.262 × 1 0 − 11 1.2524 × 1 0 − 10 1.8786 × 1 0 − 10 2.5048 × 1 0 − 10 3.131 × 1 0 − 10 3.7572 × 1 0 − 10 4.3834 × 1 0 − 10 5.0096 × 1 0 − 10 5.63582 × 1 0 − 10 6.26202 × 1 0 − 10
21 3.0876 × 1 0 − 11 6.1752 × 1 0 − 11 9.2628 × 1 0 − 11 1.235 × 1 0 − 10 1.5438 × 1 0 − 10 1.8526 × 1 0 − 10 2.1613 × 1 0 − 10 2.4701 × 1 0 − 10 2.77885 × 1 0 − 10 3.08761 × 1 0 − 10
22 1.5224 × 1 0 − 11 3.0448 × 1 0 − 11 4.5672 × 1 0 − 11 6.0896 × 1 0 − 11 7.612 × 1 0 − 11 9.1344 × 1 0 − 11 1.0657 × 1 0 − 10 1.2179 × 1 0 − 10 1.37016 × 1 0 − 10 1.5224 × 1 0 − 10
23 7.5064 × 1 0 − 12 1.5013 × 1 0 − 11 2.2519 × 1 0 − 11 3.0026 × 1 0 − 11 3.7532 × 1 0 − 11 4.5039 × 1 0 − 11 5.2545 × 1 0 − 11 6.0052 × 1 0 − 11 6.75584 × 1 0 − 11 7.50648 × 1 0 − 11
24 3.7012 × 1 0 − 12 7.4024 × 1 0 − 12 1.1104 × 1 0 − 11 1.4805 × 1 0 − 11 1.8506 × 1 0 − 11 2.2207 × 1 0 − 11 2.5908 × 1 0 − 11 2.961 × 1 0 − 11 3.33109 × 1 0 − 11 3.70121 × 1 0 − 11
25 1.825 × 1 0 − 12 3.6499 × 1 0 − 12 5.4748 × 1 0 − 12 7.2998 × 1 0 − 12 9.1248 × 1 0 − 12 1.095 × 1 0 − 11 1.2775 × 1 0 − 11 1.46 × 1 0 − 11 1.64245 × 1 0 − 11 1.82495 × 1 0 − 11
26 8.9984 × 1 0 − 13 1.7997 × 1 0 − 12 2.6995 × 1 0 − 12 3.5993 × 1 0 − 12 4.4992 × 1 0 − 12 5.399 × 1 0 − 12 6.2987 × 1 0 − 12 7.1986 × 1 0 − 12 8.09841 × 1 0 − 12 8.99825 × 1 0 − 12
27 4.4365 × 1 0 − 13 8.874 × 1 0 − 13 1.331 × 1 0 − 12 1.7747 × 1 0 − 12 2.2183 × 1 0 − 12 2.6621 × 1 0 − 12 3.1057 × 1 0 − 12 3.5494 × 1 0 − 12 3.99314 × 1 0 − 12 4.43678 × 1 0 − 12
28 2.1871 × 1 0 − 13 4.3743 × 1 0 − 13 6.5625 × 1 0 − 13 8.7497 × 1 0 − 13 1.0938 × 1 0 − 12 1.3125 × 1 0 − 12 1.5313 × 1 0 − 12 1.75 × 1 0 − 12 1.96876 × 1 0 − 12 2.18758 × 1 0 − 12
29 1.0791 × 1 0 − 13 2.1572 × 1 0 − 13 3.2363 × 1 0 − 13 4.3143 × 1 0 − 13 5.3935 × 1 0 − 13 6.4715 × 1 0 − 13 7.5506 × 1 0 − 13 8.6298 × 1 0 − 13 9.70779 × 1 0 − 13 1.07869 × 1 0 − 12
30 5.318 × 1 0 − 14 1.0636 × 1 0 − 13 1.5954 × 1 0 − 13 2.1272 × 1 0 − 13 2.659 × 1 0 − 13 3.1908 × 1 0 − 13 3.7226 × 1 0 − 13 4.2544 × 1 0 − 13 4.78617 × 1 0 − 13 5.31797 × 1 0 − 13
Table 2

Absolute error between analytical and semi-analytical solutions through MKud and VI schemes

Value of x t = 2 t = 4 t = 6 t = 8 t = 10 t = 12 t = 14 t = 16 t = 18 t = 20
0 2.61928 × 1 0 − 6 5.238 × 1 0 − 6 7.857 × 1 0 − 6 1.047 × 1 0 − 5 1.309 × 1 0 − 5 1.571 × 1 0 − 5 1.833 × 1 0 − 5 2.095 × 1 0 − 5 2.357 × 1 0 − 5 2.619 × 1 0 − 5
1 9.91091 × 1 0 − 7 1.982 × 1 0 − 6 2.973 × 1 0 − 6 3.964 × 1 0 − 6 4.955 × 1 0 − 6 5.946 × 1 0 − 6 6.937 × 1 0 − 6 7.928 × 1 0 − 6 8.919 × 1 0 − 6 9.910 × 1 0 − 6
2 3.68327 × 1 0 − 7 7.366 × 1 0 − 7 1.104 × 1 0 − 6 1.473 × 1 0 − 6 1.841 × 1 0 − 6 2.209 × 1 0 − 6 2.578 × 1 0 − 6 2.946 × 1 0 − 6 3.314 × 1 0 − 6 3.683 × 1 0 − 6
3 1.36004 × 1 0 − 7 2.720 × 1 0 − 7 4.080 × 1 0 − 7 5.440 × 1 0 − 7 6.80 × 1 0 − 7 8.160 × 1 0 − 7 9.520 × 1 0 − 7 1.088 × 1 0 − 6 1.224 × 1 0 − 6 1.360 × 1 0 − 6
4 5.01013 × 1 0 − 8 1.002 × 1 0 − 7 1.503 × 1 0 − 7 2.004 × 1 0 − 7 2.505 × 1 0 − 7 3.006 × 1 0 − 7 3.507 × 1 0 − 7 4.008 × 1 0 − 7 4.509 × 1 0 − 7 5.010 × 1 0 − 7
5 1.844 × 1 0 − 8 3.68 × 1 0 − 8 5.532 × 1 0 − 8 7.376 × 1 0 − 8 9.220 × 1 0 − 8 1.106 × 1 0 − 7 1.290 × 1 0 − 7 1.475 × 1 0 − 7 1.659 × 1 0 − 7 1.844 × 1 0 − 7
6 6.785 × 1 0 − 9 1.357 × 1 0 − 8 2.035 × 1 0 − 8 2.714 × 1 0 − 8 3.392 × 1 0 − 8 4.071 × 1 0 − 8 4.749 × 1 0 − 8 5.42 × 1 0 − 8 6.106 × 1 0 − 8 6.785 × 1 0 − 8
7 2.496 × 1 0 − 9 4.992 × 1 0 − 9 7.488 × 1 0 − 9 9.985 × 1 0 − 9 1.248 × 1 0 − 8 1.497 × 1 0 − 8 1.747 × 1 0 − 8 1.997 × 1 0 − 8 2.246 × 1 0 − 8 2.496 × 1 0 − 8
8 9.183 × 1 0 − 10 1.83 × 1 0 − 9 2.755 × 1 0 − 9 3.67 × 1 0 − 9 4.59 × 1 0 − 9 5.51 × 1 0 − 9 6.42846 × 1 0 − 9 7.346 × 1 0 − 9 8.26 × 1 0 − 9 9.183 × 1 0 − 9
9 3.37 × 1 0 − 10 6.756 × 1 0 − 10 1.013 × 1 0 − 9 1.351 × 1 0 − 9 1.689 × 1 0 − 9 2.027 × 1 0 − 9 2.364 × 1 0 − 9 2.702 × 1 0 − 9 3.040 × 1 0 − 9 3.378 × 1 0 − 9
10 1.242 × 1 0 − 10 2.485 × 1 0 − 10 3.728 × 1 0 − 10 4.971 × 1 0 − 10 6.214 × 1 0 − 10 7.457 × 1 0 − 10 8.700 × 1 0 − 10 9.942 × 1 0 − 10 1.118 × 1 0 − 9 1.242 × 1 0 − 9
11 4.572 × 1 0 − 11 9.14 × 1 0 − 11 1.371 × 1 0 − 10 1.82 × 1 0 − 10 2.28 × 1 0 − 10 2.74 × 1 0 − 10 3.201 × 1 0 − 10 3.65 × 1 0 − 10 4.11 × 1 0 − 10 4.572 × 1 0 − 10
12 1.68 × 1 0 − 11 3.36 × 1 0 − 11 5.046 × 1 0 − 11 6.728 × 1 0 − 11 8.410 × 1 0 − 11 1.009 × 1 0 − 10 1.177 × 1 0 − 10 1.345 × 1 0 − 10 1.513 × 1 0 − 10 1.682 × 1 0 − 10
13 6.188 × 1 0 − 12 1.23 × 1 0 − 11 1.85 × 1 0 − 11 2.475 × 1 0 − 11 3.093 × 1 0 − 11 3.712 × 1 0 − 11 4.331 × 1 0 − 11 4.950 × 1 0 − 11 5.569 × 1 0 − 11 6.187 × 1 0 − 11
14 2.27635 × 1 0 − 12 4.55 × 1 0 − 12 6.82 × 1 0 − 12 9.105 × 1 0 − 12 1.138 × 1 0 − 11 1.365 × 1 0 − 11 1.593 × 1 0 − 11 1.821 × 1 0 − 11 2.048 × 1 0 − 11 2.27 × 1 0 − 11
15 8.37 × 1 0 − 13 1.675 × 1 0 − 12 2.512 × 1 0 − 12 3.35 × 1 0 − 12 4.18 × 1 0 − 12 5.02 × 1 0 − 12 5.86 × 1 0 − 12 6.69 × 1 0 − 12 7.53 × 1 0 − 12 8.374 × 1 0 − 12
16 3.08 × 1 0 − 13 6.16 × 1 0 − 13 9.24 × 1 0 − 13 1.23 × 1 0 − 12 1.54 × 1 0 − 12 1.84 × 1 0 − 12 2.16 × 1 0 − 12 2.46 × 1 0 − 12 2.772 × 1 0 − 12 3.080 × 1 0 − 12
17 1.13 × 1 0 − 13 2.263 × 1 0 − 13 3.39 × 1 0 − 13 4.53138 × 1 0 − 13 5.66 × 1 0 − 13 6.79 × 1 0 − 13 7.92977 × 1 0 − 13 9.06 × 1 0 − 13 1.019 × 1 0 − 12 1.133 × 1 0 − 12
18 4.15 × 1 0 − 14 8.31 × 1 0 − 14 1.24 × 1 0 − 13 1.66 × 1 0 − 13 2.082 × 1 0 − 13 2.500 × 1 0 − 13 2.916 × 1 0 − 13 3.334 × 1 0 − 13 3.751 × 1 0 − 13 4.168 × 1 0 − 13
19 1.537 × 1 0 − 14 3.069 × 1 0 − 14 4.607 × 1 0 − 14 6.12843 × 1 0 − 14 7.677 × 1 0 − 14 9.20 × 1 0 − 14 1.0747 × 1 0 − 13 1.227 × 1 0 − 13 1.381 × 1 0 − 13 1.533 × 1 0 − 13
20 5.71765 × 1 0 − 15 1.132 × 1 0 − 14 1.698 × 1 0 − 14 2.253 × 1 0 − 14 2.831 × 1 0 − 14 3.402 × 1 0 − 14 3.969 × 1 0 − 14 4.524 × 1 0 − 14 5.095 × 1 0 − 14 5.639 × 1 0 − 14
21 2.220 × 1 0 − 15 4.163 × 1 0 − 15 6.161 × 1 0 − 15 8.326 × 1 0 − 15 1.049 × 1 0 − 14 1.249 × 1 0 − 14 1.465 × 1 0 − 14 1.659 × 1 0 − 14 1.876 × 1 0 − 14 2.076 × 1 0 − 14
22 6.661 × 1 0 − 16 1.443 × 1 0 − 15 2.164 × 1 0 − 15 2.942 × 1 0 − 15 3.663 × 1 0 − 15 4.385 × 1 0 − 15 5.273 × 1 0 − 15 5.995 × 1 0 − 15 6.716 × 1 0 − 15 7.549 × 1 0 − 15
23 2.22 × 1 0 − 16 2.22 × 1 0 − 16 7.216 × 1 0 − 16 9.992 × 1 0 − 16 1.221 × 1 0 − 15 1.498 × 1 0 − 15 1.720 × 1 0 − 15 2.109 × 1 0 − 15 2.498 × 1 0 − 15 2.609 × 1 0 − 15
24 − 5.55 × 1 0 − 17 − 5.55 × 1 0 − 17 1.665 × 1 0 − 16 3.331 × 1 0 − 16 3.331 × 1 0 − 16 3.331 × 1 0 − 16 5.551 × 1 0 − 16 7.21645 × 1 0 − 16 7.216 × 1 0 − 16 9.43 × 1 0 − 16
25 1.11 × 1 0 − 16 1.11 × 1 0 − 16 1.11 × 1 0 − 16 2.22 × 1 0 − 16 3.886 × 1 0 − 16 2.224 × 1 0 − 16 3.886 × 1 0 − 16 3.886 × 1 0 − 16 3.885 × 1 0 − 16 5.551 × 1 0 − 16
26 − 1.11 × 1 0 − 16 − 2.22 × 1 0 − 16 − 2.22 × 1 0 − 16 − 2.22 × 1 0 − 16 − 1.11 × 1 0 − 16 − 2.22 × 1 0 − 16 − 1.11 × 1 0 − 16 − 1.11 × 1 0 − 16 − 1.110 × 1 0 − 16 − 1.110 × 1 0 − 16
27 0 0 − 1.11 × 1 0 − 16 − 1.11 × 1 0 − 16 − 1.11 × 1 0 − 16 − 1.11 × 1 0 − 16 − 1.11 × 1 0 − 16 − 1.11 × 1 0 − 16 0 0
28 0 − 1.110 × 1 0 − 16 − 1.110 × 1 0 − 16 0 0 − 1.110 × 1 0 − 16 − 1.110 × 1 0 − 16 − 1.110 × 1 0 − 16 − 1.110 × 1 0 − 16 − 1.110 × 1 0 − 16
29 − 5.55 × 1 0 − 17 − 3.33 × 1 0 − 16 − 2.775 × 1 0 − 16 − 2.775 × 1 0 − 16 − 2.775 × 1 0 − 16 − 5.551 × 1 0 − 17 − 2.775 × 1 0 − 16 − 2.775 × 1 0 − 16 − 2.775 × 1 0 − 16 − 3.330 × 1 0 − 16
30 2.775 × 1 0 − 16 1.665 × 1 0 − 16 1.665 × 1 0 − 16 1.665 × 1 0 − 16 5.551 × 1 0 − 17 1.665 × 1 0 − 16 1.665 × 1 0 − 16 1.665 × 1 0 − 16 1.665 × 1 0 − 16 1.665 × 1 0 − 16
Figure 3 Absolute error between analytical and semi-analytical solutions along ESE and VI methods.
Figure 3

Absolute error between analytical and semi-analytical solutions along ESE and VI methods.

Figure 4 Absolute error between analytical and semi-analytical solutions along MKud and VI methods.
Figure 4

Absolute error between analytical and semi-analytical solutions along MKud and VI methods.

Figure 5 Matching between the calculated absolute error along ESE and MKud analytical methods and VI semi-analytical technique.
Figure 5

Matching between the calculated absolute error along ESE and MKud analytical methods and VI semi-analytical technique.

4 Conclusion

Three analytical and semi-analytical methods for solving the positive nonlinear ( 2 + 1 )-D GKP problem have been developed. Numerous new computational solutions have been found. Some have been sketched in 2D, three-dimensional, and contour plots to illustrate the physical structure of the shallow oceans’ internal solitary waves. The semi-analytical solutions have been described using the VI technique to demonstrate the power and efficacy of the analytical methods employed. The uniqueness of our study is shown by comparing our answers to those found in previously published studies.

  1. Funding information: This research was supported by Taif University Researchers Supporting Project Number (TURSP-2020/52), Taif University, Taif, Saudi Arabia.

  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.

  4. Data availability statement: The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Received: 2021-03-29
Revised: 2021-11-10
Accepted: 2021-12-15
Published Online: 2022-02-04

© 2022 Dexu Zhao et al., published by De Gruyter

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

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https://doi.org/10.1515/phys-2022-0001
Keywords for this article
positive (2 + 1)-D GKP equation; analytical and semi-analytical solutions
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  1. Regular Articles
  2. Test influence of screen thickness on double-N six-light-screen sky screen target
  3. Analysis on the speed properties of the shock wave in light curtain
  4. Abundant accurate analytical and semi-analytical solutions of the positive Gardner–Kadomtsev–Petviashvili equation
  5. Measured distribution of cloud chamber tracks from radioactive decay: A new empirical approach to investigating the quantum measurement problem
  6. Nuclear radiation detection based on the convolutional neural network under public surveillance scenarios
  7. Effect of process parameters on density and mechanical behaviour of a selective laser melted 17-4PH stainless steel alloy
  8. Performance evaluation of self-mixing interferometer with the ceramic type piezoelectric accelerometers
  9. Effect of geometry error on the non-Newtonian flow in the ceramic microchannel molded by SLA
  10. Numerical investigation of ozone decomposition by self-excited oscillation cavitation jet
  11. Modeling electrostatic potential in FDSOI MOSFETS: An approach based on homotopy perturbations
  12. Modeling analysis of microenvironment of 3D cell mechanics based on machine vision
  13. Numerical solution for two-dimensional partial differential equations using SM’s method
  14. Multiple velocity composition in the standard synchronization
  15. Electroosmotic flow for Eyring fluid with Navier slip boundary condition under high zeta potential in a parallel microchannel
  16. Soliton solutions of Calogero–Degasperis–Fokas dynamical equation via modified mathematical methods
  17. Performance evaluation of a high-performance offshore cementing wastes accelerating agent
  18. Sapphire irradiation by phosphorus as an approach to improve its optical properties
  19. A physical model for calculating cementing quality based on the XGboost algorithm
  20. Experimental investigation and numerical analysis of stress concentration distribution at the typical slots for stiffeners
  21. An analytical model for solute transport from blood to tissue
  22. Finite-size effects in one-dimensional Bose–Einstein condensation of photons
  23. Drying kinetics of Pleurotus eryngii slices during hot air drying
  24. Computer-aided measurement technology for Cu2ZnSnS4 thin-film solar cell characteristics
  25. QCD phase diagram in a finite volume in the PNJL model
  26. Study on abundant analytical solutions of the new coupled Konno–Oono equation in the magnetic field
  27. Experimental analysis of a laser beam propagating in angular turbulence
  28. Numerical investigation of heat transfer in the nanofluids under the impact of length and radius of carbon nanotubes
  29. Multiple rogue wave solutions of a generalized (3+1)-dimensional variable-coefficient Kadomtsev--Petviashvili equation
  30. Optical properties and thermal stability of the H+-implanted Dy3+/Tm3+-codoped GeS2–Ga2S3–PbI2 chalcohalide glass waveguide
  31. Nonlinear dynamics for different nonautonomous wave structure solutions
  32. Numerical analysis of bioconvection-MHD flow of Williamson nanofluid with gyrotactic microbes and thermal radiation: New iterative method
  33. Modeling extreme value data with an upside down bathtub-shaped failure rate model
  34. Abundant optical soliton structures to the Fokas system arising in monomode optical fibers
  35. Analysis of the partially ionized kerosene oil-based ternary nanofluid flow over a convectively heated rotating surface
  36. Multiple-scale analysis of the parametric-driven sine-Gordon equation with phase shifts
  37. Magnetofluid unsteady electroosmotic flow of Jeffrey fluid at high zeta potential in parallel microchannels
  38. Effect of plasma-activated water on microbial quality and physicochemical properties of fresh beef
  39. The finite element modeling of the impacting process of hard particles on pump components
  40. Analysis of respiratory mechanics models with different kernels
  41. Extended warranty decision model of failure dependence wind turbine system based on cost-effectiveness analysis
  42. Breather wave and double-periodic soliton solutions for a (2+1)-dimensional generalized Hirota–Satsuma–Ito equation
  43. First-principle calculation of electronic structure and optical properties of (P, Ga, P–Ga) doped graphene
  44. Numerical simulation of nanofluid flow between two parallel disks using 3-stage Lobatto III-A formula
  45. Optimization method for detection a flying bullet
  46. Angle error control model of laser profilometer contact measurement
  47. Numerical study on flue gas–liquid flow with side-entering mixing
  48. Travelling waves solutions of the KP equation in weakly dispersive media
  49. Characterization of damage morphology of structural SiO2 film induced by nanosecond pulsed laser
  50. A study of generalized hypergeometric Matrix functions via two-parameter Mittag–Leffler matrix function
  51. Study of the length and influencing factors of air plasma ignition time
  52. Analysis of parametric effects in the wave profile of the variant Boussinesq equation through two analytical approaches
  53. The nonlinear vibration and dispersive wave systems with extended homoclinic breather wave solutions
  54. Generalized notion of integral inequalities of variables
  55. The seasonal variation in the polarization (Ex/Ey) of the characteristic wave in ionosphere plasma
  56. Impact of COVID 19 on the demand for an inventory model under preservation technology and advance payment facility
  57. Approximate solution of linear integral equations by Taylor ordering method: Applied mathematical approach
  58. Exploring the new optical solitons to the time-fractional integrable generalized (2+1)-dimensional nonlinear Schrödinger system via three different methods
  59. Irreversibility analysis in time-dependent Darcy–Forchheimer flow of viscous fluid with diffusion-thermo and thermo-diffusion effects
  60. Double diffusion in a combined cavity occupied by a nanofluid and heterogeneous porous media
  61. NTIM solution of the fractional order parabolic partial differential equations
  62. Jointly Rayleigh lifetime products in the presence of competing risks model
  63. Abundant exact solutions of higher-order dispersion variable coefficient KdV equation
  64. Laser cutting tobacco slice experiment: Effects of cutting power and cutting speed
  65. Performance evaluation of common-aperture visible and long-wave infrared imaging system based on a comprehensive resolution
  66. Diesel engine small-sample transfer learning fault diagnosis algorithm based on STFT time–frequency image and hyperparameter autonomous optimization deep convolutional network improved by PSO–GWO–BPNN surrogate model
  67. Analyses of electrokinetic energy conversion for periodic electromagnetohydrodynamic (EMHD) nanofluid through the rectangular microchannel under the Hall effects
  68. Propagation properties of cosh-Airy beams in an inhomogeneous medium with Gaussian PT-symmetric potentials
  69. Dynamics investigation on a Kadomtsev–Petviashvili equation with variable coefficients
  70. Study on fine characterization and reconstruction modeling of porous media based on spatially-resolved nuclear magnetic resonance technology
  71. Optimal block replacement policy for two-dimensional products considering imperfect maintenance with improved Salp swarm algorithm
  72. A hybrid forecasting model based on the group method of data handling and wavelet decomposition for monthly rivers streamflow data sets
  73. Hybrid pencil beam model based on photon characteristic line algorithm for lung radiotherapy in small fields
  74. Surface waves on a coated incompressible elastic half-space
  75. Radiation dose measurement on bone scintigraphy and planning clinical management
  76. Lie symmetry analysis for generalized short pulse equation
  77. Spectroscopic characteristics and dissociation of nitrogen trifluoride under external electric fields: Theoretical study
  78. Cross electromagnetic nanofluid flow examination with infinite shear rate viscosity and melting heat through Skan-Falkner wedge
  79. Convection heat–mass transfer of generalized Maxwell fluid with radiation effect, exponential heating, and chemical reaction using fractional Caputo–Fabrizio derivatives
  80. Weak nonlinear analysis of nanofluid convection with g-jitter using the Ginzburg--Landau model
  81. Strip waveguides in Yb3+-doped silicate glass formed by combination of He+ ion implantation and precise ultrashort pulse laser ablation
  82. Best selected forecasting models for COVID-19 pandemic
  83. Research on attenuation motion test at oblique incidence based on double-N six-light-screen system
  84. Review Articles
  85. Progress in epitaxial growth of stanene
  86. Review and validation of photovoltaic solar simulation tools/software based on case study
  87. Brief Report
  88. The Debye–Scherrer technique – rapid detection for applications
  89. Rapid Communication
  90. Radial oscillations of an electron in a Coulomb attracting field
  91. Special Issue on Novel Numerical and Analytical Techniques for Fractional Nonlinear Schrodinger Type - Part II
  92. The exact solutions of the stochastic fractional-space Allen–Cahn equation
  93. Propagation of some new traveling wave patterns of the double dispersive equation
  94. A new modified technique to study the dynamics of fractional hyperbolic-telegraph equations
  95. An orthotropic thermo-viscoelastic infinite medium with a cylindrical cavity of temperature dependent properties via MGT thermoelasticity
  96. Modeling of hepatitis B epidemic model with fractional operator
  97. Special Issue on Transport phenomena and thermal analysis in micro/nano-scale structure surfaces - Part III
  98. Investigation of effective thermal conductivity of SiC foam ceramics with various pore densities
  99. Nonlocal magneto-thermoelastic infinite half-space due to a periodically varying heat flow under Caputo–Fabrizio fractional derivative heat equation
  100. The flow and heat transfer characteristics of DPF porous media with different structures based on LBM
  101. Homotopy analysis method with application to thin-film flow of couple stress fluid through a vertical cylinder
  102. Special Issue on Advanced Topics on the Modelling and Assessment of Complicated Physical Phenomena - Part II
  103. Asymptotic analysis of hepatitis B epidemic model using Caputo Fabrizio fractional operator
  104. Influence of chemical reaction on MHD Newtonian fluid flow on vertical plate in porous medium in conjunction with thermal radiation
  105. Structure of analytical ion-acoustic solitary wave solutions for the dynamical system of nonlinear wave propagation
  106. Evaluation of ESBL resistance dynamics in Escherichia coli isolates by mathematical modeling
  107. On theoretical analysis of nonlinear fractional order partial Benney equations under nonsingular kernel
  108. The solutions of nonlinear fractional partial differential equations by using a novel technique
  109. Modelling and graphing the Wi-Fi wave field using the shape function
  110. Generalized invexity and duality in multiobjective variational problems involving non-singular fractional derivative
  111. Impact of the convergent geometric profile on boundary layer separation in the supersonic over-expanded nozzle
  112. Variable stepsize construction of a two-step optimized hybrid block method with relative stability
  113. Thermal transport with nanoparticles of fractional Oldroyd-B fluid under the effects of magnetic field, radiations, and viscous dissipation: Entropy generation; via finite difference method
  114. Special Issue on Advanced Energy Materials - Part I
  115. Voltage regulation and power-saving method of asynchronous motor based on fuzzy control theory
  116. The structure design of mobile charging piles
  117. Analysis and modeling of pitaya slices in a heat pump drying system
  118. Design of pulse laser high-precision ranging algorithm under low signal-to-noise ratio
  119. Special Issue on Geological Modeling and Geospatial Data Analysis
  120. Determination of luminescent characteristics of organometallic complex in land and coal mining
  121. InSAR terrain mapping error sources based on satellite interferometry
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