Application of HAM to seventh order KdV equations

Arora, Rajan; Sharma, Himanshu

doi:10.1007/s13198-016-0490-7

Cited by 18 publications

(16 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this article, all the figures and tables are included by considering the values of the parameter as a = 0.1. 36 The achieved outcomes are compared with the solution of Arora and Sharma 36 and El-Sayed and Kaya 37 for α = 1, which shows the validation of the present study. Calculated results are displayed in terms of plots.…”

Section: Numerical Results and Discussionsupporting

confidence: 72%

See 1 more Smart Citation

On the wave solutions of time‐fractional Sawada‐Kotera‐Ito equation arising in shallow water

Jena

Sedighi

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

A widescale description of various phenomena in science and engineering such as physics, chemical, acoustics, control theory, finance, economics, mechanical engineering, civil engineering, and social sciences is well described by nonlinear fractional differential equations (NLFDEs). In turbulence, fluid dynamics, and nonlinear biological systems, applications of NLFDEs can also be found. NLFDEs are believed to be powerful tools to describe real-world problems more precisely than the differential equation of the integer-order. In this research, we have used the fractional reduced differential transform method (FRDTM) to find the solution of the time-fractional Sawada-Kotera-Ito seventh-order equation. The novelty of the FRDTM is that it does not require any discretization, transformation, perturbation, or any restrictive conditions. In addition, compared to other methods, this approach needs less calculation. For special cases of an integer and noninteger orders, computed results are compared with existing results. Present results are in good agreement with the existing solutions. Here, the fractional derivatives are considered in the Caputo sense. Convergence analysis of the results has also been studied with the increasing number of terms of the solution.

show abstract

Section: Numerical Results and Discussionsupporting

confidence: 72%

“…Comparison of the present result of Equation 3 with the existing results at α = 1 are demonstrated in Table 2. Solution plot of Equation 3 has been com- 36 Absolute error…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

On the wave solutions of time‐fractional Sawada‐Kotera‐Ito equation arising in shallow water

Jena

Sedighi

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…The concept of the physical phenomena, depends upon the nature of the solutions of PDEs, so to find the solutions of PDEs, the analytical, semi analytical and numerical methods are introduced. It has been a requirement to choose a competent mechanism that adopts such mathematical models which cover physical processes as well [ 1 ].…”

Section: Introductionmentioning

confidence: 99%

“…So, the scientists have introduced many techniques like perturbation methods, approximation methods and numerical methods to overcome the problems of such DEs. Considering the effect of nonlinearity, many analytical techniques concentrate on linearizing the physical phenomena, that yields the variation in the nature of these analytical solutions from the actual ones and becomes a cause of change in the real physics of the given phenomena [ 1 ].…”

Section: Introductionmentioning

confidence: 99%

“…In literature, there exist many analytical and numerical schemes like He’s variation iteration method, pseudospectral method, Adomian decomposition method (ADM), Bäcklund transformations to solve such problems, finite difference method (FDM), finite element method (FEM), finite volume method (FVM), homotopy analysis method (HAM), Fourier spectral method (FSM) and variation iteration method (VIM) to solve such problems [ 1 , 2 , 14 – 17 ], some of them are given by:…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A reliable algorithm to compute the approximate solution of KdV-type partial differential equations of order seven

2021

View full text Add to dashboard Cite

The approximate solution of KdV-type partial differential equations of order seven is presented. The algorithm based on one-dimensional Haar wavelet collocation method is adapted for this purpose. One-dimensional Haar wavelet collocation method is verified on Lax equation, Sawada-Kotera-Ito equation and Kaup-Kuperschmidt equation of order seven. The approximated results are displayed by means of tables (consisting point wise errors and maximum absolute errors) to measure the accuracy and proficiency of the scheme in a few number of grid points. Moreover, the approximate solutions and exact solutions are compared graphically, that represent a close match between the two solutions and confirm the adequate behavior of the proposed method.

show abstract

Fractional residual series for conformable time‐fractional Sawada–Kotera–Ito, Lax, and Kaup–Kupershmidt equations of seventh order

Al‐Smadi

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

The purpose of the present work is to analyze and investigate the analytical solution of the seventh‐order fractional Sawada–Kotera–Ito, Lax, and Kaup–Kupershmidt equations under the conformable derivatives. Using the residual series expansion algorithm, the posed fractional models are studied analytically and numerically. Simultaneously, convergence analysis and error estimation are performed for the proposed technique. For this purpose, an advanced numerical algorithm is designed to handle the complexity of nonlinear fractional terms. Some experiments are provided to support theoretical analysis in a one‐dimensional finite compact regime in light of the conformable sense. Eventually, numerical comparison is performed with other existing techniques along with some two‐ and three‐dimensional representative graphs. Physical interpretation of the solution behaviors is discussed as well for various α values in an adequate time duration. Comparative results indicate the superiority and great flexibility of the presented method for fractional evolution systems arising in nonlinear wave propagation phenomena.

show abstract

Application of HAM to seventh order KdV equations

Cited by 18 publications

References 11 publications

On the wave solutions of time‐fractional Sawada‐Kotera‐Ito equation arising in shallow water

On the wave solutions of time‐fractional Sawada‐Kotera‐Ito equation arising in shallow water

A reliable algorithm to compute the approximate solution of KdV-type partial differential equations of order seven

Fractional residual series for conformable time‐fractional Sawada–Kotera–Ito, Lax, and Kaup–Kupershmidt equations of seventh order

Contact Info

Product

Resources

About