In this article, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the Exp-function method are employed for constructing the exact solutions of nonlinear time fractional partial diff...In this article, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the Exp-function method are employed for constructing the exact solutions of nonlinear time fractional partial differential equations in mathematical physics. As a result, some new exact solutions for them are successfully established. It is indicated that the solutions obtained by the Exp-function method are reliable, straightforward and effective method for strongly nonlinear fractional partial equations with modified Riemann-Liouville derivative by Jumarie's. This approach can also be applied to other nonlinear time and space fractional differential equations.展开更多
We applied the multiple exp-function scheme to the（2＋1）-dimensional Sawada-Kotera（SK） equation and（3＋1）-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analyti...We applied the multiple exp-function scheme to the（2＋1）-dimensional Sawada-Kotera（SK） equation and（3＋1）-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota＇s perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.展开更多
This paper focuses on the application of Exp-function method to obtain generalized solutions of the KdV-Burgers-Kuramoto equation and the Kuramoto-Sivashinsky equation.It is demonstrated that the Exp-function method p...This paper focuses on the application of Exp-function method to obtain generalized solutions of the KdV-Burgers-Kuramoto equation and the Kuramoto-Sivashinsky equation.It is demonstrated that the Exp-function method provides a mathematical tool for solving the nonlinear evolution equation in mathematical physics.展开更多
In this paper, we apply Exp-function method to give traveling wave solutions of second order sine-Bratu type equations. This method is straightforward, concise and effective.
In this article, we propose a generalized exp(-Φ(ξ))-expansion method and successfully implement it to find exact traveling wave solutions to the fifth order standard Sawada-Kotera (SK) equation. The exact traveling...In this article, we propose a generalized exp(-Φ(ξ))-expansion method and successfully implement it to find exact traveling wave solutions to the fifth order standard Sawada-Kotera (SK) equation. The exact traveling wave solutions are established in the form of trigonometric, hyperbolic, exponential and rational functions with some free parameters. It is shown that this method is standard, effective and easily applicable mathematical tool for solving nonlinear partial differential equations arises in the field of mathematical physics and engineering.展开更多
In this work, the exp(-φ (ξ )) -expansion method is used for the first time to investigate the exact traveling wave solutions involving parameters of nonlinear evolution equations. When these parameters are taken to...In this work, the exp(-φ (ξ )) -expansion method is used for the first time to investigate the exact traveling wave solutions involving parameters of nonlinear evolution equations. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. The validity and reliability of the method are tested by its applications to Nano-ionic solitons wave’s propagation along microtubules in living cells and Nano-ionic currents of MTs which play an important role in biology.展开更多
In this study,we investigate a variety of exact soliton solutions of general(2+1)-dimensional Bogoyavlensky–Konopelchenko equation via the exp(-Φ(η))-expansion method and modified Kudryashov method.The exact soluti...In this study,we investigate a variety of exact soliton solutions of general(2+1)-dimensional Bogoyavlensky–Konopelchenko equation via the exp(-Φ(η))-expansion method and modified Kudryashov method.The exact solutions are characterized in the form of hyperbolic,trigonometric and rational function solutions using exp(-Φ(η))-expansion method,whereas the solution in the form of hyperbolic function expression is obtained by the modified Kudryashov method.These exact solutions also include kink,bright,dark,singular and periodic soliton solutions.The graphical interpretation of the exact solutions is addressed for specific choices of the parameters appearing in the solutions.展开更多
A hyperbolic Lindstedt-Poincare method is presented to determine the homoclinic solutions of a kind of nonlinear oscillators, in which critical value of the homoclinic bifurcation parameter can be determined. The gene...A hyperbolic Lindstedt-Poincare method is presented to determine the homoclinic solutions of a kind of nonlinear oscillators, in which critical value of the homoclinic bifurcation parameter can be determined. The generalized Lienard oscillator is studied in detail, and the present method＇s predictions are compared with those of Runge-Kutta method to illustrate its accuracy.展开更多
Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet ...Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.展开更多
文摘In this article, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the Exp-function method are employed for constructing the exact solutions of nonlinear time fractional partial differential equations in mathematical physics. As a result, some new exact solutions for them are successfully established. It is indicated that the solutions obtained by the Exp-function method are reliable, straightforward and effective method for strongly nonlinear fractional partial equations with modified Riemann-Liouville derivative by Jumarie's. This approach can also be applied to other nonlinear time and space fractional differential equations.
文摘We applied the multiple exp-function scheme to the（2＋1）-dimensional Sawada-Kotera（SK） equation and（3＋1）-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota＇s perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.
基金Supported by the National Natural Science Foundation of China(91024026,10975126)Supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China(200934021100 32)
文摘This paper focuses on the application of Exp-function method to obtain generalized solutions of the KdV-Burgers-Kuramoto equation and the Kuramoto-Sivashinsky equation.It is demonstrated that the Exp-function method provides a mathematical tool for solving the nonlinear evolution equation in mathematical physics.
文摘In this paper, we apply Exp-function method to give traveling wave solutions of second order sine-Bratu type equations. This method is straightforward, concise and effective.
文摘In this article, we propose a generalized exp(-Φ(ξ))-expansion method and successfully implement it to find exact traveling wave solutions to the fifth order standard Sawada-Kotera (SK) equation. The exact traveling wave solutions are established in the form of trigonometric, hyperbolic, exponential and rational functions with some free parameters. It is shown that this method is standard, effective and easily applicable mathematical tool for solving nonlinear partial differential equations arises in the field of mathematical physics and engineering.
文摘In this work, the exp(-φ (ξ )) -expansion method is used for the first time to investigate the exact traveling wave solutions involving parameters of nonlinear evolution equations. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. The validity and reliability of the method are tested by its applications to Nano-ionic solitons wave’s propagation along microtubules in living cells and Nano-ionic currents of MTs which play an important role in biology.
文摘In this study,we investigate a variety of exact soliton solutions of general(2+1)-dimensional Bogoyavlensky–Konopelchenko equation via the exp(-Φ(η))-expansion method and modified Kudryashov method.The exact solutions are characterized in the form of hyperbolic,trigonometric and rational function solutions using exp(-Φ(η))-expansion method,whereas the solution in the form of hyperbolic function expression is obtained by the modified Kudryashov method.These exact solutions also include kink,bright,dark,singular and periodic soliton solutions.The graphical interpretation of the exact solutions is addressed for specific choices of the parameters appearing in the solutions.
基金supported by the National Natural Science Foundation of China (10672193)Sun Yat-sen University (Fu Lan Scholarship)the University of Hong Kong (CRGC grant).
文摘A hyperbolic Lindstedt-Poincare method is presented to determine the homoclinic solutions of a kind of nonlinear oscillators, in which critical value of the homoclinic bifurcation parameter can be determined. The generalized Lienard oscillator is studied in detail, and the present method＇s predictions are compared with those of Runge-Kutta method to illustrate its accuracy.
基金the National Natural Science Foundation of China (Nos.11571238,11601332,91130014,11471312 and 91430216).
文摘Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.
