This paper studies the generalized Kawahara equation in terms of the approximate homotopy symmetry method and the approximate homotopy direct method. Using both methods it obtains the similarity reduction solutions an...This paper studies the generalized Kawahara equation in terms of the approximate homotopy symmetry method and the approximate homotopy direct method. Using both methods it obtains the similarity reduction solutions and similarity reduction equations of different orders, showing that the approximate homotopy direct method yields more general approximate similarity reductions than the approximate homotopy symmetry method. The homotopy series solutions to the generalized Kawahara equation are consequently derived.展开更多
In this work, an algorithm based on method of lines coupled with radial basis functions namely meshless method of lines (MMOL) is presented for the numerical solution of Kawahara, modified Kawahara and KdV Kawahara eq...In this work, an algorithm based on method of lines coupled with radial basis functions namely meshless method of lines (MMOL) is presented for the numerical solution of Kawahara, modified Kawahara and KdV Kawahara equations. The motion of a single solitary wave, interaction of two and three solitons and the phenomena of wave generation is discussed. The results are compared with the exact solution and with the results in the relevant literature to show the efficiency of the method.展开更多
The Kawahara equation is studied through the approximate homotopy symmetry method. Under this method we get the similarity reduction solutions of the Kawahara equation, leading to the corresponding homotopy series sol...The Kawahara equation is studied through the approximate homotopy symmetry method. Under this method we get the similarity reduction solutions of the Kawahara equation, leading to the corresponding homotopy series solutions. Furthermore, the similarity solutions of the corresponding reduced linear ordinary differential equations are also considered.展开更多
We derive a new multisymplectic integrator for the Kawahara-type equation which is a fully explicit scheme and thus needs less computation cost. Multisympecticity of such scheme guarantees the long-time numerical beha...We derive a new multisymplectic integrator for the Kawahara-type equation which is a fully explicit scheme and thus needs less computation cost. Multisympecticity of such scheme guarantees the long-time numerical behaviors. Nu- merical experiments are presented to verify the accuracy of this scheme as well as the excellent performance on invariant preservation for three kinds of Kawahara-type equations.展开更多
In this paper, we present a combination method of constructing the explicit and exact solutions of nonlinear partial differential equations. And as an illustrative example, we apply the method to the Benney-Kawahara-L...In this paper, we present a combination method of constructing the explicit and exact solutions of nonlinear partial differential equations. And as an illustrative example, we apply the method to the Benney-Kawahara-Lin equation and derive its many explicit and exact solutions which are all new solutions.展开更多
In this paper,the generalized dissipative Kawahara equation in the sense of conformable fractional derivative is presented and solved by applying the tanh-coth-expansion and sine-cosine function techniques.The quadrat...In this paper,the generalized dissipative Kawahara equation in the sense of conformable fractional derivative is presented and solved by applying the tanh-coth-expansion and sine-cosine function techniques.The quadratic-case and cubic-case are investigated for the proposed model.Expected solutions are obtained with highlighting to the effect of the presence of the alternative fractional-derivative and the effect of the added dissipation term to the generalized Kawahara equation.Some graphical analysis is presented to support the findings of the paper.Finally,we believe that the obtained results in this work will be important and valuable in nonlinear sciences and ocean engineering.展开更多
In this paper, Adomian decomposition method (ADM) is implemented to approximate the solution of the Korteweg-de Vries (KdV) equations of seventh order, which are Kaup-Kuperschmidt equation and seventh order Kawahara e...In this paper, Adomian decomposition method (ADM) is implemented to approximate the solution of the Korteweg-de Vries (KdV) equations of seventh order, which are Kaup-Kuperschmidt equation and seventh order Kawahara equation. The results obtained by the ADM are compared with the exact solutions. It is found that the ADM is very efficient and convenient and can be applied to a large class of problems. The conservation properties of solution are examined by calculating the first three invariants.展开更多
In this paper, we develop a method to calculate numerical and approximate solution of some fifth-order Korteweg-de Vries equations with initial condition with the help of Laplace Decomposition Method (LDM). The techni...In this paper, we develop a method to calculate numerical and approximate solution of some fifth-order Korteweg-de Vries equations with initial condition with the help of Laplace Decomposition Method (LDM). The technique is based on the application of Laplace transform to some fifth-order Kdv equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of four examples and results of the present technique have closed agreement with approximate solutions obtained with the help of (LDM).展开更多
In this study,we investigate the seventh-order nonlinear Caputo time-fractional KdV equation.The suggested model's solutions,which have a series form,are obtained using the hybrid ZZ-transform under the aforementi...In this study,we investigate the seventh-order nonlinear Caputo time-fractional KdV equation.The suggested model's solutions,which have a series form,are obtained using the hybrid ZZ-transform under the aforementioned fractional operator.The proposed approach combines the homotopy perturbation method(HPM)and the ZZ-transform.We consider two specific examples with suitable initial conditions and find the series solution to test their applicability.To demonstrate the utility of the presented technique,we explore its applications to the fractional Sawada–Kotera–Ito problem and the Lax equation.We observe the impact of a few fractional orders on the wave solution evolution for the problems under consideration.We provide the efficiency and reliability of the ZZHPM by calculating the absolute error between the series solution and the exact solution of both the Sawada–Kotera–Ito and Lax equations.The convergence and uniqueness of the solution are portrayed via fixed-point theory.展开更多
基金Project supported by the National Natural Science Foundations of China(Grant Nos.10735030,10475055,10675065 and 90503006)the National Basic Research Program of China(Grant No.2007CB814800)
文摘This paper studies the generalized Kawahara equation in terms of the approximate homotopy symmetry method and the approximate homotopy direct method. Using both methods it obtains the similarity reduction solutions and similarity reduction equations of different orders, showing that the approximate homotopy direct method yields more general approximate similarity reductions than the approximate homotopy symmetry method. The homotopy series solutions to the generalized Kawahara equation are consequently derived.
文摘In this work, an algorithm based on method of lines coupled with radial basis functions namely meshless method of lines (MMOL) is presented for the numerical solution of Kawahara, modified Kawahara and KdV Kawahara equations. The motion of a single solitary wave, interaction of two and three solitons and the phenomena of wave generation is discussed. The results are compared with the exact solution and with the results in the relevant literature to show the efficiency of the method.
基金Supported by the Program for the Young Yeachers of Henan Province (2020GGJS210)Key Scientific Research Projects of Henan Province (22B110014,23A110021)the Program for Innovative Research Team of Henan Province(23IRTSTHN018)。
基金Supported by the National Natural Science Foundations of China under Grant Nos.10735030,10475055,10675065,and 90503006National Basic Research Program of China (973 Program 2007CB814800)
文摘The Kawahara equation is studied through the approximate homotopy symmetry method. Under this method we get the similarity reduction solutions of the Kawahara equation, leading to the corresponding homotopy series solutions. Furthermore, the similarity solutions of the corresponding reduced linear ordinary differential equations are also considered.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11271195 and 11271196)the Project of Graduate Education Innovation of Jiangsu Province,China(Grant No.CXZZ12-0385)
文摘We derive a new multisymplectic integrator for the Kawahara-type equation which is a fully explicit scheme and thus needs less computation cost. Multisympecticity of such scheme guarantees the long-time numerical behaviors. Nu- merical experiments are presented to verify the accuracy of this scheme as well as the excellent performance on invariant preservation for three kinds of Kawahara-type equations.
基金Project supported by the National Natural Science Foundation of China (Grant No 10672053)
文摘In this paper, we present a combination method of constructing the explicit and exact solutions of nonlinear partial differential equations. And as an illustrative example, we apply the method to the Benney-Kawahara-Lin equation and derive its many explicit and exact solutions which are all new solutions.
文摘In this paper,the generalized dissipative Kawahara equation in the sense of conformable fractional derivative is presented and solved by applying the tanh-coth-expansion and sine-cosine function techniques.The quadratic-case and cubic-case are investigated for the proposed model.Expected solutions are obtained with highlighting to the effect of the presence of the alternative fractional-derivative and the effect of the added dissipation term to the generalized Kawahara equation.Some graphical analysis is presented to support the findings of the paper.Finally,we believe that the obtained results in this work will be important and valuable in nonlinear sciences and ocean engineering.
文摘In this paper, Adomian decomposition method (ADM) is implemented to approximate the solution of the Korteweg-de Vries (KdV) equations of seventh order, which are Kaup-Kuperschmidt equation and seventh order Kawahara equation. The results obtained by the ADM are compared with the exact solutions. It is found that the ADM is very efficient and convenient and can be applied to a large class of problems. The conservation properties of solution are examined by calculating the first three invariants.
文摘In this paper, we develop a method to calculate numerical and approximate solution of some fifth-order Korteweg-de Vries equations with initial condition with the help of Laplace Decomposition Method (LDM). The technique is based on the application of Laplace transform to some fifth-order Kdv equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of four examples and results of the present technique have closed agreement with approximate solutions obtained with the help of (LDM).
文摘In this study,we investigate the seventh-order nonlinear Caputo time-fractional KdV equation.The suggested model's solutions,which have a series form,are obtained using the hybrid ZZ-transform under the aforementioned fractional operator.The proposed approach combines the homotopy perturbation method(HPM)and the ZZ-transform.We consider two specific examples with suitable initial conditions and find the series solution to test their applicability.To demonstrate the utility of the presented technique,we explore its applications to the fractional Sawada–Kotera–Ito problem and the Lax equation.We observe the impact of a few fractional orders on the wave solution evolution for the problems under consideration.We provide the efficiency and reliability of the ZZHPM by calculating the absolute error between the series solution and the exact solution of both the Sawada–Kotera–Ito and Lax equations.The convergence and uniqueness of the solution are portrayed via fixed-point theory.
