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An efficient approach for the numerical solution of fifth-order KdV equations

  • Hijaz Ahmad , Tufail A. Khan and Shao-Wen Yao EMAIL logo
Published/Copyright: July 22, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

Abstract

The main aim of this article is to use a new and simple algorithm namely the modified variational iteration algorithm-II (MVIA-II) to obtain numerical solutions of different types of fifth-order Korteweg-de Vries (KdV) equations. In order to assess the precision, stability and accuracy of the solutions, five test problems are offered for different types of fifth-order KdV equations. Numerical results are compared with the Adomian decomposition method, Laplace decomposition method, modified Adomian decomposition method and the homotopy perturbation transform method, which reveals that the MVIA-II exceptionally productive, computationally attractive and has more accuracy than the others.

Keywords: Korteweg-de Vries equation; variational iteration algorithm-II; fifth-order KdV equation; Caudrey-Dodd-Gibbon equation; Lax equation; Kawahara equation; Sawada-Kotera equation
MSC 2010: 35Q53

1 Introduction

Nonlinear partial differential equations (PDEs) have turned into a helpful tool for delineating a large number of physical problems that arise in many fields of mathematics and science, including fluid dynamics, chemical physics, hydrodynamics, fluid mechanics, heat and mass transfer, solid-state physics, chemical kinematics, plasma physics, etc. [1,2,3,4,5,6,7,8,9,10,11,12]. The Korteweg-de Vries (KdV) equation is a nonlinear PDE and assumes a significant role with numerous applications, for example, in illustration of magneto-acoustic waves, ion-acoustic waves, nonlinear LC circuit’s waves and shallow water waves in plasmas. This equation has experienced a few extensions and modifications leading to numerous types of KdV equations showing up in three, five, seven or more order forms deferential equations. However, some notable types of them are specifically noteworthy in physical phenomena. These types of the KdV equation of fifth order are as follows:

  1. The Sawada-Kotera (SK) equation [13]:

    (1) ∂ u ∂ t + 45 u 2 ∂ u ∂ x + 15 ∂ u ∂ x ∂ 2 u ∂ x 2 + 15 u ∂ 3 u ∂ x 3 + ∂ 5 u ∂ x 5 = 0 .

  2. The Caudrey-Dodd-Gibbon (C-D-G) equation [14]:

    (2) ∂ u ∂ t + 180 u 2 ∂ u ∂ x + 30 ∂ u ∂ x ∂ 2 u ∂ x 2 + 30 u ∂ 3 u ∂ x 3 + ∂ 5 u ∂ x 5 = 0 .

  3. The Lax equation [15,16]:

    (3) ∂ u ∂ t + 30 u 2 ∂ u ∂ x + 30 ∂ u ∂ x ∂ 2 u ∂ x 2 + 10 u ∂ 3 u ∂ x 3 + ∂ 5 u ∂ x 5 = 0 .

  4. The fifth-order KdV equation [17]:

    (4) ∂ u ∂ t + u ∂ u ∂ x − u ∂ 3 u ∂ x 3 + ∂ 5 u ∂ x 5 = 0 .

  5. The Kawahara equation [18]:

    (5) ∂ u ∂ t + u ∂ u ∂ x + ∂ 3 u ∂ x 3 − ∂ 5 u ∂ x 5 = 0 .

The fifth-order KdV equation, also called generalized KdV, is the indispensable model for numerous physical phenomena in quantum mechanics and nonlinear optics. Its general solution is not known while the exact solution for the particular case of solitary waves of the KdV equation of fifth order is reported in [19]. To solve this model numerically, several methods can be found in the literature, such as Wazwaz [16] has employed the tanh and sine-cosine methods, Djidjeli et al. [20] used finite difference schemes, Chun [21] utilized the Exp-function method, Abbasbandy and Zakaria [22] employed the homotopy analysis method for soliton solution, Goswami et al. [18] applied homotopy perturbation transform method (HPTM), Kaya [23,24] used the decomposition method as well as Adomian decomposition method, Rafei and Ganji [25] provided a comparative study of homotopy perturbation method and Adomian decomposition method, Lei et al. used the homogeneous balance method [26], Helal and Mehanna [27] provided a comparative investigation of the Adomian decomposition method and Crank-Nicholas method, Handibag and Karande [17] have developed the Laplace decomposition method and Bakodah [28] has used the modified Adomian decomposition method.

The main objective of this article is to employ modified variational iteration algorithm-II (MVIA-II) for the numerical treatment of nonlinear PDEs in physical sciences and engineering via the KdV equation of fifth order. The article is prepared as follows. In Section 2, MVIA-II is described. In Section 3, five different forms of KdV equations have been investigated to show the applicability and precision of the modified algorithm, and in Section 4, conclusions are given.

2 Implementation of MVIA-II

In this section, we will describe MVIA-II for finding the numerical solutions of fifth-order KdV equations. Suppose the generalized form of the KdV equation of fifth order:

(6) ∂ u ∂ t + α u ∂ 3 u ∂ x 3 + β ∂ u ∂ x ∂ 2 u ∂ x 2 + γ u 2 ∂ u ∂ x + ∂ 5 u ∂ x 5 = 0 ,

where γ , β and α are nonzero arbitrary constants. Approximate solution u k + 1 ( x ) of Eq. (6) for given initial condition u 0 ( x ) can be acquired as follows:

(7) u k + 1 ( x , t , h ) = u 0 ( x , t , h ) + h ∫ 0 t λ ( η ) ∂ u k ( x , η , h ) ∂ ( η ) + α u k ( x , η , h ) ∂ 3 u k ( x , η , h ) ∂ ( x ) 3 ˜ + β ∂ u k ( x , η , h ) ∂ ( x ) ∂ 2 u k ( x , η , h ) ∂ ( x ) 2 ˜ + γ u k 2 ( x , η , h ) ∂ u k ( x , η , h ) ∂ ( x ) ˜ + ∂ 5 u k ( x , η , h ) ∂ ( x ) 5 ˜ d η ,

where λ and h are two unknown parameters, the first one is known as the Lagrange multiplier [29], while the second one is an auxiliary parameter, which was used to accelerate the convergence in different methods [30,31,32,33,34,35,36,37], but we are the first to use it in VIA-II. The Lagrange multiplier can be found by taking δ on both sides of the recurrence relation (7) w.r.t. the variable u k ( x ) ,

(8) δ u k + 1 ( x , t , h ) = δ u 0 ( x , t , h ) + h δ ∫ 0 t λ ( η ) ∂ u k ( x , η , h ) ∂ ( η ) + α u k ( x , η , h ) ∂ 3 u k ( x , η , h ) ∂ ( x ) 3 ˜ + β ∂ u k ( x , η , h ) ∂ ( x ) ∂ 2 u k ( x , η , h ) ∂ ( x ) 2 ˜ + γ u k 2 ( x , η , h ) ∂ u k ( x , η , h ) ∂ ( x ) ˜ + ∂ 5 u k ( x , η , h ) ∂ ( x ) 5 ˜ d η ,

where u k ( x , η , h ) ˜ is a restricted term, i.e., δ u k ( x , η , h ) ˜ = 0 . The significant value of λ ( η ) can be identified by making use of the variation theory [38,39,40]. While h is an auxiliary term which is utilized to ensure convergence of numerically obtained solution ideally by limiting norm 2 of residual error over the space of the given system. The ideal decision of this h improves the precision and proficiency of the algorithm. For the optimal solution of the auxiliary parameter, we define a residual function for the approximated solution:

(9) r p ( x , t , h ) = ∂ u p ( x , t , h ) ∂ ( t ) + α u p ( x , t , h ) ∂ 3 u p ( x , t , h ) ∂ ( x ) 3 + β ∂ u p ( x , t , h ) ∂ ( x ) ∂ 2 u p ( x , t , h ) ∂ ( x ) 2 + γ u p 2 ( x , t , h ) ∂ u p ( x , t , h ) ∂ ( x ) + ∂ 5 u p ( x , t , h ) ∂ ( x ) 5 ,

where p is the number of approximation. The square of function (9) for the pth-order approximation with respect to parameter h for ( x , t ) ϵ [ a , b ] × [ a , b ] is

(10) 1 ( b + 1 ) 2 ∑ i = a b ∑ j = a b ( r p ( i , j , h ) ) 2 1 / 2 .

The lowest value of function (10) occurs at the point which should be selected and that will be the value of auxiliary parameter h. In small domains where the standard VIA-II gives high-order accuracy, this technology gives the value of h as 1. Putting values of both the parameters in the recurrence relation, we get an iterative formula:

(11) u k + 1 ( x , t , h ) = u 0 ( x , t , h ) + h ∫ 0 t λ ( η ) α u k ( x , η , h ) ∂ 3 u k ( x , η , h ) ∂ ( x ) 3 + β ∂ u k ( x , η , h ) ∂ ( x ) ∂ 2 u k ( x , η , h ) ∂ ( x ) 2 + γ u k 2 ( x , η , h ) ∂ u k ( x , η , h ) ∂ ( x ) + ∂ 5 u k ( x , η , h ) ∂ ( x ) 5 d η .

Introducing with a proper initial guess, one can get the different approximations by utilizing the iterative structure (11). This gives an exact solution u ( x ) , when

(12) u ( x ) = lim k → ∞ u k ( x ) .

This algorithm is named as MVIA-II. We utilize this modified algorithm for the solution of the different types of KdV equations of fifth order, which is able to provide numerical results for nonlinear and linear problems, in a direct way very accurately.

3 Numerical examples

In this section, MVIA-II is employed for the numerical treatment of the SK equation, C-D-G equation, a fifth-order KdV equation, Lax equation and Kawahara equation. Numerical and graphical results gained from the modified algorithm are very encouraging, empowering, noteworthy and significant. Illustrated test problems revealed the effectiveness and power of the suggested algorithm.

3.1 Test Problem 1

The SK equation (1) having the following initial condition

(13) u ( x , 0 ) = 2 k 2 sech 2 [ k ( x − c ) ] ,

and the exact solution of SK equation (1) with condition (13) was given by [28]

(14) u ( x , t ) = 2 k 2 sech 2 [ k ( x − 16 k 4 t − c ) ] .

The numerical solutions for Test Problem 3.1 using MVIA-II are reported in Table 1 of SK equation (1). To show the efficiency and applicability of the proposed algorithm in comparison with adomian decomposition method (ADM) [24] and modified ADM [28], the absolute errors are reported in Tables 1 and 2 for various values of t and x and k = 0.01 and c = 0.0 . A full agreement between the results of MVI-II and exact solution is observed, which confirms the validity of the proposed algorithm. In comparison with [24,28] results, one can ensure that the results of MVIA-II are more efficient and reliable. Results are also shown through illustrations in Figure 1 at different times t and for different x, while the behaviour of approximate and exact solutions can be seen in Figure 2.

Table 1

Numerical results of SK equation (1) in terms of absolute error using MVIA-II for Test Problem 3.1

x/t t = 0.2 t = 0.4 t = 5.0
MVIA-II [28] MVIA-II [28] MVIA-II [28]
2 0 1.54499 × 10−18 2.71050 × 10−20 4.52654 × 10−18 2.71050 × 10−20 6.6435 × 10−15
4 2.71050 × 10−20 5.36680 × 10−18 5.42101 × 10−20 1.12757 × 10−17 8.13151 × 10−20 1.32874 × 10−14
6 2.71050 × 10−20 1.13841 × 10−17 2.71050 × 10−20 2.02746 × 10−17 8.13151 × 10−20 1.99336 × 10−14
8 2.71050 × 10−20 1.97054 × 10−17 5.42101 × 10−20 3.15232 × 10−17 5.42101 × 10−20 2.65820 × 10−14
10 5.42101 × 10−20 3.02492 × 10−17 5.42101 × 10−20 4.50486 × 10−17 5.42101 × 10−20 3.32327 × 10−14
Table 2

Comparison of numerical solutions of SK equation (1) for Test Problem 3.1

x/t t = 0.2 t = 0.4 t = 0.5
MVIA-II [24] MVIA-II [24] MVIA-II [24]
0.1 0 9.59980 × 10−16 2.71050 × 10−20 1.91996 × 10−15 8.13151 × 10−20 6.6435 × 10−15
0.2 2.71050 × 10−20 1.91996 × 10−15 8.13151 × 10−20 3.83989 × 10−14 5.42101 × 10−20 1.32874 × 10−14
0.3 2.71050 × 10−20 2.87980 × 10−15 2.71050 × 10−20 5.75966 × 10−14 5.42101 × 10−20 1.99336 × 10−14
0.4 2.71050 × 10−20 3.83973 × 10−15 2.71050 × 10−20 7.67932 × 10−14 5.42101 × 10−20 2.65820 × 10−14
0.5 2.71050 × 10−20 4.79941 × 10−15 5.42101 × 10−20 9.59871 × 10−14 5.42101 × 10−20 3.32327 × 10−14
Figure 1 The exact and numerical solutions of Test Problem 3.1 for t = 10 t=10 and k = 0.01 k=0.01 .
Figure 1

The exact and numerical solutions of Test Problem 3.1 for t = 10 and k = 0.01 .

Figure 2 Approximate (MVIA-II) (left) and exact (right) solutions of Test Problem 3.1 for k = 0.01 k=0.01 in space-time graph form.
Figure 2

Approximate (MVIA-II) (left) and exact (right) solutions of Test Problem 3.1 for k = 0.01 in space-time graph form.

3.2 Test Problem 2

The C-D-G equation (22) having the initial condition:

(15) u ( x , 0 ) = k 2 e k x ( 1 + e k x ) 2 ,

and the exact solution of equation (22) with condition (15) was given by [28]

(16) u ( x , t ) = k 2 e k ( x − k 4 t ) ( 1 + e k ( x − k 4 t ) ) 2 .

The obtained numerical results from the MVIA-II for Test Problem 3.2 are reported in Table 3. Results by MVIA-II show improvement over previous ones reported by [28]. Comparison of approximate and exact solutions is also shown through curves in Figure 3 for k = 0.01 and time t = 10 . While Figure 4 shows the behaviour of the approximate and exact solution, which indicates that the proposed algorithm is a powerful mathematical tool for getting an accurate numerical solution of the C-D-G equation. The results of the proposed algorithm (MVIA-II) are compared with the results of the modified Adomian decomposition method [28]. It is well defined from the numerical and graphical results that the performance of the proposed algorithm is more precise.

Table 3

Comparison of numerical solutions of C–D–G equation (22) for Test Problem 3.2

x/t t = 0.4 t = 0.8 t = 5.0
MVIA-II [28] MVIA-II [28] MVIA-II [28]
2 6.77626 × 10−21 3.11957 × 10−21 6.77626 × 10−21 1.7893 × 10−20 1.01643 × 10−20 2.48025 × 10−18
4 3.38813 × 10−21 2.70138 × 10−21 6.77626 × 10−21 2.01336 × 10−20 1.35525 × 10−20 5.70681 × 10−18
6 3.38813 × 10−21 5.50257 × 10−21 0 3.91461 × 10−20 1.01643 × 10−20 8.93996 × 10−18
8 6.77626 × 10−21 4.74685 × 10−21 6.77626 × 10−21 5.79897 × 10−20 6.77626 × 10−21 1.21797 × 10−17
10 3.38813 × 10−21 7.21049 × 10−21 6.77626 × 10−21 5.63358 × 10−20 3.38813 × 10−21 1.54092 × 10−17
Figure 3 The exact and numerical solutions’ comparison of Test Problem 3.2 for k = 0.01 k=0.01 and time t = 10 t=10 .
Figure 3

The exact and numerical solutions’ comparison of Test Problem 3.2 for k = 0.01 and time t = 10 .

Figure 4 Approximate (MVIA-II) (left) and exact (right) solutions of Test Problem 3.2 for k = 0.01 k=0.01 .
Figure 4

Approximate (MVIA-II) (left) and exact (right) solutions of Test Problem 3.2 for k = 0.01 .

3.3 Test Problem 3

The Lax equation (3) having the following initial condition

(17) u ( x , 0 ) = 2 k 2 [ 2 − 3 tanh 2 k ( x − c ) ] ,

and the exact solution of Lax’s equation with condition (17) was given by [28]

(18) u ( x , t ) = 2 k 2 [ 2 − 3 tanh 2 k ( x − 56 k 4 t − c ) ] .

From the numerical results of Test Problem 3.3 using MVIA-II reported in Table 4, we conclude that the obtained results are in quite agreement with the exact and with the results given in [28]. Figure 5 shows the comparison of approximate and exact solutions of Test Problem 3.3 for t = 10 and k = 0.01 .

Table 4

Comparison of numerical solutions of Lax equation (3) for Test Problem 3.3

x/t t = 0.2 t = 0.8 t = 5.0
MVIA-II [28] MVIA-II [28] MVIA-II [28]
2 5.75976 × 10−14 5.76197 × 10−14 2.30320 × 10−13 2.30342 × 10−13 1.42090 × 10−12 1.42104 × 10−12
4 1.15193 × 10−13 1.15281 × 10−13 4.60638 × 10−13 4.60727 × 10−13 2.84181 × 10−12 2.84213 × 10−12
6 1.72787 × 10−13 1.72985 × 10−13 6.90954 × 10−13 6.91153 × 10−13 4.26271 × 10−12 4.26326 × 10−12
8 2.30378 × 10−13 2.30730 × 10−13 9.21269 × 10−13 9.21621 × 10−13 5.68361 × 10−12 5.68444 × 10−12
10 2.87968 × 10−13 2.88518 × 10−13 1.15158 × 10−12 1.15213 × 10−12 7.10452 × 10−12 7.10566 × 10−12
Figure 5 Exact and approximate solutions’ comparison of Test Problem 3.3 for t = 10 t=10 and k = 0.01 k=0.01 .
Figure 5

Exact and approximate solutions’ comparison of Test Problem 3.3 for t = 10 and k = 0.01 .

3.4 Test Problem 4

Consider the following KdV equation of fifth order (4)

(19) ∂ u ∂ t + u ∂ u ∂ x − u ∂ 3 u ∂ x 3 + ∂ 5 u ∂ x 5 = 0 ,

having the initial condition:

(20) u ( x , 0 ) = e x ,

and the exact solution of this fKdV equation (4) with condition (20) was given by [18]

(21) u ( x , t ) = e x − t .

The numerical simulation for different values of times t and x of Test Problem 3.4 using the MVIA-II is reported in Table 5. In the Tables 5 and 6, the comparison of numerical results of the MVIA-II, Laplace decomposition method [17] and homotopy perturbation transform method [18] for Test Problem 3.4 has been carried out. Comparison of numerical results is shown in Table 5 in terms of absolute errors for second-, fourth- and sixth-order approximations, as the order of approximation increases, the order of accuracy increases and results converge to exact solution and to show the efficiency and reliability of the MVIA-II for different values of x and t, the comparison of errors with a well-known method for nonlinear PDEs, the Laplace decomposition method, is shown in Table 6.

Table 5

Comparison of numerical solutions of fKdV equation (4) for Test Problem 3.4

x/t E 2 E 4 E 6
MVIA-II [18] MVIA-II [18] MVIA-II [18]
0.1 4.03000 × 10−7 4.03001 × 10−7 2.03168 × 10−10 2.04000 × 10−10 4.85726 × 10−14 2.00000 × 10−12
0.2 3.14615 × 10−6 3.14615 × 10−6 6.39581 × 10−9 6.39800 × 10−9 6.14129 × 10−12 8.00000 × 10−12
0.3 1.03656 × 10−5 1.03656 × 10−5 4.77886 × 10−8 4.77890 × 10−8 1.03657 × 10−10 1.03000 × 10−10
0.4 2.39942 × 10−5 2.39942 × 10−5 1.98186 × 10−7 1.98186 × 10−7 7.67223 × 10−10 7.66000 × 10−10
0.5 4.57809 × 10−5 4.57809 × 10−5 5.95330 × 10−7 5.95333 × 10−7 3.61487 × 10−9 3.61500 × 10−9
Table 6

Comparison of absolute errors of the MVIA-II and the Laplace decomposition method for sixth approximation

t/x x = 1.0 x = 1.5 x = 2.5
MVIA-II [17] MVIA-II [17] MVIA-II [17]
0.01 0 4.38198 × 10−11 0 3.32204 × 10−10 3.55271 × 10−15 4.25802 × 10−9
0.02 8.88178 × 10−16 7.22467 × 10−11 8.88178 × 10−16 5.47711 × 10−10 3.55271 × 10−15 7.02029 × 10−9
0.03 1.19904 × 10−14 1.19114 × 10−10 1.95399 × 10−14 9.03023 × 10−10 4.97379 × 10−14 1.15745 × 10−8
0.04 8.79296 × 10−14 1.96387 × 10−10 1.43884 × 10−13 1.48883 × 10−9 3.94351 × 10−13 1.90831 × 10−8
0.05 4.19220 × 10−13 3.23787 × 10−10 6.90114 × 10−13 2.45467 × 10−9 1.87405 × 10−12 3.14627 × 10−8

3.5 Test Problem 5

Consider the Kawahara equation (5)

(22) ∂ u ∂ t + u ∂ u ∂ x + ∂ 3 u ∂ x 3 − ∂ 5 u ∂ x 5 = 0 ,

having the initial condition:

(23) u ( x , 0 ) = 105 169 sech 4 x − x 0 2 13 ,

and the exact solution of equation (5) with condition (23) was given by [18]

(24) u ( x , t ) = 105 169 sech 4 1 2 13 x − x 0 + 36 t 169 .

To demonstrate the applicability and efficiency of our modified algorithm, we report the absolute errors for different values of x and t in Table 7. From the tabulated data and Figure 6, one can observe that the numerical and exact solutions by MVIA-II for the Kawahara equation (5) for different time levels with x = 6.0 are in good agreement for second-order approximation.

Table 7

Comparison of numerical solutions of Kawahara equation (5) for Test Problem 3.5

x/t MVIA-II [18]
0.1 2.96656 × 10−8 1.64944 × 10−6
0.2 2.37783 × 10−7 6.67437 × 10−6
0.3 8.04066 × 10−7 1.51864 × 10−5
0.4 1.90959 × 10−6 2.72977 × 10−5
0.5 3.73681 × 10−6 4.31169 × 10−5
Figure 6 Absolute erorr graph for the Test Problem 3.5.
Figure 6

Absolute erorr graph for the Test Problem 3.5.

4 Conclusion

This article shows that the MVIA-II is very proficient, reliable and practically well suited for use in finding new traveling wave solutions for the higher order differential equations of KdV type. The reliability and accuracy of the method, as well as the decrease in the size of computational work, give this modified algorithm a more extensive pertinency. The results demonstrate that the algorithm is reliable and effective and gives more accurate solutions. This modified algorithm makes easy the computational work for solving nonlinear problems that emerge in science and engineering, and results of high degree accuracy can be obtained in few iterations as compared to earlier methods, and we hope that the obtained results will be useful for further studies in scientific materials science and engineering.

Acknowledgments

Hijaz Ahmad thanks Prof. Dr. Ji-Huan He for appreciation and continuous helpful discussions and advice. All the authors thank the Editor-in-Chief and unknown referees for the fruitful comments and significant remarks that helped them in improving the quality and readability of the paper, which led to a significant improvement of the paper. This work was supported by the National Natural Science Foundation of China (No. 71601072) and the Key Scientific Research Project of Higher Education Institutions in Henan Province of China (No. 20B110006).

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Received: 2019-11-15
Revised: 2020-04-28
Accepted: 2020-05-22
Published Online: 2020-07-22

© 2020 Hijaz Ahmad et al., published by De Gruyter

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

Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
https://doi.org/10.1515/math-2020-0036
Keywords for this article
Korteweg-de Vries equation; variational iteration algorithm-II; fifth-order KdV equation; Caudrey-Dodd-Gibbon equation; Lax equation; Kawahara equation; Sawada-Kotera equation
Creative Commons
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Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
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